geometry 1 i

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TERMINOLOGY

4 Geometry 1

Altitude Height Any line segment from a vertex to theopposite side of a polygon that is perpendicular to that side

Congruent triangles Identical triangles that are the sameshape and size Corresponding sides and angles areequal The symbol is

Interval Part of a line including the endpoints

Median A line segment that joins a vertex to theopposite side of a triangle that bisects that side

Perpendicular A line that is at right angles to anotherline The symbol is =

Polygon General term for a many sided plane 1047297gure Aclosed plane (two dimensional) 1047297gure with straight sides

Quadrilateral A four-sided closed 1047297gure such as a squarerectangle trapezium etc

Similar triangles Triangles that are the same shape butdifferent sizes The symbol is yz

Vertex The point where three planes meet The corner ofa 1047297gure

Vertically opposite angles Angles that are formedopposite each other when two lines intersect



141Chapter 4 Geometry 1

INTRODUCTION

GEOMETRY IS USED IN many areas including surveying building and graphics

These fields all require a knowledge of angles parallel lines and so on and

how to measure them In this chapter you will study angles parallel linestriangles types of quadrilaterals and general polygons

Many exercises in this chapter on geometry need you to prove something

or give reasons for your answers The solutions to geometry proofs only give

one method but other methods are also acceptable

DID YOU KNOW

Geometry means measurement of the earth and comes from Greek Geometry was used in ancient

civilisations such as Babylonia However it was the Greeks who formalised the study of geometry

in the period between 500 BC and AD 300

Notation

In order to show reasons for exercises you must know how to name figures

correctly

bull B

The point is called B

The interval (part of a line) is called AB or BA

If AB and CD are parallel lines we write AB CDlt

This angle is named BAC+ or CAB+ It can sometimes be named A+

Angles can also be written as BAC^

or BAC

This triangle is named ABC3



142 Maths In Focus Mathematics Extension 1 Preliminary Course

This quadrilateral is called ABCD

Line AB is produced to C

DB bisects ABC+

AM is a median of ABCD

AP is an altitude of ABCD

Types of Angles

Acute angle

0 90xc c c1 1

To name a quadrilateral

go around it for example

BCDA is correct but ACBDis not

Producing a line is the same

as extending it

ABD+ and DBC + are

equal




Right angle

A right angle is 90c

Complementary angles are angles whose sum is 90c

Obtuse angle

x90 180c c c1 1

Straight angle

A straight angle is 180c

Supplementary angles are angles whose sum is 180c

Re1047298ex angle

x180 360c c c1 1

Angle of revolution

An angle of revolution is 360c

Vertically opposite angles

AEC+ and DEB+ are called vertically opposite angles AED+ and CEB+ are

also vertically opposite angles




Proof

( )

( ) ( )

( )

AEC x

AED x CED

DEB x AEB

x

CEB x CED

AEC DEB AED CEB

180 180

180 180 180

180 180

Let

Then straight angle

Now straight angle

Also straight angle

and`

c

c c c

c c c c

c

c c c

+

+ +

+ +

+ +

+ + + +

=

= -

= - -

=

= -

= =

EXAMPLES

Find the values of all pronumerals giving reasons

1

Solution

( )x ABC

x

x

154 180 180

154 180

26

154 154

is a straight angle

`

c++ =

+ =

=

- -

2

Solution( )x

x

x

x

x

x

2 142 90 360 360

2 232 360

2 232 360

2 128

2 128

64

232 232

2 2

angle of revolution c+ + =

+ =

+ =

=

=

=

- -

Vertically opposite angles are equal

That is AEC DEB+ += and AED CEB+ +=




3

Solution

( ) y y

y

y

y

y

y

2 30 90 90

3 30 90

3 30 90

3 60

3 60

20

30 30

3 3

right angle c+ + =

+ =

+ =

=

=

=

- -

4

Solution

(

( )

(

x WZX YZV

x

x

y XZY

w WZY XZV

50 165

50 165

115

180 165 180

15

15

50 50

and vertically opposite)

straight angle


c

+ +

+

+ +

+ =

+ =

=

= -

=

=

- -

5

CONTINUED




Solution

( )

( )

( )

( )

a

b

b

b

b

d

c

90

53 90 180 180

143 180

143 180

37

37

53

143 143

vertically opposite angles

straight angle


similarly

c

=

+ + =

+ =

+ =

=

=

=

- -

6 Find the supplement of 57 12c l

Solution

Supplementary angles add up to 180c

So the supplement of 57 12c l is180 57 12 1 2 482c c c- =l l

7 Prove that AB and CD are straight lines

Solution

x x x x

x

x

x

x

6 10 30 5 30 2 10 360

14 80 360

14 280

14 280

20

80 80

14 14

angle of revolution+ + + + + + + =

+ =

=

=

=

- -

^ h

( )

( )

AEC

DEB

20 30

50

2 20 10

50

c

c

c

c

+

+

= +

=

= +

=

These are equal vertically opposite angles

AB and CD are straight lines

C

D A

B

E x 10)

( x +

(5 x + 3 )

x + 30)




41 Exercises

1 Find values of all pronumerals

giving reasons

yc 133c

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Find the supplement of

(a) 59c (b) 107 31c l

(c) 45 12c l

3 Find the complement of

(a) 48c

(b) 34 23c l

(c) 16 57c l

4 Find the (i) complement and

(ii) supplement of

(a) 43c 81c(b)

27c(c)

(d) 55c

(e) 38c

(f) 74 53c l

(g) 42 24c l

(h) 17 39c l

(i) 63 49c l

(j) 51 9c l

5 (a) Evaluate x Find the complement of(b) x

Find the supplement of(c) x

(2 x +30)c

142c




6 Find the values of all

pronumerals giving reasons for

each step of your working

(a)

(b)

(c)

(d)

(e)

(f)

7

Prove that AC and DE are straight

lines

8

Prove that CD bisects AFE+

9 Prove that AC is a straight line

A

B

C

D

(110-3 x )c

(3 x + 70)c

10 Show that + AED is a right angle

A B

C

D E

(50- 8 y)c

(5 y- 20)c

(3 y+ 60)c




Parallel Lines

When a transversal cuts two lines it forms pairs of angles When the two

lines are parallel these pairs of angles have special properties

Alternate angles

Alternate angles form

a Z shape Can you

1047297nd another set of

alternate angles

Corresponding angles form

an F shape There are 4 pairs

of corresponding angles Can

you 1047297nd them

If the lines are parallel then alternate angles are equal

Corresponding angles

If the lines are parallel then corresponding angles are equal




Cointerior angles

Cointerior angles form

a U shape Can you 1047297nd

another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt

If BEF DFE 180 c+ ++ =

then AB CDlt

If the lines are parallel cointerior angles are supplementary (ie their sum

is 180c )

Tests for parallel lines

If alternate angles are equal then the lines are parallel

If corresponding angles are equal then the lines are parallel

If cointerior angles are supplementary then the lines are parallel




EXAMPLES

1 Find the value of y giving reasons for each step of your working

Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle

corresponding angles`

c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =

But CBF + and HCD+ are corresponding angles

EF GH ` lt Can you prove this

in a different way

If 2 lines are both parallel to a third line then the 3 lines are parallel to

each other That is if AB CDlt and EF CDlt then AB EF lt





(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)

42 ExercisesThink about the reasons for

each step of your calculations




Types of Triangles

Names of triangles

A scalene triangle has no two sides or angles equal

A right (or right-angled) triangle contains a right angle

The side opposite the right angle (the longest side) is called the

hypotenuse

An isosceles triangle has two equal sides

A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)

