chapter09 geometric figures

Geometric figures

9Space and geometry

There are many different shapes that you see every day, in buildings, on roads, in manufacturing and in artwork. Triangles and quadrilaterals seem to be more common than circular shapes or other polygons. This chapter looks at the language of geometry, geometric properties and constructions involving triangles and quadrilaterals.

recognise different types of polygons, including convex and regular polygons

label and name points, intervals, equal intervals, equal angles, lines, parallel lines, perpendicular lines, triangles and quadrilaterals

recognise and classify triangles using sides and angles recognise and classify quadrilaterals, including convex and non-convex

quadrilaterals construct perpendicular lines and parallel lines using set squares and

rulers construct various types of triangles and quadrilaterals using compasses,

protractors, set squares and rulers investigate the properties of triangles and quadrilaterals, including sides,

angles, diagonals, axes of symmetry, and order of rotational symmetry.

polygon Any plane shape with straight sides. diagonal An interval joining two non-adjacent vertices of a polygon. regular polygon A polygon with all sides equal and all angles equal. convex polygon A polygon whose vertices point outwards, not inwards,

where any interval joining two points on the polygon lies completelyinside it.

interval Part of a line, with a starting point, an end point and a definite length.

obtuse-angled triangle A triangle with one obtuse angle. included angle The angle between two given sides of a polygon. set square A ruling instrument in the shape of a right-angled triangle.

Why is it impossible to construct a triangle with sides of length 7 cm, 15 cm and 5 cm?

In this chapter you will:

Wordbank

Think!

GEOMETRIC FIGURES 283

CHAPTER 9

284

NEW CENTURY MATHS 7

1 Draw:a a pair of perpendicular lines b a pair of parallel lines.

2 Draw a rectangle and mark in any axes of symmetry.3 For the quadrilateral shown on the right:

a name two intervals that are parallelb name two intervals that are perpendicularc are the diagonals AC and DB equal in length?d what is the size of DEC?e if DAB and ABC are cointerior and

DAB = 115, what is the size of ABC?4 a What type of triangle has three equal sides?

b What type of quadrilateral has opposite sides parallel and all angles measuring 90?5 Copy these shapes and mark in the axes of symmetry on each one.

6 Which of the following shapes have rotational symmetry? State the order of rotational symmetry for those that do.

7 For the rectangular prism shown on the right decide whether each of the following is true (T) or false (F):a KL LP b NR II MQc NM OP d KO II PQ.

8 a Draw a triangle that has an obtuse angle.b Draw a scalene triangle.

C

A B

E

D

a b c d

e hf g

a b c

d e f

O

L

M

P

K

R Q

N

Start up

Worksheet 9-02

Symmetry

Worksheet 9-01

Brainstarters 9

Skillsheet 9-01

Line and rotational symmetry

GEOMETRIC FIGURES

285

CHAPTER 9

Polygons

A polygon is a closed plane shape made up of straight sides. The word polygon means many angles. These shapes are all polygons:

A polygon is named by the number of sides it has.

Convex and non-convex polygons

In Chapter 4, you learned about convex and non-convex solids. We can also describe convex and non-convex polygons.Convex polygons have vertices that point

outwards

while non-convex (or concave) polygons have vertices that point or cave

inwards

.

Name Number of sides

Pentagon 5Hexagon 6Heptagon 7

Octagon 8Nonagon 9Decagon 10Undecagon 11Dodecagon 12

a b c d

e f g

h i j k l

SkillBuilder 23-01

Shapes I

A convex polygon A non-convex polygon

286

NEW CENTURY MATHS 7

A simple test to determine if a polygon is convex or non-convex is to draw any interval joining two points on the polygon, or any diagonal joining two vertices of the polygon.

If the interval lies completely

inside

If all or part of the interval lies

outside

the polygon, then it is

convex

. of the polygon, then it is

non-convex

.

1 Name each of the polygons (a to l) on page 285, and state which one is non-convex.2 How many sides has:

a a hexagon? b a quadrilateral? c a nonagon?d a decagon? e a heptagon? f a pentagon?g a dodecagon? h an octagon? i an undecagon?

3 Regular polygons have all sides equal and all angles equal. Which of the polygons from Question 1 are regular?

4 Draw:a a regular hexagon b a non-regular hexagon c a regular triangled a non-regular heptagon e a convex pentagon f a non-convex dodecagon.

5 Which of the following shapes are not polygons?a trapezium b ellipse c squared diamond e prism f circle

6 a Draw a pentagon with one axis of symmetry.b Draw a quadrilateral with four axes of symmetryc Draw a hexagon with six axes of symmetryd Draw a decagon with two axes of symmetry.

7 Geometry software, such as Cabri Geometry or Geometers Sketchpad, can be used to demonstrate that you understand the words you are required to learn in this chapter. Use this link to go to a drawing exercise.

8 How many diagonals has a:a kite? b pentagon? c hexagon?

9 What shapes have been put together to form each of these composite shapes?a b c

d e f

Exercise 9-01

Geometry 9-01

The vocabulary of geometry

GEOMETRIC FIGURES 287 CHAPTER 9

10 Copy these composite shapes into your book and divide them into the shapes requested.

11 Use Geometers Sketchpad or Cabri Geometry to create shapes that a partner can try to draw using your instructions.

a b c

d e f g

Two pentagonsTwo triangles and one rectangle

One trapezium and one hexagon

Two trapeziums Four triangles One triangle and one trapezium

One square and one heptagon

Geometry 9-02

Creative copying

Working mathematically

Communicating: Logos and designs1 Find examples of company

logos. Draw them in your book. Discuss the shapes used to make them.

2 Research some Islamic or Grecian art in the library or on the Internet. Bring some pictures to class.

288 NEW CENTURY MATHS 7

Classifying trianglesA polygon is two-dimensional (at) because it has length and breadth (but not thickness). A triangle, having three sides, is the simplest type of polygon. It is an important shape that has been used throughout history and civilisations in building, construction, packaging, and even as a cultural or religious symbol.Triangles can be classied in two ways: by their sides (equilateral, isosceles or scalene) by their angles (acute-angled, obtuse-angled, right-angled).

Sides

EquilateralThree equal sides

Three equal angles

IsoscelesTwo equal sides

Two equal angles

ScaleneNo equal sides

No equal angles

Angles

Acute-angledAll three angles acute

Right-angledOne right angle

Obtuse-angledOne obtuse angle

Example 1Classify this triangle using sides and angles.

SolutionThis triangle has two equal sides and one obtuse angle.It is isosceles and obtuse-angled.

R

ST

1 Rule up a table with these headings:

Place the letter of each of the following triangles under the headings that match. (The same triangle may appear under more than one heading.)

Acute-angled Obtuse-angled Right-angled Equilateral Isosceles Scalene

Exercise 9-02

Worksheet 9-03

Properties of triangles

Example 1


2 Draw the following triangles:a a scalene triangle b a right-angled scalene trianglec an isosceles triangle d an equilateral trianglee a right-angled isosceles triangle f an acute scalene triangle.

3 Use Geometers Sketchpad or Cabri Geometry to explore and draw special trianglesand form your own denitions.

4 Is it possible to draw an equilateral right-angled triangle? Why?5 Copy these triangles into your book and draw in all axes of symmetry.

6 Do any triangles have rotational symmetry? Give examples to support your answer.7 Is it possible to draw a triangle with two obtuse angles? Why?8 The prex tri means three. Find the meaning of each of the following mathematical

tri words:a trisect b trilateral c triangulate.

a b c

d e f

g h i

j k l

3088

62

140

20

3 cm

3 cm3 cm

12 cm

6 cm

15 cm

12 cm 12 cm

20

60

60 60

a b c

d e f

SkillBuilder 23-02

Shapes II

Geometry 9-03

Triangles


Naming geometric figuresPoints, lines and intervalsA point is a position represented by a dot which is labelled by a capital letter. The points on the right are labelled P, A and O.A line is a straight edge that continues innitely (forever) in both directions, so it is usually drawn with arrowheads at both ends. A line is named by any two points on it. The line in this diagram is labelled LM.An interval is a part of a line. It is a section of the line with a starting point, an end point, and a denite length. An interval is labelled by its two end-points. The interval in this diagram is RS.

Reasoning and reecting: Building with shapes1 When you look at the shapes of buildings and other constructions, you will notice

that some shapes are more common than others. Write the names of the most commonly used shapes.

2 When building any structure, strength is important. Which is the strongest shape?

3 a Use ice block sticks or geo-sticks to make a triangle, a square and a pentagon as shown above.

b Stand each shape up and push one corner. What happens?4 You saw in Question 3 that a triangular framework is very strong or rigid, which is

why that shape is used in many types of constructions. How can you make the other shapes in Question 3 stronger?

5 Find as many pictures as you can of triangular frameworks in everyday use. The ANZAC Bridge in Sydney is a good example.


pointsP A

O

line

LM

intervalR S


TrianglesA triangle is identied by the capital letters that label its vertices or angles. The triangle in this diagram is labelled ABC or BCA or CAB (gures are usually labelled in clockwise order).The angles of a triangle can be labelled by one letter or three letters. The marked angle, shown, is T or RTS or STR.

The sides of a triangle can be described in two ways: by two capital letters labelling their endpoints by a small letter that matches the capital letter

naming its opposite angle.This diagram shows the triangle ABC.The angles are labelled A, B, C.The sides are labelled a, b, c, where side a is opposite A, side b is opposite B, and side c is opposite C.The red side can be called CB, BC, or a.

QuadrilateralsA quadrilateral is any plane shape with four sides and is identied by the capital letters that label its vertices or angles. The quadrilateral in this diagram is labelled PQRS or QRSP or RSPQ or SPQR.The angles of a quadrilateral can be labelled by one letter or three letters. The marked angle in the quadrilateral PQRS is R or QRS or SRQ.The sides of a quadrilateral can be identied by the two capital letters labelling their endpoints. The red side can be labelled PS or SP.