The angles (called the base angles) opposite the equal sides in an

isosceles triangle are equal

An equilateral triangle has three equal sides and angles




All the angles are acute in an acute-angled triangle

An obtuse-angled triangle contains an obtuse angle

Angle sum of a triangle

The sum of the interior angles in any triangle is 180c

that is a b c 180+ + =

Proof

YXZ a XYZ b YZX c Let andc c c+ + += = =

( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line

Then alternate angles

similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =




Exterior angle of a triangle

Class Investigation

Could you prove the base angles in an isosceles triangle are equal1

Can there be more than one obtuse angle in a triangle2

Could you prove that each angle in an equilateral triangle is3 60c

Can a right-angled triangle be an obtuse-angled triangle4

Can you 1047297nd an isosceles triangle with a right angle in it5

The exterior angle in any triangle is equal to the sum of the two opposite

interior angles That is

x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y

corresponding angles

alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES

Find the values of all pronumerals giving reasons for each step

1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135

angle sum of cD+ + =

+ =

+ =

=

- -

2

Solution

( ) A C x base angles of isosceles+ + D= =

( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2

angle sum in a cD+ + =

+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106

35 35(exterior angle of

`

D+ =+ =

=

- -

This example can be done using the interior sum of angles

( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

2 Show that each angle in an

equilateral triangle is 60c

3 Find ACB+ in terms of x

43 ExercisesThink of the reasons

for each step of your

calculations




4 Prove AB EDlt

5 Show ABCD is isosceles

6 Line CE bisects BCD+ Find the

value of y giving reasons

7 Evaluate all pronumerals giving

reasons for your working

(a)

(b)

(c)

(d)

8 Prove IJLD is equilateral and

JKLD is isosceles

9 In triangle BCD below BC BD= Prove AB ED

A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles

Two triangles are congruent if they are the same shape and size All pairs of

corresponding sides and angles are equal

For example

We write ABC XYZ D D

Tests

To prove that two triangles are congruent we only need to prove that certain

combinations of sides or angles are equal

Two triangles are congruent if

bull SSS all three pairs of corresponding sides are equal

bull SAS two pairs of corresponding sides and their included angles are

equal

bull AAS two pairs of angles and one pair of corresponding sides are equal

bull RHS both have a right angle their hypotenuses are equal and one

other pair of corresponding sides are equal

EXAMPLES

1 Prove that OTS OQP D D where O is the centre of the circle

CONTINUED

The included angle

is the angle between

the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)

(vertically opposite angles)

(equal radii)

by SAS`

+ +

D D

=

=

=

2 Which two triangles are congruent

Solution

To 1047297nd corresponding sides look at each side in relation to the angles

For example one set of corresponding sides is AB DF GH and JL

ABC JKL A(by S S)D D

3 Show that triangles ABC and DEC are congruent Hence prove that

AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS

corresponding sides in congruent s

`

`

+ +

+ +

lt

D D

D

=

=

=

=




1 Are these triangles congruent

If they are prove that they are

congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e

2 Prove that these triangles are

congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that

(a) ∆ ABD is congruent to ∆ ACD

(b) AB bisects BC given ABCD is

isosceles with AB AC=

4 Prove that triangles ABD and CDB

are congruent Hence prove that

AD BC=

5 In the circle below O is the centre

of the circle

O

A

B

D

C

Prove that(a) OABT and OCDT

are congruent

Show that(b) AB CD=

6 In the kite ABCD AB AD= and

BC DC=

A

B D

C

Prove that(a) ABCT and ADCT

are congruent

Show that(b) ABC ADC+ +=

7 The centre of a circle is O and AC

is perpendicular to OB

O

A

B

C

Show that(a) OABT and OBCT

are congruent

Prove that(b) ABC 90c+ =

8 ABCF is a trapezium with

AF BC= and FE CD= AE and BD

are perpendicular to FC

D

A B

C F E

Show that(a) AFET and BCDT

are congruent

Prove that(b) AFE BCD+ +=




9 The circle below has centre O and

OB bisects chord AC

O

A

B

C

Prove that(a) OABT is congruent

to OBCT

Prove that(b) OB is perpendicular

to AC

10 ABCD is a rectangle as shown

below

D

A B

C

Prove that(a) ADCT is

congruent to BCDT

Show that diagonals(b) AC and

BD are equal

Investigation

The triangle is used in many

structures for example trestle

tables stepladders and roofs

Find out how many different ways

the triangle is used in the building

industry Visit a building site orinterview a carpenter Write a

report on what you 1047297nd

Similar Triangles

Triangles for example ABC and XYZ are similar if they are the same shape but

different sizes

As in the example all three pairs of corresponding angles are equal

All three pairs of corresponding sides are in proportion (in the same ratio)




Application

Similar 1047297gures are used in many areas including maps scale drawings models

and enlargements

EXAMPLE

1 Find the values of x and y in similar triangles CBA and XYZ

Solution

First check which sides correspond to one another (by looking at their

relationships to the angles)

YZ and BA XZ and CA and XY and CB are corresponding sides

CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=

We write XYZ D ABC ltD

XYZ D is three times larger than ABCD

AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =

This shows that all 3 pairs

of sides are in proportion




y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=

Two triangles are similar if

three pairs ofbull corresponding angles are equal

three pairs ofbull corresponding sides are in proportion

two pairs ofbull sides are in proportion and their included angles

are equal

If 2 pairs of angles are

equal then the third

pair must also be equal

EXAMPLES

1Prove that triangles(a) ABC and ADE are similar

Hence 1047297nd the value of(b) y to 1 decimal place

Solution

(a) A+ is common

ADE D

( )( )

( )

ABC ADE BC DE ACB AED

ABC

corresponding anglessimilarly

3 pairs of angles equal`

+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests

There are three tests for similar triangles




AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=

2 Prove WVZ D XYZ ltD

Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=

` since two pairs of sides are in proportion and their included angles are

equal the triangles are similar

Ratio of intercepts

The following result comes from similar triangles

When two (or more) transversals cut a series of parallel lines the

ratios of their intercepts are equal

AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof

Draw DG and EH parallel to AC

`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2

Then opposite sides of a parallelogram

Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES

1 Find the value of x to 3 signi1047297cant 1047297gures

Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44

ratios of intercepts on parallel lines

=

=

=

=

^ h

CONTINUED




2 Evaluate x and y to 1 decimal place

Solution

Use either similar triangles or ratios of intercepts to 1047297nd x You must use

similar triangles to 1047297nd y

x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=

1 Find the value of all pronumerals

to 1 decimal place where

appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines

These ratios come from

similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)