Equal angles and intervalsIn geometric diagrams, equal angles are marked by identical symbols, while equal intervals are marked by identical strokes. In this diagram, DF, DE and EG are all the same length while F and DEF are the same size.

Parallel and perpendicular intervalsIn Chapter 2, we learned about parallel and perpendicular lines. We can use the same language and symbols to describe parallel and perpendicular intervals.Parallel intervals point in the same direction and do not intersect. In the rectangle WXYZ, shown, WZ is parallel to XY, which is written WZ II XY.Perpendicular intervals meet at right angles (90). In the rectangle WXYZ, WX is perpendicular to XY, which is written as WX XY.

C B

A

S

R

TA

B

C

bc

aThe side labelled a is opposite the angle labelled A.

P

R

S

Q

D

F

EG

W X

Z Y


Example 2JKL is an isosceles triangle.The interval JM divides JKL into two smaller triangles.a Name the two smaller triangles.b What can be said about sides JL and JK?c Name the two equal angles in JKL.d Explain the meaning of this sentence:

If JM LK, then LM = MK.Solutiona The two smaller triangles are JML and JKM.b JL and JK have equal length.c JLM and JKM (or L and K)d If side JM is perpendicular to side LK, then the lengths of intervals LM and MK are equal.

a Draw a parallelogram and label it DEFG. b Mark both pairs of parallel sides.c Name both pairs of parallel sides. d Mark the equal angles D and F.e Mark the equal sides DG and EF.Solutiona The answer should resemble the diagram on the right.b On the diagram, one pair of parallel sides is marked by

arrows, and the other pair is marked by double arrows.c DE II GF and DG II EF.d The equal angles D and F are marked by equal

arcs on the diagram.e The equal sides DG and EF are marked by dashes on the diagram.

L

J

KM

Example 3

G

D

F

E

1 In this diagram, name a pair of intervals that are:a equalb perpendicularc parallel.

2 a Draw two lines, PQ and RS, intersecting at T.b Mark the equal angles PTR and STQ.c PTR = STQ. Why?

3 Copy the triangle KLM shown on the right, and correctly label its sides k, l, and m.

G

D

FE

JIH

LM

K

Exercise 9-03Example 2

Example 3


4 What is the difference between the line EF and the interval EF?

5 a Name the two triangles in the diagram on the right.b Name the interval that is equal to:

i QR ii PT.c If TQRS is a trapezium, name the parallel sides.d Copy the diagram and mark the parallel sides.e Mark the equal angles PTQ and TSR.f PTQ = TSR. Why?

6 CDEF is a kite.a Copy the diagram and mark in the equal sides.b What side is equal to DE?c Mark the equal angles F and D.d Draw the two diagonals FD and CE.e Show on your diagram that FD CE.

7 a Draw an isosceles triangle EFG where EF = EG.b Label the sides of the triangle e, f, and g.c What is another name for the side EF?d Mark on the triangle the equal angles F and G.

8 a Draw parallel lines AB and CD.b Draw a transversal EF crossing both lines AB and CD, where EF AB.c CD EF. True or false?

9 a What type of quadrilateral is STUV?b VS = ST. True or false?c VS II ST. True or false?d = . True or false?e Name the marked pair of equal sides.

10 a Draw a square, WXYZ, and mark all equal sides and angles.b Name the point where side XY meets side ZY.c Name a pair of parallel sides and mark them.d Name a pair of perpendicular sides.e Explain the meaning of:

i WX XY ii WX = XY 11 a Draw a trapezium, UVWX, where UX = VW.

b Mark the equal angles UXW and VWX.c UX II VW. True or false?d UV II XW. True or false?

EF

S

T

P

R

Q

F

E

D

C

T

S

V

U

SV T VTU

W X

Z Y


Constructing trianglesTo construct a triangle we need to know the length of its sides and the size of its angles. We also need a ruler, a protractor and compasses.The following examples will show you how to construct triangles. Hint: Draw a rough sketch before beginning the construction.

Example 41 Construct a triangle with sides 3 cm,

5 cm and 4 cm.

SolutionStep 1: Draw an interval 5 cm long. (It is easier

to start with the longest side.)Step 2: Open the compasses to a 3 cm radius and

draw an arc from one end of the interval. (Every point on this arc is 3 cm from the end of the interval.)

Step 3: Open the compasses to 4 cm and draw an arc from the other end of the interval. (Every point on this second arc is 4 cm from the other end of the interval.)

Step 4: Complete the triangle by joining the intersecting point of the arcs to the ends of the interval.

2 Construct ABC where a = 5 cm, C = 30 and B = 70.

5 cm

3 cm 4 cmRough sketch

5 cm

5 cm3 c

m

5 cm

4 cm

4 cm

5 cm

3 cm

7030

a = 5 cm

A

B

C

bRough sketch c


SolutionStep 1: Draw an interval 5 cm long.

Step 2: Draw a 70 angle at B.Step 3: At C, draw a 30 angle.

Step 4: Join the arms to complete the triangle.

3 Construct a triangle with one side measuring 6 cm, another side measuring 4 cm and an angle between them of 35.

SolutionStep 1: Draw an interval 6 cm long.

Step 2: Draw an angle of 35 at one end.

Step 3: Measure an interval of 4 cm onthe new arm.

Step 4: Complete the triangle.

Note: 35 is called the included angle because it is between the two sides.

5 cm BC

70305 cm BC

70305 cm BC

A

4 cm

356 cm

Rough sketch

6 cm

356 cm

4 cm

356 cm

4 cm

356 cm


1 Construct each of these triangles accurately:

2 Geometers Sketchpad or Cabri Geometry can be used to accurately draw triangles. This link will take you to an activity that shows you how.

3 For each of the triangles constructed in Question 1:i name the largest angle and the longest side

ii name the smallest angle and the shortest side.4 Copy and complete this sentence:

In any triangle, the longest side is always the largest angle, while the shortest side is always the angle.

5 a Which triangle in Question 1 is equilateral?b Measure its angles. What do you notice?

6 a On a sheet of paper, construct an equilateral triangle of side length 5 cm, and cut it out.b By folding along each of its axes of symmetry, what do you observe about the sizes

of the triangles angles?c Measure the angles. What do you notice?d Copy and complete:

An equilateral triangle has three sides, and three angles each of size .7 Construct each of the following triangles. Draw a rough sketch rst.

a ABC with a = 4 cm, b = 3 cm, and C = 50.b RST with r = 5 cm, s = 3 cm, and t = 3 cm.c PQR with P = 60, Q = 60, and PQ = 4 cm.d LMN with LN = 5 cm, ML = 4 cm, and NLM = 25.

8 a Which triangle in Question 7 is isosceles?b Measure its angles. What do you notice?

9 a On a sheet of paper, construct an isosceles triangle with two sides of length 6 cm, and cut it out.

b By folding, what do you observe about the sizes of the triangles angles?c Measure the angles. What do you notice?d Copy and complete:

An isosceles triangle has two sides, and two angles opposite them.

a b c

d e f

4 cm

Q

S R

3 cm

4 cm40

M

O N6 cm80

C

E D3 cm

F

H G3 cm

3 cm3 cm

K

M L3 cm120

3 cm

X

Z

Y

3040

mm

130

2 cmExercise 9-04

Example 4

Geometry 9-04

Constructing triangles

Worksheet 9-03

Properties of triangles


Classifying quadrilateralsA quadrilateral is any shape with four sides, but there are six special quadrilaterals that you need to know. These are listed in the table below.

Name A quadrilateral with Diagrams

Trapezium one pair of opposite sides parallel

Parallelogram two pairs of opposite sides parallel

Rhombus(or diamond)

four equal sides

Rectangle four right angles

Square four equal sides and four right angles

Kite two pairs of adjacent sides equal

10 The triangle inequality rule says that if you add any two sides of a triangle, the combined length is always greater than the length of the third side. (This inequality can be written as a + b c).a Test that this inequality is true for all of the triangles you constructed in

Question 7.b Why is it impossible to construct a triangle with sides of length 7 cm, 15 cm

and 5 cm?


Velcro sticksAll types of uses have been found for Velcro fasteners in our world. They are used in the clothing industry, and for attaching the chambers in artificial hearts, while astronauts use them to fasten equipment so that it does not float away within their space capsules.The idea for Velcro came to Georges de Mestral, a Swiss engineer, in 1948. It is made of two surfaces, one with hooks and one with loops. A thumbsize piece of Velcro contains about 750 hooks and, on the other side, about 12 500 loops. De Mestral conceived the idea of Velcro when he noticed tiny seed pods caught in his socks after a walk in a forest.

Find four uses for Velcro.

Just for the record

Example 51 PQRS is a parallelogram, as shown on the right.

a Measure the lengths of its sides. Are opposite sides equal?

b Measure the size of its angles. Are opposite angles equal?

c Does a parallelogram have line symmetry? If so, draw its axes of symmetry.

d Does a parallelogram have rotational symmetry? If so, state the order.e Draw the diagonals PR and QS and measure them. Are the diagonals equal?Solutiona By measurement, PQ = SR = 4.2 cm and PS = QR = 2 cm. Opposite sides are equal.b P = R = 100 and Q = S = 80. Opposite angles are equal.c A parallelogram has no axes of symmetry

(you cannot fold it in half).d A parallelogram has rotational symmetry of

order 2. You can rotate it 180 so that it maps on to itself. The centre of symmetry is marked O.

e PR = 4.2 cm while QS = 5 cm. The diagonals are not equal.

2 This diagram illustrates the properties of the diagonals of a rectangle.a Are the diagonals equal?b Do the diagonals bisect each other?c Do the diagonals intersect at right angles?d Do the diagonals bisect the angles of the rectangle?