2 Evaluate a and b to 2 decimal

places

3 Show that ABCD and CDED are

similar

4 EF bisects GFD+ Show that

DEF D

and FGED

are similar

5 Show that ABCD and DEF D are

similar Hence 1047297nd the value of y

42

49

686

13

588182

A

C

B D

E F

yc87c

52c

6 The diagram shows two

concentric circles with centre O

Prove that(a) D OCDOAB ltD

If radius(b) OC 5 9 c m= and

radius OB 8 3 cm= and the

length of CD 3 7 cm= 1047297nd the

length of AB correct to 2 decimal

places

7 (a) Prove that ADED ABC ltD

Find the values of(b) x and y

correct to 2 decimal places

8 ABCD is a parallelogram with

CD produced to E Prove that

CEBD ABF ltD




9 Show that ABC D AED ltD Find

the value of m

10 Prove that ABCD and ACDD are

similar Hence evaluate x and y


pronumerals to 1 decimal place

(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=

13 Evaluate a and b correct to

1 decimal place

14 Find the value of y to 2

signi1047297cant 1047297gures

15 Evaluate x and y correct to

2 decimal places




Pythagorasrsquo Theorem

DID YOU KNOW

The triangle with sides in the

proportion 345 was known to be

right angled as far back as ancient

Egyptian times Egyptian surveyors

used to measure right angles by

stretching out a rope with knots tied

in it at regular intervals

They used the rope for forming

right angles while building and

dividing 1047297elds into rectangular plots

It was Pythagoras (572ndash495 BC)

who actually discovered the

relationship between the sides of the

right-angled triangle He was able to

generalise the rule to all right-angled triangles

Pythagoras was a Greek mathematician

philosopher and mystic He founded the Pythagorean

School where mathematics science and philosophy

were studied The school developed a brotherhood and

performed secret rituals He and his followers believed

that the whole universe was based on numbers

Pythagoras was murdered when he was 77 and the

brotherhood was disbanded

The square on the hypotenuse in any right-angled triangle is equal to the

sum of the squares on the other two sides

c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof

Draw CD perpendicular to AB

Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC

equal corresponding s+

ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES

1 Find the value of x correct to 2 decimal places

Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=

c a b ABCIf then must be right angled2 2 2D= +




x 65

8 06 to 2 decimal places

=

=

2 Find the exact value of y

Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=

3 Find the length of the diagonal in a square with sides 6 cm Answer to

1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=

So the length of the diagonal is 85 cm

Leave the answer in

surd form for the exact

answer

CONTINUED





correct to 1 decimal place(a)

(b)

(c)

(d)

2 Find the exact value of all

pronumerals(a)

(b)

(c)

(d)

46 Exercises

4 A triangle has sides 51 cm 68 cm and 85 cm Prove that the triangle

is right angled

Solution

68 cm

85 cm

51 cm

Let c 8 5= (largest side) and a and b the other two smaller sides

a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +

So the triangle is right angled




3 Find the slant height s of a

cone with diameter 68 m and

perpendicular height 52 m to

1 decimal place

4 Find the length of CE correct

to 1 decimal place in this

rectangular pyramid 86 AB cm=

and 159 CF cm=

5 Prove that ABCD is a right-angled

triangle

6 Show that XYZ D is a right-angled

isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=

8 (a) Find the length of diagonal

AC in the 1047297gure

Hence or otherwise prove(b)

that AC is perpendicular to DC

9 Find the length of side AB in

terms of b

10 Find the exact ratio of YZ XY

in

terms of x and y in XYZ D




11 Show that the distance squared

between A and B is given by

d t t 13 180 6252 2= - +

12 An 850 mm by 1200 mm gate

is to have a diagonal timber

brace to give it strength To what

length should the timber be cut

to the nearest mm

13 A rectangular park has a length of620 m and a width of 287 m If I

walk diagonally across the park

how far do I walk

14 The triangular garden bed below

is to have a border around it

How many metres of border are

needed to 1 decimal place

15 What is the longest length of

stick that will 1047297t into the box

below to 1 decimal place

16 A ramp is 45 m long and 13 m

high How far along the ground

does the ramp go Answer correct

to one decimal place

45 m

13 m

17 The diagonal of a television

screen is 72 cm If the screen is

58 cm high how wide is it

18 A property has one side 13 km

and another 11 km as shown

with a straight road diagonally

through the middle of the

property If the road is 15 km

long show that the property is

not rectangular

13 km

11 km

15 km

19 Jodie buys a ladder 2 m long and

wants to take it home in the boot

of her car If the boot is 12 m by

07 m will the ladder 1047297t




Types of Quadrilaterals

A quadrilateral is any four-sided 1047297gure

In any quadrilateral the sum of the interior angles is 360c

20 A chord AB in a circle with

centre O and radius 6 cm has a

perpendicular line OC as shown

4 cm long

A

B

O

C

6 cm

4 cm

By 1047297nding the lengths of(a) AC

and BC show that OC bisects the

chord

By proving congruent(b)

triangles show that OC bisects

the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB

ADC DCA CAD ABC BCA CAB

ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =




opposite sidesbull of a parallelogram are equal

bull opposite angles of a parallelogram are equal

bull diagonals in a parallelogram bisect each other

each diagonal bisects the parallelogram into twobull

congruent triangles

A quadrilateral is a parallelogram if

both pairs ofbull opposite sides are equal

both pairs ofbull opposite angles are equal

onebull pair of sides is both equal and parallel

thebull diagonals bisect each other

These properties can

all be proven

Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel

EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94

angle sum of quadrilaterali

i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus

A rectangle is a parallelogram with one angle a right angle

the same as for a parallelogram and alsobull

diagonals are equalbull

A quadrilateral is a rectangle if its diagonals are equal

Application

Builders use the property of equal diagonals to check if a rectangle is accurate

For example a timber frame may look rectangular but may be slightly slantingChecking the diagonals makes sure that a building does not end up like the

Leaning Tower of Pisa

It can be proved that

all sides are equal

If one angle is a right

angle then you can

prove all angles are

right angles

A rhombus is a parallelogram with a pair of adjacent sides equal

the same as for parallelogram and alsobull

diagonals bisect at right anglesbull

diagonals bisect the angles of the rhombusbull

Rectangle

PROPERTIES

PROPERTIES

TEST




Square

A square is a rectangle with a pair of adjacent sides equal

bull the same as for rectangle and also

diagonals are perpendicularbull

diagonals make angles ofbull 45c with the sides

Trapezium

A trapezium is a quadrilateral with one pair of sides parallel

Kite

A kite is a quadrilateral with two pairs of adjacent sides equal

A quadrilateral is a rhombus if

all sides are equalbull

diagonals bisect each other at right anglesbull

TESTS

PROPERTIES




EXAMPLES

1 Find the values of i x and y giving reasons

Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram

cm opposite sides in gram


c + lt

lt

lt

i =

=

=

2 Find the length of AB in square ABCD as a surd in its simplest form if

6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let

Since is a square adjacent sides equal

Also by definitionc+

=

= =

=

By Pythagorasrsquo theorem

3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED




1 Find the value of all pronumeralsgiving reasons

(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3

Two equal circles have centres(a) O and P respectively Prove that OAPB

is a rhombus

Hence or otherwise show that(b) AB is the perpendicular bisector

of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii

similarlySince the circles are equal

=

=

= = =

` since all sides are equal OAPB is a rhombus

The diagonals in any rhombus are perpendicular bisectors(b)

Since OAPB is a rhombus with diagonals AB and OP AB is the

perpendicular bisector of OP




2 Given AB AE= prove CD is

perpendicular to AD

3 (a) Show that C xc+ = and

( ) B D x180 c+ += = -

Hence show that the sum of(b)

angles of ABCD is 360c

4 Find the value of a and b


pronumerals giving reasons

(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)

6 In the 1047297gure BD bisects

ADC+ Prove BD also bisects

ABC+

7 Prove that each 1047297gure is a

parallelogram

(a)

(b)




(c)

(d)

8 Evaluate all pronumerals

(a)

(b)

ABCD is a kite

(c)

(d)

(e)