P Q

S R

P Q

S R

O

A B

D C


Solutiona The diagonals, AC and BD, have equal length.b The diagonals bisect each other (cut each other in half), shown by the equal markings.c The diagonals do not intersect at right angles.d The diagonals do not bisect the angles of the rectangle, that is, the right-angled

vertices (A, B, C, and D) are not cut into halves (45) by the diagonals.

1 Find what the prex quad means. List other words beginning with quad.2 Label each of these quadrilaterals as convex or non-convex.

3 Make an enlarged copy of each of the quadrilaterals in the diagram below or print out a copy. Cut out each quadrilateral and name it.

a b c

d e f

Exercise 9-05

Worksheet9-04

Properties of quadrilaterals


4 Use Geometers Sketchpad or Cabri Geometry to explore the quadrilaterals in Question 3 and form your own denitions.

5 a Copy or print out this table.

b Test the properties of each quadrilateral listed in the table by folding and measuring them with a ruler, protractor and set square. If the listed property is true, then place a tick in the appropriate space. Write appropriate numbers in the last two rows.

c Check your results with your teacher.d You should have noticed that there are no ticks for the kite. Write two properties of the

kite (that is two things that are always true about its sides, angles or diagonals).6 Draw each of the following quadrilaterals and mark all axes of symmetry.

a rectangle b square c parallelogramd rhombus e trapezium f kite

7 List the quadrilaterals in Question 6 that have rotational symmetry and mark the centre of symmetry, O, each time.

8 Use a ruler and protractor with the quadrilaterals you cut out in Question 3, to discover the properties of the diagonals of each one, as listed in the table below. Copy or print out this table. Place ticks in the appropriate spaces.

9 Which quadrilateral am I? (There may be more than one answer.)a My diagonals are equal. b All my sides are equal.c My opposite sides are equal. d My diagonals bisect each other.e I have four right angles. f I have two pairs of opposite sides parallel.g I have rotational symmetry, but no axes of symmetry.h My diagonals bisect each other at right angles.

10 a Does a square have the same properties as a rectangle? Why do you think?b Does a rhombus have the same properties as a parallelogram? Why do you think?

Trapezium Parallelogram Rhombus Rectangle Square Kite

Opposite sides are equal

Opposite sides are parallel

Opposite angles are equal

All angles are 90

Diagonals are equal

Number of axes of symmetry

Order of rotational symmetry

Trapezium Parallelogram Rhombus Rectangle Square Kite

Diagonals are equal

Diagonals bisect each other

Diagonals intersect at right angles

Diagonals bisect the angles of the quadrilateral

Example 5

Geometry 9-05

Quadrilaterals

SkillBuilder 23-11Axes of

symmetry

SkillBuilder 23-05

4-sided figures

Worksheet 9-04


Worksheet 9-04



Properties of triangles and quadrilaterals: a summaryShape Properties

Equilateral triangle

All three sides equal All three angles 60 Three axes of symmetry

Isosceles triangle

Two sides equal Two angles equal (opposite the equal sides) One axis of symmetry

Scalene triangle

No sides or angles equal No axes of symmetry

Trapezium One pair of parallel sides No axes of symmetry

Kite Two pairs of adjacent sides equal One pair of opposite angles equal One axis of symmetry Diagonals intersect at right angles

Parallelogram Opposite sides equal and parallel Opposite angles equal No axes of symmetry Diagonals bisect each other

Rhombus All four sides equal Opposite sides parallel Opposite angles equal Two axes of symmetry Diagonals bisect at right angles Diagonals bisect the angles of the rhombus

Rectangle All four angles 90 Opposite sides equal and parallel Two axes of symmetry Diagonals are equal Diagonals bisect each other

Square All four sides equal, all four angles 90 Four axes of symmetry Diagonals are equal and bisect each other

at right angles Diagonals bisect the angles of the square

60 60

60

= 45


Applying strategies and reasoning: Shape puzzlesAll of the shapes used in this activity can be printed out. Use the link to nd them.

1 a How many squares can you nd in this shape? (The answer is not 16!)

b How many rectangles can you nd?

2 Can you trace this shape without going over any line twice and without lifting your pencil off the paper?

3 There are 12 different ways ve squares can be arranged. These shapes are called pentominoes. The rst ve have been done for you. Draw the other seven different arrangements.

4 Copy this equilateral triangle, cut out the four pieces and rearrange them into a square.

5 How many triangles can you nd in each of these shapes?

AB

D

C

a b c d


Worksheet 9-05

Shape puzzles


Constructing perpendicular and parallel linesThe set squareA set square is made in the shape of a right-angled triangle. It is used for measuring and drawing right angles and for constructing perpendicular and parallel lines. There are two types of set squares, named according to their angle sizes:

Perpendicular lines and parallel lines can be constructed using a set square or a protractor.

6 Copy this hexagon twice on to a piece of paper and then:a cut the rst hexagon into two pieces

and rearrange them to make a parallelogram

b cut the second hexagon into three pieces and rearrange them to make a rhombus.