9 The diagonals of a rhombus

are 8 cm and 10 cm long Find

the length of the sides of the

rhombus

10 ABCD is a rectangle with

EBC 59c+ = Find ECB EDC+ +

and ADE+

11 The diagonals of a square are

8 cm long Find the exact lengthof the side of the square

12 In the rhombus ECB 33c+ =

Find the value of x and y

Polygons

A polygon is a closed plane 1047297gure with straight sides

A regular polygon has all sides and all interior angles equal




Proof

Draw any n -sided polygon and divide it into n triangles as

shown Then the total sum of angles is n 180 c or 180 n

But this sum includes all the angles at O So the sum of

interior angles is 180 360 n c-

That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW

Carl Gauss (1777ndash1855) was a famous German mathematician physicist and astronomer When

he was 19 years old he showed that a 17-sided polygon could be constructed using a ruler and

compasses This was a major achievement in geometryGauss made a huge contribution to the study of mathematics and science including

correctly calculating where the magnetic south pole is and designing a lens to correct

astigmatism

He was the director of the Goumlttingen Observatory for 40 years It is said that he did not

become a professor of mathematics because he did not like teaching

The sum of the interior angles of an n -sided polygon is given by

( 2) 180

S n

S n

180 360

or c

= -

= -

The sum of the exterior angles of any polygon is 360c

Proof

Draw any n -sided polygon Then the sum of both the

exterior and interior angles is n 180 c

n

n n

n n

180

180 180 360

180 180 360

360

Sum of exterior angles sum of interior angles c

c

c

c

= -

= - -

= - +

=

] g




EXAMPLES

1 Find the sum of the interior angles of a regular polygon with 15 sides

How large is each angle

Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=

Each angle has size 2340 15 156c c=

2 Find the number of sides in a regular polygon whose interior angles

are 140c

Solution

Let n be the number of sides

Then the sum of interior angles is 140n

( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=

So the polygon has 9 sides

There are n sides and so n

angles each 140 c

1 Find the sum of the interior

angles of

a pentagon(a)

a hexagon(b)

an octagon(c)a decagon(d)

a 12-sided polygon(e)

an 18-sided polygon(f)

2 Find the size of each interior

angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)

3 Find the size of each exterior

angle of a regular

hexagon(a)

decagon(b)

octagon(c)15-sided polygon(d)

4 Calculate the size of each

interior angle in a regular 7-sided

polygon to the nearest minute

5 The sum of the interior angles of

a regular polygon is 1980c

How many sides has the(a)

polygon

Find the size of each interior(b)

angle to the nearest minute

48 Exercises




6 Find the number of sides of a

regular polygon whose interior

angles are 157 30c l


angles of a regular polygon whose

exterior angles are 18c

8 A regular polygon has interior

angles of 156c Find the sum of its

interior angles


angle in a regular polygon if

the sum of the interior angles is

5220c

10 Show that there is no regular

polygon with interior angles of

145c


regular polygon with exterior

angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c

12 ABCDEF is a regular hexagon

F

E D

A B

C

Show that triangles(a) AFE and

BCD are congruent

Show that(b) AE and BD are

parallel

13 A regular octagon has a

quadrilateral ACEG inscribed as

shown

D

A

B

E

C

F

G

H

Show that ACEG is a square

14 In the regular pentagon below

show that EAC is an isosceles

triangle

D

A

B E

C

15 (a) Find the size of each exterior

angle in a regular polygon with

side p

Hence show that each interior(b)

angle is

( )

p

p180 2-




Areas

Most areas of plane 1047297gures come from the area of a rectangle

Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b

Draw rectangle ABCD where b length= and h breadth=

A square is a

special rectangle

The area of a triangle

is half the area of a

rectangle




bharea

21

21

21

21

` =

DEF AEFD CEF EBCF Area area and area areaD D= =

CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof

In parallelogram ABCD produce DC to E and draw BE perpendicular to CE

Then ABEF is a rectangle

Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90

opposite sides of a rectangle

opposite sides of a parallelogram

by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a

parallelogram is the

same as the area of

two triangles

A xy 21

=

(x and y are lengths of diagonals)

Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof

Let AC x= and BD y =

By properties of a rhombus

AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area

total area of rhombus

21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES

1 Find the area of this trapezium

Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=

2 Find the area of the shaded region in this 1047297gure

8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2

Area trapezium area area rectangle area

21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle

You will study the circle in

more detail in Chapter 9




Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15

Area large rectangle

cm

Area small rectangle

cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=

3 A park with straight sides of length 126 m and width 54 m has semi-

circular ends as shown Find its area correct to 2 decimal places

2 m

5 4 m

Solution

-Area of 2 semicircles area of 1 circle=

2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=

1 Find the area of each 1047297gure

(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)

2 Find the area of a rhombus with

diagonals 23 m and 42 m

3 Find each shaded area(a)

(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)

5 Find the exact area of the 1047297gure

6 Find the area of this 1047297gure

correct to 4 signi1047297cant 1047297gures

The arch is a semicircle

7 Jenny buys tiles for the 1047298oor of

her bathroom (shown top next

column) at $4550 per m2 How

much do they cost altogether

8 The dimensions of a battleaxe

block of land are shown below

Find its area(a)

A house in the district where(b)

this land is can only take up 55

of the land How large (to the

nearest m2 ) can the area of the

house beIf the house is to be a(c)

rectangular shape with width

85 m what will its length be

9 A rhombus has one diagonal

25 cm long and its area is

600 cm2 Find the length of

its other diagonal and(a)

its side to the nearest cm(b)

10 The width w of a rectangle is

a quarter the size of its length

If the width is increased by 3units while the length remains

constant 1047297nd the amount of

increase in its area in terms of w




Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure

1 Find the values of all pronumerals

(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)

2 Prove that AB and CD are parallel lines

3 Find the area of the 1047297gure to 2 decimalplaces

4 (a) Prove that triangles ABC and ADE are

similar

Evaluate(b) x and y to 1 decimal place

5 Find the size of each interior angle in a

regular 20-sided polygon

6 Find the volume of a cylinder with radius

57 cm and height 10 cm correct to

1 decimal place

7 Find the perimeter of the triangle below




8 (a) Prove triangles ABC and ADC are

congruent in the kite below

Prove triangle(b) AOB and COD are

congruent (O is the centre of the circle)

9 Find the area of the 1047297gure below

10 Prove triangle ABC is right angled

11 Prove AG AF

AC AB

=

12 Triangle ABC is isosceles and AD bisects

BC

Prove triangles(a) ABD and ACD are

congruent

Prove(b) AD and BC are perpendicular

13 Triangle ABC is isosceles with AB AC=

Show that triangle ACD is isosceles

14 Prove that opposite sides in any

parallelogram are equal

15 A rhombus has diagonals 6 cm and 8 cm

Find the area of the rhombus(a)

Find the length of its side(b)

16 The interior angles in a regular polygon

are 140c How many sides has the

polygon

17 Prove AB and CD are parallel






7 Prove that the diagonals in a square

make angles of 45c with the sides

8 Prove that the diagonals in a kite are

perpendicular

9 Prove that MN is parallel to XY

10 Evaluate x

11 The letter Z is painted on a billboard

Find the area of the letter(a)

Find the exact perimeter of the letter(b)

12 Find the values of x and y correct to

1 decimal place


2 decimal places

14 ABCD is a square and BD is produced to

E such that DE BD21

=

Show that(a) ABCE is a kite

Prove that(b) DE x

2

2= units when

sides of the square are x units long




INTRODUCTION

GEOMETRY IS USED IN many areas including surveying building and graphics

These fields all require a knowledge of angles parallel lines and so on and

how to measure them In this chapter you will study angles parallel linestriangles types of quadrilaterals and general polygons

Many exercises in this chapter on geometry need you to prove something

or give reasons for your answers The solutions to geometry proofs only give

one method but other methods are also acceptable

DID YOU KNOW

Geometry means measurement of the earth and comes from Greek Geometry was used in ancient

civilisations such as Babylonia However it was the Greeks who formalised the study of geometry

in the period between 500 BC and AD 300

Notation

In order to show reasons for exercises you must know how to name figures

correctly

bull B

The point is called B

The interval (part of a line) is called AB or BA

If AB and CD are parallel lines we write AB CDlt

This angle is named BAC+ or CAB+ It can sometimes be named A+

Angles can also be written as BAC^

or BAC

This triangle is named ABC3






DB bisects ABC+



Types of Angles

Acute angle

0 90xc c c1 1





as extending it

ABD+ and DBC + are

equal




Right angle



Obtuse angle

x90 180c c c1 1

Straight angle



Re1047298ex angle

x180 360c c c1 1

Angle of revolution








Proof

( )

( ) ( )

( )

AEC x

AED x CED

DEB x AEB

x

CEB x CED

AEC DEB AED CEB

180 180

180 180 180

180 180

Let

Then straight angle

Now straight angle

Also straight angle

and`

c

c c c

c c c c

c

c c c

+

+ +

+ +

+ +

+ + + +

=

= -

= - -

=

= -

= =

EXAMPLES


1

Solution

( )x ABC

x

x

154 180 180

154 180

26

154 154

is a straight angle

`

c++ =

+ =

=

- -

2

Solution( )x

x

x

x

x

x

2 142 90 360 360

2 232 360

2 232 360

2 128

2 128

64

232 232

2 2


+ =

+ =

=

=

=

- -






3

Solution

( ) y y

y

y

y

y

y

2 30 90 90

3 30 90

3 30 90

3 60

3 60

20

30 30

3 3

right angle c+ + =

+ =

+ =

=

=

=

- -

4

Solution

(

( )

(

x WZX YZV

x

x

y XZY

w WZY XZV

50 165

50 165

115

180 165 180

15

15

50 50


straight angle


c

+ +

+

+ +

+ =

+ =

=

= -

=

=

- -

5

CONTINUED




Solution

( )

( )

( )

( )

a

b

b

b

b

d

c

90

53 90 180 180

143 180

143 180

37

37

53

143 143


straight angle


similarly

c

=

+ + =

+ =

+ =

=

=

=

- -


Solution




Solution

x x x x

x

x

x

x

6 10 30 5 30 2 10 360

14 80 360

14 280

14 280

20

80 80

14 14


+ =

=

=

=

- -

^ h

( )

( )

AEC

DEB

20 30

50

2 20 10

50

c

c

c

c

+

+

= +

=

= +

=



C

D A

B

E x 10)

( x +

(5 x + 3 )

x + 30)




41 Exercises


giving reasons

yc 133c

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)


(a) 59c (b) 107 31c l

(c) 45 12c l


(a) 48c

(b) 34 23c l

(c) 16 57c l


(ii) supplement of

(a) 43c 81c(b)

27c(c)

(d) 55c

(e) 38c

(f) 74 53c l

(g) 42 24c l

(h) 17 39c l

(i) 63 49c l

(j) 51 9c l



(2 x +30)c

142c







(a)

(b)

(c)

(d)

(e)

(f)

7


lines

8



A

B

C

D

(110-3 x )c

(3 x + 70)c


A B

C

D E

(50- 8 y)c

(5 y- 20)c

(3 y+ 60)c




Parallel Lines



Alternate angles


a Z shape Can you


alternate angles




you 1047297nd them







Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when







DB bisects ABC+



Types of Angles

Acute angle

0 90xc c c1 1





as extending it

ABD+ and DBC + are

equal




Right angle



Obtuse angle

x90 180c c c1 1

Straight angle



Re1047298ex angle

x180 360c c c1 1

Angle of revolution








Proof

( )

( ) ( )

( )

AEC x

AED x CED

DEB x AEB

x

CEB x CED

AEC DEB AED CEB

180 180

180 180 180

180 180

Let

Then straight angle

Now straight angle

Also straight angle

and`

c

c c c

c c c c

c

c c c

+

+ +

+ +

+ +

+ + + +

=

= -

= - -

=

= -

= =

EXAMPLES


1

Solution

( )x ABC

x

x

154 180 180

154 180

26

154 154

is a straight angle

`

c++ =

+ =

=

- -

2

Solution( )x

x

x

x

x

x

2 142 90 360 360

2 232 360

2 232 360

2 128

2 128

64

232 232

2 2


+ =

+ =

=

=

=

- -






3

Solution

( ) y y

y

y

y

y

y

2 30 90 90

3 30 90

3 30 90

3 60

3 60

20

30 30

3 3

right angle c+ + =

+ =

+ =

=

=

=

- -

4

Solution

(

( )

(

x WZX YZV

x

x

y XZY

w WZY XZV

50 165

50 165

115

180 165 180

15

15

50 50


straight angle


c

+ +

+

+ +

+ =

+ =

=

= -

=

=

- -

5

CONTINUED




Solution

( )

( )

( )

( )

a

b

b

b

b

d

c

90

53 90 180 180

143 180

143 180

37

37

53

143 143


straight angle


similarly

c

=

+ + =

+ =

+ =

=

=

=

- -


Solution




Solution

x x x x

x

x

x

x

6 10 30 5 30 2 10 360

14 80 360

14 280

14 280

20

80 80

14 14


+ =

=

=

=

- -

^ h

( )

( )

AEC

DEB

20 30

50

2 20 10

50

c

c

c

c

+

+

= +

=

= +

=



C

D A

B

E x 10)

( x +

(5 x + 3 )

x + 30)




41 Exercises


giving reasons

yc 133c

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)


(a) 59c (b) 107 31c l

(c) 45 12c l


(a) 48c

(b) 34 23c l

(c) 16 57c l


(ii) supplement of

(a) 43c 81c(b)

27c(c)

(d) 55c

(e) 38c

(f) 74 53c l

(g) 42 24c l

(h) 17 39c l

(i) 63 49c l

(j) 51 9c l



(2 x +30)c

142c







(a)

(b)

(c)

(d)

(e)

(f)

7


lines

8



A

B

C

D

(110-3 x )c

(3 x + 70)c


A B

C

D E

(50- 8 y)c

(5 y- 20)c

(3 y+ 60)c




Parallel Lines



Alternate angles


a Z shape Can you


alternate angles




you 1047297nd them







Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Right angle



Obtuse angle

x90 180c c c1 1

Straight angle



Re1047298ex angle

x180 360c c c1 1

Angle of revolution








Proof

( )

( ) ( )

( )

AEC x

AED x CED

DEB x AEB

x

CEB x CED

AEC DEB AED CEB

180 180

180 180 180

180 180

Let

Then straight angle

Now straight angle

Also straight angle

and`

c

c c c

c c c c

c

c c c

+

+ +

+ +

+ +

+ + + +

=

= -

= - -

=

= -

= =

EXAMPLES


1

Solution

( )x ABC

x

x

154 180 180

154 180

26

154 154

is a straight angle

`

c++ =

+ =

=

- -

2

Solution( )x

x

x

x

x

x

2 142 90 360 360

2 232 360

2 232 360

2 128

2 128

64

232 232

2 2


+ =

+ =

=

=

=

- -






3

Solution

( ) y y

y

y

y

y

y

2 30 90 90

3 30 90

3 30 90

3 60

3 60

20

30 30

3 3

right angle c+ + =

+ =

+ =

=

=

=

- -

4

Solution

(

( )

(

x WZX YZV

x

x

y XZY

w WZY XZV

50 165

50 165

115

180 165 180

15

15

50 50


straight angle


c

+ +

+

+ +

+ =

+ =

=

= -

=

=

- -

5

CONTINUED




Solution

( )

( )

( )