60

30

45

45

The 6030 set square

The 45 set square

Example 6Use a set square to construct a line perpendicular to XY through point W.

X

Y

W


Solution(A perpendicular line can also be constructed using a protractor, by measuring a 90 angle.)

Use a set square to construct a line perpendicular to XY through point Z (where Z is not on XY).

SolutionStep 1: Place the set square on the line XY.Step 2: Place the ruler through the point Z.Step 3: Slide the set square along XY until it

meets the ruler.Step 4: Slide the ruler until it ts the edge

of the set square and is perpendicular to XY.

Step 5: Rule the perpendicular line.

Use a set square to construct a line parallel to AB through point P.

X

Y

W

Example 7

X

Y

Z

X

Y

Z

Example 8

P

A

B


SolutionStep 1: Place the set square on the line AB.Step 2: Place the ruler next to the set square,

at right angles to AB.Step 3: Hold the ruler rmly and slide the set

square until its edge passes through point P.

Step 4: Rule the parallel line.(This construction can also be done with a protractor replacing the set square, using the 90 mark.)

P

A

B

1 What is the name of the set square that is:a tall and thin?b short and wide, and half of a square?c an isosceles, right-angled triangle?d a scalene, right-angled triangle?

2 Why do you think the word square in set square is used to describe a right angle? What other types of squares are used to draw or measure right angles?

3 Draw a line and mark a point L on it. Construct a perpendicular line through L.

4 Draw a line and mark a point X below it. Construct a perpendicular line to the line through X.

5 Draw a line and mark a point P above it. Construct a parallel line through P.

6 a Draw an interval, AB, 4 cm long. This will be the base of a triangle.

b Mark X, the midpoint of AB, and construct a perpendicular interval XC of length 5 cm at X.

c Join C to A and then to B to make a triangle, CBA.d What special type of triangle is CBA?

7 a Draw a line XY and mark a point A above it. b Draw a line parallel to XY through A.c Draw a line from A to XY perpendicular to the line drawn in part b. Label the line AB.d What do you notice about AB and XY?

L

X

P

X BA

C

Exercise 9-06

Example 6

Example 7

Example 8


8 Are parallel lines always the same distance apart?a Draw a pair of parallel lines and mark the

points D and E on one of them. b Draw perpendiculars from D and E to the other

line. Where the lines intersect, mark the points F and G.

c Measure the lengths of DG and EF. What do you notice?

9 Draw two intervals that are parallel and of different lengths. Join their ends to make a quadrilateral. What type of quadrilateral have you constructed?

10 Draw an interval and mark its midpoint. Draw another interval of a different length through this midpoint, perpendicular to the rst interval. Join the ends of both intervals to make a quadrilateral. What type of quadrilateral have you constructed?

11 Use the link to go to an exercise which uses Cabri Geometry or Geometers Sketchpad to draw parallel and perpendicular lines.

G

D

E

F

Geometry 9-06

Parallel and perpendicular

lines

Communicating and reecting: Vertical linesMany people use the word perpendicular when they really mean vertical. A vertical line is a line that is perpendicular to the Earths surface (or the horizon). Vertical means up and down, while horizontal means at or across.Brick walls are vertical.

Vertical lines are important when building homes, hanging wallpaper and positioning pictures.1 Find out how builders use a plumb bob to set out vertical lines.2 In your own words, write the meanings of perpendicular and vertical.


Right angle


Constructing quadrilateralsExample 9Construct a square, FGHI, of side length 4 cm.

SolutionStep 1: Construct the base, IH, of length 4 cm.Step 2: Use a set square to construct the

perpendiculars, FI and GH, of length 4 cm.Step 3: Join FG.

Construct this kite, PQRS.

SolutionStep 1: Draw PS of length 3.5 cm.Step 2: Measure 105 at S.Step 3: Construct SR of length 5 cm.Step 4: At R, use your compass to draw an arc

of radius 5 cm.Step 5: At P, use your compass to draw an arc

of radius 3.5 cm.Step 6: The arcs cross at Q, the fourth vertex of

the kite. Join P and R to Q.

Rough sketch

4 cmHI

GF

4 cm HI

GF

Example 10

P

R

QS

5 cm

3.5 cm

105

5 cm

P

R

QS

3.5 cm

105

3.5 cm

5 cm

Worksheet9-06

Constructions in diagrams

Worksheet9-08

Try drawing these!

Worksheet9-07

Constructionsin words


1 a Draw BA measuring 6 cm. b Construct a perpendicular, BC, 3 cm long.c Complete the rectangle, ABCD.

2 Construct a square, KLMN, of side length 5 cm.

3 Construct this trapezium.

4 Construct a parallelogram with sides of 6 cm and 4 cm and an included angle of 65.

5 a Construct two parallel intervals 4 cm long and 3 cm apart. b Join the ends to make a quadrilateral. What type of

quadrilateral is it?c Measure the lengths of the two new sides.d Are these sides both equal and parallel?