( )

a

b

b

b

b

d

c

90

53 90 180 180

143 180

143 180

37

37

53

143 143


straight angle


similarly

c

=

+ + =

+ =

+ =

=

=

=

- -


Solution




Solution

x x x x

x

x

x

x

6 10 30 5 30 2 10 360

14 80 360

14 280

14 280

20

80 80

14 14


+ =

=

=

=

- -

^ h

( )

( )

AEC

DEB

20 30

50

2 20 10

50

c

c

c

c

+

+

= +

=

= +

=



C

D A

B

E x 10)

( x +

(5 x + 3 )

x + 30)




41 Exercises


giving reasons

yc 133c

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)


(a) 59c (b) 107 31c l

(c) 45 12c l


(a) 48c

(b) 34 23c l

(c) 16 57c l


(ii) supplement of

(a) 43c 81c(b)

27c(c)

(d) 55c

(e) 38c

(f) 74 53c l

(g) 42 24c l

(h) 17 39c l

(i) 63 49c l

(j) 51 9c l



(2 x +30)c

142c







(a)

(b)

(c)

(d)

(e)

(f)

7


lines

8



A

B

C

D

(110-3 x )c

(3 x + 70)c


A B

C

D E

(50- 8 y)c

(5 y- 20)c

(3 y+ 60)c




Parallel Lines



Alternate angles


a Z shape Can you


alternate angles




you 1047297nd them







Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Proof

( )

( ) ( )

( )

AEC x

AED x CED

DEB x AEB

x

CEB x CED

AEC DEB AED CEB

180 180

180 180 180

180 180

Let

Then straight angle

Now straight angle

Also straight angle

and`

c

c c c

c c c c

c

c c c

+

+ +

+ +

+ +

+ + + +

=

= -

= - -

=

= -

= =

EXAMPLES


1

Solution

( )x ABC

x

x

154 180 180

154 180

26

154 154

is a straight angle

`

c++ =

+ =

=

- -

2

Solution( )x

x

x

x

x

x

2 142 90 360 360

2 232 360

2 232 360

2 128

2 128

64

232 232

2 2


+ =

+ =

=

=

=

- -






3

Solution

( ) y y

y

y

y

y

y

2 30 90 90

3 30 90

3 30 90

3 60

3 60

20

30 30

3 3

right angle c+ + =

+ =

+ =

=

=

=

- -

4

Solution

(

( )

(

x WZX YZV

x

x

y XZY

w WZY XZV

50 165

50 165

115

180 165 180

15

15

50 50


straight angle


c

+ +

+

+ +

+ =

+ =

=

= -

=

=

- -

5

CONTINUED




Solution

( )

( )

( )

( )

a

b

b

b

b

d

c

90

53 90 180 180

143 180

143 180

37

37

53

143 143


straight angle


similarly

c

=

+ + =

+ =

+ =

=

=

=

- -


Solution




Solution

x x x x

x

x

x

x

6 10 30 5 30 2 10 360

14 80 360

14 280

14 280

20

80 80

14 14


+ =

=

=

=

- -

^ h

( )

( )

AEC

DEB

20 30

50

2 20 10

50

c

c

c

c

+

+

= +

=

= +

=



C

D A

B

E x 10)

( x +

(5 x + 3 )

x + 30)




41 Exercises


giving reasons

yc 133c

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)


(a) 59c (b) 107 31c l

(c) 45 12c l


(a) 48c

(b) 34 23c l

(c) 16 57c l


(ii) supplement of

(a) 43c 81c(b)

27c(c)

(d) 55c

(e) 38c

(f) 74 53c l

(g) 42 24c l

(h) 17 39c l

(i) 63 49c l

(j) 51 9c l



(2 x +30)c

142c







(a)

(b)

(c)

(d)

(e)

(f)

7


lines

8



A

B

C

D

(110-3 x )c

(3 x + 70)c


A B

C

D E

(50- 8 y)c

(5 y- 20)c

(3 y+ 60)c




Parallel Lines



Alternate angles


a Z shape Can you


alternate angles




you 1047297nd them







Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





3

Solution

( ) y y

y

y

y

y

y

2 30 90 90

3 30 90

3 30 90

3 60

3 60

20

30 30

3 3

right angle c+ + =

+ =

+ =

=

=

=

- -

4

Solution

(

( )

(

x WZX YZV

x

x

y XZY

w WZY XZV

50 165

50 165

115

180 165 180

15

15

50 50


straight angle


c

+ +

+

+ +

+ =

+ =

=

= -

=

=

- -

5

CONTINUED




Solution

( )

( )

( )

( )

a

b

b

b

b

d

c

90

53 90 180 180

143 180

143 180

37

37

53

143 143


straight angle


similarly

c

=

+ + =

+ =

+ =

=

=

=

- -


Solution




Solution

x x x x

x

x

x

x

6 10 30 5 30 2 10 360

14 80 360

14 280

14 280

20

80 80

14 14


+ =

=

=

=

- -

^ h

( )

( )

AEC

DEB

20 30

50

2 20 10

50

c

c

c

c

+

+

= +

=

= +

=



C

D A

B

E x 10)

( x +

(5 x + 3 )

x + 30)




41 Exercises


giving reasons

yc 133c

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)


(a) 59c (b) 107 31c l

(c) 45 12c l


(a) 48c

(b) 34 23c l

(c) 16 57c l


(ii) supplement of

(a) 43c 81c(b)

27c(c)

(d) 55c

(e) 38c

(f) 74 53c l

(g) 42 24c l

(h) 17 39c l

(i) 63 49c l

(j) 51 9c l



(2 x +30)c

142c







(a)

(b)

(c)

(d)

(e)

(f)

7


lines

8



A

B

C

D

(110-3 x )c

(3 x + 70)c


A B

C

D E

(50- 8 y)c

(5 y- 20)c

(3 y+ 60)c




Parallel Lines



Alternate angles


a Z shape Can you


alternate angles




you 1047297nd them







Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Solution

( )

( )

( )

( )

a

b

b

b

b

d

c

90

53 90 180 180

143 180

143 180

37

37

53

143 143


straight angle


similarly

c

=

+ + =

+ =

+ =

=

=

=

- -


Solution




Solution

x x x x

x

x

x

x

6 10 30 5 30 2 10 360

14 80 360

14 280

14 280

20

80 80

14 14


+ =

=

=

=

- -

^ h

( )

( )

AEC

DEB

20 30

50

2 20 10

50

c

c

c

c

+

+

= +

=

= +

=



C

D A

B

E x 10)

( x +

(5 x + 3 )

x + 30)




41 Exercises


giving reasons

yc 133c

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)


(a) 59c (b) 107 31c l

(c) 45 12c l


(a) 48c

(b) 34 23c l

(c) 16 57c l


(ii) supplement of

(a) 43c 81c(b)

27c(c)

(d) 55c

(e) 38c

(f) 74 53c l

(g) 42 24c l

(h) 17 39c l

(i) 63 49c l

(j) 51 9c l



(2 x +30)c

142c







(a)

(b)

(c)

(d)

(e)

(f)

7


lines

8



A

B

C

D

(110-3 x )c

(3 x + 70)c


A B

C

D E

(50- 8 y)c

(5 y- 20)c

(3 y+ 60)c




Parallel Lines



Alternate angles


a Z shape Can you


alternate angles




you 1047297nd them







Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





41 Exercises


giving reasons

yc 133c

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)


(a) 59c (b) 107 31c l

(c) 45 12c l


(a) 48c

(b) 34 23c l

(c) 16 57c l


(ii) supplement of

(a) 43c 81c(b)

27c(c)

(d) 55c

(e) 38c

(f) 74 53c l

(g) 42 24c l

(h) 17 39c l

(i) 63 49c l

(j) 51 9c l



(2 x +30)c

142c







(a)

(b)

(c)

(d)

(e)

(f)

7


lines

8



A

B

C

D

(110-3 x )c

(3 x + 70)c


A B

C

D E

(50- 8 y)c

(5 y- 20)c

(3 y+ 60)c




Parallel Lines



Alternate angles


a Z shape Can you


alternate angles




you 1047297nd them







Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when








(a)

(b)

(c)

(d)

(e)

(f)

7


lines

8



A

B

C

D

(110-3 x )c

(3 x + 70)c


A B

C

D E

(50- 8 y)c

(5 y- 20)c

(3 y+ 60)c




Parallel Lines



Alternate angles


a Z shape Can you


alternate angles




you 1047297nd them







Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Parallel Lines



Alternate angles


a Z shape Can you


alternate angles




you 1047297nd them







Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Cointerior angles



another pair

If AEF EFD+ +=

then AB CDlt

If BEF DFG+ +=

then AB CDlt


then AB CDlt


is 180c )








EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





EXAMPLES


Solution

( )

55 ( )

AGF FGH

y AGF CFE AB CD

180 125

55

is a straight angle


c c

c

c

+ +

+ + lt

= -

=

=

2 Prove EF GH lt

Solution

( )CBF ABC

CBF HCD

180 120

60

60

is a straight angle

`

c c

c

c

+ +

+ +

= -

=

= =



in a different way







(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when






(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2 Prove AB CDlt

(a)

(b)

A

B C

D

E 104c

76c

(c)






Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Types of Triangles

Names of triangles




hypotenuse


A

B

C

D

E

F

52c

128c

(d) A B

C

D E

F

G

H

138c

115c

23c

(e)












Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when










Proof


( )

( )

( )

AB YZ

BXZ c BXZ XZY AB YZ

AXY b

YXZ AXY BXZ AXB

a b c

180

180

Draw line


similarly

is a straight angle

`

c

c

c

+ + +

+

+ + + +

lt

lt=

=

+ + =

+ + =





Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when






Class Investigation








x y z+ =

Proof

ABC x BAC y ACD z

CE AB

Let and

Draw line

c c c+ + +

lt

= = =

( )

( )

z ACE ECD

ECD x ECD ABC AB CE

ACE y ACE BAC AB CE

z x y


alternate angles

`

c

c

c

+ +

+ + +

+ + +

lt

lt

= +

=

=

= +

EXAMPLES


1

CONTINUED




Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Solution

( )x

x

xx

53 82 180 180

135 180

135 18045

135 135


+ =

+ =

=

- -

2

Solution


( )x x

x

x

x

x

x

48 180 180

2 48 180

2 48 180

2 132

2 132

66

48 48

2 2


+ =

+ =

=

=

=

- -

3

Solution

) y y

y

35 14135 141

106


`

D+ =+ =

=

- -


( )

( )

BCA BCD

y

y

y

y

180 141 180

39

39 35 180 180

74 180

74 180

106

74 74

is a straight angle

angle sum of

`

c c c

c

c

+ +

D

= -

=

+ + =

+ =

+ =

=

- -





pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when






pronumerals

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)






calculations




4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





4 Prove AB EDlt






(a)

(b)

(c)

(d)


JKLD is isosceles


A

B

C

D

E

88c

46c

10 Prove that MN QP

P

N

M

O

Q

32c

75c

73c




Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Congruent Triangles



For example


Tests






equal




EXAMPLES


CONTINUED

The included angle


the 2 sides




Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Solution

S

A

S

OS OQ

TOS QOP

OT OP

OTS OQP

(equal radii)


(equal radii)

by SAS`

+ +

D D

=

=

=


Solution





AB ED=

Solution

( )

( )

( )

( )

A

A

S

BAC CDE AB ED

ABC CED

AC CD

ABC DEC

AB ED

alternate angles

similarly

given

by AAS


`

`

+ +

+ +

lt

D D

D

=

=

=

=






congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when







congruent

(a)

(b)

X

Z

Y

B

C

A

4 7 m

2 3 m

2 3 m

4 7 m 110c 1 1 0

c

(c)

(d)

(e)(e


congruent

(a)

(b)

(c)

(d)

(e)

44 Exercises




3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





3 Prove that






AD BC=


of the circle

O

A

B

D

C


are congruent

Show that(b) AB CD=


BC DC=

A

B D

C


are congruent




O

A

B

C


are congruent





D

A B

C F E


are congruent






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when






OB bisects chord AC

O

A

B

C


to OBCT


to AC


below

D

A B

C


congruent to BCDT


BD are equal

Investigation








Similar Triangles


different sizes






Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Application


and enlargements

EXAMPLE


Solution




CA XZ

CB XY

y

y 4 9 3 6

5 4

3 6 4 9 5 4

`

=

=

=



AB XY

AC XZ

BCYZ

AB XY

AC XZ

BCYZ

26

3

412

3

515 3

`

= =

= =

= =

= =






y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





y

BAYZ

CB XY

x

x

x

3 6

4 9 5 4

7 35

2 3 3 65 4

3 6 2 3 5 4

3 6

2 3 5 4

3 45

=

=

=

=

=

=

=





are equal




EXAMPLES



Solution

(a) A+ is common

ADE D

( )( )

( )


ABC



+ +

+ +

lt

ltD

=

=

(b)

CONTINUED

Tests





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





AE

BC DE

AC AE

y

y

y

2 4 1 9

4 3

3 7 2 42 4 3 7 4 3

2 43 7 4 3

6 6

4 3

= +

=

=

=

=

=

=


Solution

( )

ZV XZ

ZW YZ

ZV XZ

ZW YZ

XZY WZV

3515

73

146

73


`

+ +

= =

= =

=

=



Ratio of intercepts




AB BC DE EF

BC AB

EF DE

That is

or

=

=




Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Proof


`

EHF D

`

`

( )

( )

( )

( )

( )

( )

DG AB

EH BC

BC AB

EH DG

GDE HEF DG EH

DEG EFH BE CF

DGE EHF

DGE

EH DG

EF DE

BC AB

EF DE

1

2


Also (similarly)

corresponding s

corresponding s

angle sum of s

So

From (1) and (2)

+ + +

+ + +

+ +

lt

lt

lt

D

D

=

=

=

=

=

=

=

=

EXAMPLES


Solution

x

x

x

8 9 9 31 5

9 3 8 9 1 5

9 3

8 9 1 5

1 44


=

=

=

=

^ h

CONTINUED





Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when






Solution



x

x

y

y

5 8 3 42 7

3 4

2 7 5 8

4 6

7 1 3 4

2 7 3 4

3 46 1 7 1

12 7

=

=

=

= +

=

=



appropriate

(a)

(b)

(c)

(d)

(e)

45 Exercises

These ratios come

from intercepts on

parallel lines


similar triangles

Why




(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





(f)

143

a

4 6 c

1 9 c

1 1 5 c

4 6 c

x c

91

257

89 y

(g)


places


similar


DEF D

and FGED

are similar



42

49

686

13

588182

A

C

B D

E F

yc87c

52c








places






CEBD ABF ltD





the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when






the value of m





(a)

(b)

(c)

(d)

(e)