6 a Draw two joined intervals of the same length and use your instruments to complete these shapes.

b What are these shapes called?

7 A trapezium with two equal (non-parallel) sides is called an isosceles trapezium.a Name the equal sides in the isosceles

trapezium, PQRS, shown on the right. b Construct the isosceles trapezium PQRS.c Measure all four angles of the trapezium.d Name all pairs of equal angles.

8 Construct a rhombus with sides of 6 cm and an included angle of 50.

6 cmB A

C

3 cm

6 cmU T

R

4 cm

S2 cm

60

3 cm

4 cm

4 cm

i ii

3 cm

3 cm

25 mm

25 mm

3 cmP Q

SR5 cm

4 cm4 cm

1 cm

Exercise 9-07

Example 9


9 a Construct this quadrilateral. b What type of quadrilateral is JKLM?

10 Construct this kite.

11 Construct the quadrilateral ABCD where AB BC, AB = 7 cm, BC = 3 cm, DC = 5 cm and AD = 4.5 cm.

12 Construct the trapezium DEFG where DE II GF, DE = 6 cm, EF = GF = 3 cm and F = 135.

5.5 cmK100

J

M

3 cm

4 cm L80

Z X

Y

W

40

6 cm

4 cm

Example 10

1 Name each of the following polygons and state whether it is convex or non-convex.

2 Explain the difference between parallel lines, perpendicular lines and skew lines.3 Use the denitions of the quadrilaterals on page 297 to help you answer these questions.

a Is the square a special type of rhombus?b Is the rhombus a special type of square?c Is the parallelogram a special type of trapezium?d Is the rectangle a special type of parallelogram?e Is the parallelogram a special type of kite?f Is the rectangle a special type of square?

a b c

d e f

Power plus


4 a What additional property makes a parallelogram into a rectangle? b What makes a kite into a rhombus?c What makes a rectangle a square?

5 How many diagonals has:a a quadrilateral? b an octagon?c a dodecagon?

6 Name all the quadrilaterals whose diagonals:a bisect each other at right angles b bisect each otherc intersect at right angles d have equal lengthe bisect the angles of the quadrilateral f are equal and bisect each other.

7 a Draw an angle of any size, ABC.b Using only a ruler and compasses, construct a rhombus

from ABC, with one vertex at B.c Bisect the angle ABC by drawing one diagonal of the

rhombus.

8 Name the most general quadrilateral in which:a opposite angles are equal b diagonals intersect at 90c diagonals are equal d all angles are 90e opposite sides are parallel f diagonals bisect each other.

9 a Construct a regular hexagon inside a circle of radius 5 cm. b Construct a regular octagon inside a circle of radius 7 cm.

10 a Draw a line, AB, and a point, X, above it.

b Using only a ruler and compasses, construct a line through X parallel to AB by creating a rhombus with one vertex at X and two vertices on AB.

11 a Construct an interval AB and mark its midpoint M. b Construct another interval CD perpendicular to AB through M, so that M is also the

midpoint of CD.c Join the ends of both intervals to make a quadrilateral. What type of quadrilateral have

you constructed?

12 a Repeat Question 11 but make sure that CD is the same length as AB.b What type of quadrilateral have you constructed?

B C

A

X

AB


Topic overview How useful do you think this chapter will be to you in the future? Can you name any jobs which use some of the concepts covered in this chapter? Did you have any problems with any sections of this chapter? Discuss any problems with

a friend or your teacher.

Language of mathsacute-angled bisect compasses constructconvex decagon diagonal equilateralincluded angle interval isosceles kiteline symmetry obtuse-angled octagon orderparallel parallelogram perpendicular polygonprotractor quadrilateral rectangle regular polygonrhombus right-angled rotational symmetry scaleneset square square trapezium vertex/vertices1 Draw a non-convex hexagon.2 What is the difference between a line and an interval?3 The word isosceles comes from Greece. Use a dictionary to nd out what it

means in Greek.4 What word in geometry means to cut in half?5 What is a set square and what is it used for?6 What is the more common name for a regular quadrilateral?

Worksheet 9-09

Geometryfind-a-word

GEOMETRIC FIGURES

Quadrilaterals

Polygons

Naming geometric

figures

Constructing figures

TrianglesA ______ O______ R ______ E ______S ______ I ______


Chapter 9 Review Topic testChapter 9

1 What type of polygon has 10 sides?

2 Name a shape that is not a polygon.

3 Draw:a a regular pentagon b a non-regular pentagonc a convex quadrilateral d a non-convex quadrilateral.

4 Classify these triangles, by sides and angles.

5 a Classify FGH by sides and angles.b Which angles in FGH are equal?

6 a Name a pair of parallel sides in this gure.b Name a pair of perpendicular sides.c What type of quadrilateral is ABCD?

7 Construct the following triangles.a

b PQR with P = 20, PR = 3 cm and PQ = 4 cm.c MNO with MN = 4 cm, NO = 5 cm and OM = 6 cm.

8 a Draw an obtuse-angled triangle, XYZ, and label its sides x, y and z.b What is the relationship between the triangles longest side and its largest angle?

Ex 9-01

Ex 9-01

Ex 9-01

Ex 9-02

a b c

d e f

Ex 9-02 F

H G

5 cm 4 cm

4 cm

Ex 9-03

C

B

D

A

Ex 9-04

A

C B

406 cm

40

Ex 9-04


9 Name each of the following polygons:

10 a Copy each shape in Question 9 and mark all the axes of symmetry.b List the shapes in Question 9 that have rotational symmetry, and state the order

of rotational symmetry of each one.

11 What is the denition of a rhombus?

12 Write two properties of a parallelogram.

13 What polygon am I? (There may be more than one answer.)a I have three sides and all of my angles are equal.b I am a quadrilateral with opposite sides parallel.c I have ve sides.d I have four sides and my diagonals bisect each other.e I am a quadrilateral with one pair of parallel sides.f I have three sides. My angles are 60, 80 and 40.

14 Copy this diagram and use a set square and ruler to construct a line, through P:a perpendicular to QRb parallel to QR.

15 Construct this parallelogram.

16 Construct this quadrilateral.

Ex 9-05a b c

d e f

Ex 9-05

Ex 9-05

Ex 9-05

Ex 9-05

Ex 9-06Q

R

P

6 cm

4 cm

80

Ex 9-07

P5 cm

3 cm

Q

NM

55

4 cm

4 cm

Ex 9-07

Student textImprint pageTable of contentsPrefaceHow to use this bookHow to use the CD-ROMAcknowledgementsSyllabus reference grid1 The history of numbersDifferent number systemsThe HinduArabic number systemPlace valueExpanded notationThe four operationsArithmagonsDividing by a two-digit numberOrder of operationsThe symbols of mathematicsTopic overviewChapter review

2 AnglesNaming anglesComparing angle sizeThe protractorDrawing anglesAngle geometryNaming linesAngles and parallel linesFinding parallel linesTopic overviewChapter review

3 Exploring numbersSpecial number patternsTests for divisibilityFactorsPrime and composite numbersPrime factorsIndex notationSquares, cubes and rootsTopic overviewChapter review

Mixed revision 14 SolidsNaming solidsConvex and non-convex solidsPolyhedraPrisms and pyramidsCylinders, cones and spheresClassifying solidsEulers ruleEdges of a solidThe Platonic solidsDrawing and building solidsDifferent views of solidsTopic overviewChapter review

5 IntegersNumber linesNumbers above and below zeroDirected numbersOrdering directed numbersAdding and subtracting integersMultiplying integersDividing integersThe four operations with integersReading a map gridThe number planeThe number plane with negative numbersTopic overviewChapter review

6 Patterns and rulesNumber rules from geometric patternsUsing pattern rulesThe language of algebraTables of valuesFinding the ruleFinding harder rulesFinding rules for geometric patternsAlgebraic abbreviationsSubstitutionSubstitution with negative numbersTopic overviewChapter review

Mixed revision 27 DecimalsPlace valueUnderstanding the pointOrdering decimalsDecimals are special fractionsAdding and subtracting decimalsMultiplying and dividing by powers of 10Multiplying decimalsCalculating changeDividing decimalsDecimals at workConverting common fractions to decimalsRecurring decimalsRounding decimalsMore decimals at workTopic overviewChapter review

8 Length and areaThe history of measurementThe metric systemConverting units of lengthReading measurement scalesThe accuracy of measuring instrumentsEstimating and measuring lengthPerimeterAreaConverting units of areaArea of squares, rectangles and trianglesAreas of composite shapesMeasuring large areasTopic overviewChapter review

9 Geometric figuresPolygonsClassifying trianglesNaming geometric figuresConstructing trianglesClassifying quadrilateralsConstructing perpendicular and parallel linesConstructing quadrilateralsTopic overviewChapter review

Mixed revision 310 FractionsHighest common factor and lowest common multipleNaming fractionsEquivalent fractionsOrdering fractionsAdding and subtracting fractionsAdding and subtracting mixed numeralsFractions of quantitiesMultiplying fractionsDividing fractionsTopic overviewChapter review

11 Volume, mass and timeVolumeVolume of a rectangular prismCapacity and liquid measureMassTimelinesConverting units of timeTime calculationsWorld standard timesTimetablesTopic overviewChapter review

12 AlgebraAlgebraic expressionsAlgebraic abbreviationsFrom words to algebraic expressionsLike termsMultiplying algebraic termsExpanding an expressionExpanding and simplifyingAlgebraic substitutionTopic overviewChapter review

13 Interpreting graphs and tablesPicture graphsColumn graphs and divided bar graphsSector graphsLine graphsTravel graphs and conversion graphsStep graphsReading tablesTopic overviewChapter review

Mixed revision 4General revisionAnswersIndex

GlossaryABCDEFG HI JK LMNOPQRSTU VW X Y Z

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Using technology menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 13

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