12 Show that

(a) BC AB

FG AF

=

(b) AC AB

AG AF

=

(c)CE BD

EG DF

=


1 decimal place




2 decimal places





DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when






DID YOU KNOW


























c a b

c a b

That is

or

2 2 2

2 2

= +

= +




Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Proof


Let AD x DB y = =

Then x y c + =

In ADCD and ABCD

A+ is common

D

D

( ) ABC

ABC


ADC ACB

ADC

AB AC

AC AD

c b

bx

b xc

BDC

BC DB

AB BC

a

y

c a

a yc

a b yc xc

c y x

c c

c

90

Similarly

Now

2

2

2 2

2

`

c+ +

lt

lt

D

D

= =

=

=

=

=

=

=

+ = +

= +

=

=

^]

hg

EXAMPLES


Solution

c a b

x 7 4

49 16

65

2 2 2

2 2 2

= +

= +

= +

=





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





x 65


=

=


Solution

c a b

y

y y

y

8 4

64 1648

48

16 3

4 3

2 2 2

2 2 2

2

2

`

= +

= +

= +

=

=

=

=


1 decimal place

Solution

6 cm

cm

c a b

c

6 6

72

72

8 5

2 2 2

2 2

= +

= +

=

=

=


Leave the answer in


answer

CONTINUED






(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when







(b)

(c)

(d)


pronumerals(a)

(b)

(c)

(d)

46 Exercises


is right angled

Solution

68 cm

85 cm

51 cm


a b

c

c a b

5 1 6 8

72 25

8 5

72 25

2 2 2 2

2 2

2 2 2`

+ = +

=

=

=

= +








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when








1 decimal place




and 159 CF cm=


triangle


isosceles triangle

X

Y Z 1

12

7 Show that AC BC2=






terms of b


in







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when







d t t 13 180 6252 2= - +





to the nearest mm



how far do I walk












45 m

13 m










not rectangular

13 km

11 km

15 km














4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when











4 cm long

A

B

O

C

6 cm

4 cm



chord



the chord

Proof

Draw in diagonal AC

180 ( )

( )

ADC DCA CAD

ABC BCA CAB


ADC DCB CBA BAD

180

360

360

angle sum of

similarly

That is

`

c

c

c

c

+ + +

+ + +

+ + + + + +

+ + + +

D+ + =

+ + =

+ + + + + =

+ + + =








congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when









congruent triangles







all be proven

Parallelogram


EXAMPLE

Find the value of i

Solution

120 56 90 360

266 360

94


i

i

+ + + =

+ =

=

^ h

PROPERTIES

TESTS




Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Rhombus





Application





all sides are equal


angle then you can


right angles





Rectangle

PROPERTIES

PROPERTIES

TEST




Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Square





Trapezium


Kite





TESTS

PROPERTIES




EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





EXAMPLES


Solution

( )

( )

( )

x

y

83

6 7

2 3

opposite s in gram



c + lt

lt

lt

i =

=

=


6 BD cm=

Solution

( )

( )

AB x

ABCD AB AD x

A 90

Let



=

= =

=


3

c a b

x x

x

x

x

6

36 2

18

182 cm

2 2 2

2 2 2

2

2

`

= +

= +

=

=

=

=

CONTINUED





(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when






(a)

(b)

(c)

(d)

(e)

(f)

(g)

47 Exercises

3


is a rhombus


of OP

Solution

(a) ( )

( )

OA OB

PA PBOA OB PA PB

equal radii


=

=

= = =









perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when






perpendicular to AD


( ) B D x180 c+ += = -






(a)

(b)

(c)

(d)

(e)

7

y

3 x

x + 6

(f)



ABC+


parallelogram

(a)

(b)




(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





(c)

(d)


(a)

(b)

ABCD is a kite

(c)

(d)

(e)




rhombus



and ADE+





Polygons






Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Proof





That is S n

n

180 360

2 180 c

= -

= -] g

EXAMPLES

4-sided(square)

3-sided(equilateral

triangle)

5-sided(pentagon)

6-sided(hexagon)

8-sided(octagon)

10-sided(decagon)

DID YOU KNOW





astigmatism




( 2) 180

S n

S n

180 360

or c

= -

= -


Proof



n

n n

n n

180

180 180 360

180 180 360

360


c

c

c

= -

= - -

= - +

=

] g




EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





EXAMPLES



Solution

( )

( )

n

S n

15

0

15 0

0

2340

2 18

2 18

13 18

c

c

c

c

=

= -

= -

=

=



are 140c

Solution



( )

( )

S n

n n

n

n

n

2 180

140 2 180

180 360

360 40

9

But

So

c

c

= -

= -

= -

=

=



angles each 140 c


angles of

a pentagon(a)

a hexagon(b)





angle of a regular

pentagon(a)

octagon(b)

12-sided polygon(c)

20-sided polygon(d)

15-sided polygon(e)


angle of a regular

hexagon(a)

decagon(b)








polygon



48 Exercises












interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when













interior angles




5220c



145c



angles

(a) 40c

(b) 03 c

(c) 45c

(d) 36c

(e) 12c


F

E D

A B

C


BCD are congruent


parallel



shown

D

A

B

E

C

F

G

H




triangle

D

A

B E

C



side p


angle is

( )

p

p180 2-




Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Areas


Rectangle

A lb=

Square

A x2=

Triangle

A bh21

=

Proof

h

b


A square is a

special rectangle



rectangle




bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





bharea

21

21

21

21

` =


CDE ABCDarea` D =

A bhThat is =

area

A bh=

Proof



Area ABEF bh=

In ADF D and BCED

( )

( )

AFD BEC

AF BE h

AD BC

ADF BCE

ADF BCE

ABCD ABEF

bh

90



by RHS

area area

So area area

`

`

c+ +

D D

D D

= =

= =

=

=

=

=

Rhombus

The area of a


same as the area of

two triangles

A xy 21

=


Parallelogram




( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





( ) A h a b21

= +

Proof

DE x

DF x a

FC b x a

b x a

Let

Then

`

=

= +

= - +

= - -

] g

Proof



AE EC x21

= = and DE EB y 21

= =

Also AEB 90c+ =

ABC x y

xy

ADC x y

xy

xy xy

xy

Area

Area


21

21

41

21

21

41

41

41

21

`

D

D

=

=

=

=

= +

=

Trapezium




A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





A r 2r=

EXAMPLES


Solution

( )

( ) ( )

24

A h a b

4 7 5

2 12

m2

21

21

= +

= +

=

=


8 c

m

7 c m

21 cm

42 cm

CONTINUED

( )

( )

( )

ADE ABFE BFC

xh ah b x a h

h x a b x a

h a b

2


21

21

21

2

1

D D= + +

= + + - -

= + + - -

= +

Circle






Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Solution

lb

lb

8 9 12 1

107 69

3 7 4 2

15 54

107 69 15 54

92 15


cm


cm

shaded area

cm

2

2

2

`

=

=

=

=

=

=

= -

=



2 m

5 4 m

Solution


2

( )

r

A r

254

27

27

2290 22 m

2

2

r

r

=

=

=

=

=

126 54

6804

2290 22 6804

9094 22

Area rectangle

Total area

m2

=

=

= +

=


(a)

(b)

49 Exercises




(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





(c)

(d)

(e)

(f)

(g)




(b)

(c)

(d)

(e)

6 c m

2 cm


(a)

(b)




(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





(c)

(d)

(e)











Find its area(a)





















Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when





Test Yourself 4

The perimeter

is the distance

around the outs

of the 1047297gure


(a)

(b)

(c)

x (d)

O is the centre

of the circle)

(e)

(f)

(g)




similar






1 decimal place











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when











11 Prove AG AF

AC AB

=


BC


congruent











polygon










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when










perpendicular


10 Evaluate x





1 decimal place


2 decimal places


E such that DE BD21

=


Prove that(b) DE x

2

2= units when


geometry 1 i

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