geometry and trig test thursday 1/25 - mr. kleckner's...

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Geometry and Trig Test

Thursday 1/25

Warm-up

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8 5 3) 3

2 1 1

2 4 6) 2

3 0 3

a m

b m

a

b

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Line AB contains the points A(1, 5) and B(-2, 8) Line DC contains the points D(2, 4) and C(-1, 7)

Are the two lines parallel, perpendicular, or neither?

8 5 31

2 1 3

7 4 31

1 2 3

AB

DC

m

m

parallel

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7 7 0) ,

12 1 11

3 7 10) ,

1 1 0

5 7 2)

2 1 1

a m horizontal x axis

b m vertical y axis

c m neither

Find the gradient of any line perpendicular to the line passing through the points A(5,10) and B(6,16)

16 10 66

6 5 1

1

6perp

m

m

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Line L1 passes through the points P(1,4) and Q(-3,a). a) Find an expression for the gradient of L1 in terms of a.

Line L1 is perpendicular to line L2. The gradient of L2 is 3.

b) Write down the gradient of L1 c) Find the value of a.

4 4) 1

3 1 4 4

1)

3

1 4)

3 4

44

3

16

3

a a aa m or

b

ac

a

a

1 1

1 1

Point-gradient form of a line with a gradient of

through the point ( , )

( )

m x y

y y m x x

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1 1( )

7 5( 1)

7 5 5

5 2

y y m x x

y x

y x

y x

1 1

1 ( 4) 5) 5

1 2 1

) ( )

1 5( 1)

1 5 5

5 6

a m

b y y m x x

y x

y x

y x

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2 5 3)

1 ( 3) 4

32 ( 1)

4

4 8 3( 1)

4 8 3 3

3 4 11 0

)3(5 / 3) 4 11 0

5 4 11 0

4 6 0

3

2

a m

y x

y x

y x

x y

b t

t

t

t

(2 1) 4 0

2 1 4 0

3 3 0

3 3

1

2(1) 1

3

(1,3)

x x

x x

x

x

x

y

y

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Assignment pg. 90 #1

pg. 91 #3,4

pg. 93 #4,5

pg. 94 #2,4,7

pg. 97 #3,4,6,7ab

pg. 99 #7

pg. 102 #2acd,3


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Warm-up

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The Rule of Pythagoras

2 2 2a b c

2 2 2

2

2

7 8

49 64

15

15

x

x

x

x


a2 + b2 = c2

2 22

2

2

3 2

3 2

5

5

x

x

x

x m

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a2 + b2 = c2

22 2

2 2

2

2

15 (2 ) (3 )

15 4 9

15 5

3

3

x x

x x

x

x

x


a2 + b2 = c2

5cm

y

2 2 2

2

22 2

2

2

5 6

11

11

11 5

11 25

14

14

y

y

y

x

x

x

x

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89000 m 12000 m

2 2 2

2 2 2

2

120000 89000

89000 12000

7777000000

88.2

x

x

x

x km

Right Triangles in other shapes

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12cm

13cm

2 2 2

2

2

12 13

25

5

(12 )(5 )

60

x

x

x cm

Area cm cm

Area cm


3cm

3cm

4cm

4cm

2 2 2

2

3 4

25

5

x

x

x cm

x

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Converse of the Rule of Pythagoras

right triangle a2 + b2 = c2

acute triangle a2 + b2 > c2

obtuse triangle a2 + b2 < c2

If you know the sides of a triangle

you can use the Rule of Pythagoras

to classify the triangle as right,

acute, or obtuse

Converse of the Rule of Pythagoras

2 2 24 5 ?6

16 24?36

40 36

acute

2 2 26 8 ?10

36 64?100

100 100

right

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True Bearings

True bearings measure the direction of travel by

comparing it with the true north direction. The

degree measurement is always taken in the

clockwise direction.

True Bearings

368km 472km

90o

2 2 2

2

472 368

358208

599

x

x

x km

A

C x

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Assignment on the website


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Warm-up

Warm-up

2 2 2420 297

514.4

297 297 420 420 514.4

1948.4

x

x mm

total

total mm

2 2 23.2 7.1

7.79

AB

AB m

2 2 28 ? 4 5

64 41

not a right triangle

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a2 + b2 = c2

2 2 2

2

90 73

13429

116

l

l

l mm

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2 2 2

2

8 10

36

6

r

r

r cm

810

An ice cream cone is 8cm tall and its slant height is 10cm. Find the radius of the circle at the top of the cone.

2 2 2

2

22 2

2

10 10

200

200

10 200

300

17.3

x

x

x

d

d

x m

x10

10

10

A cubic iron grid has sides of length 10m. Find the length of a diagonal brace of the cube.

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2 2 2

2

22 2

2

1 2

5

5

3 5

14

3.74

x

x

x

AB

AB

AB cm

x

A rectangular box has the dimensions shown. B is an LED at the centre of the top face. Find the direct distance from A to B.

xy

2 2 2

2

22 2

2

2.8 5.2

34.88

34.88

27.7 34.88

802.17

28.3

x

x

x

y

y

y m

A pipe runs diagonally across a roof 2.8m high, 27.7m long, and 10.4m wide. How long is the pipe?

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2 2 2

2

14.1 25

426.19

20.6

h

h

h m

20

25

A symmetrical square-based pyramid has base edges 20m and slant edges 25m. Find the height of the pyramid.

Assignment on the website

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Thursday 1/25

Warm-up

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Warm-up

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Right Triangle Trig

There are 3 trig ratios that you use to find the sides and angles of right

triangles

Find the exact value of sin and cos S S

28sin

53

45cos

53

S

S

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Find the value of x

0sin 3520

0.573620

20(0.5736)

11.5

x

x

x

x

0 10sin 36

100.5879

10

0.5789

17.3

x

x

x

x

0 10tan 25

100.4663

10

0.4663

21.4

x

x

x

x

Find the value of x

1

5sin

14

sin 0.3571

sin (0.3571)

20.9o

x

x

x

x

1

8tan

5

tan 1.6

tan (1.6)

58.0o

x

x

x

x

1

9cos

13

cos 0.6923

cos (0.6923)

46.2o

x

x

x

x

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Find the value of x

2cos 61

20.4848

2

0.4848

4.13

o

x

x

x

x cm

sin 6510

9.06

9.06sin 48

12.2

o

o

y

y

x

x cm

2 2

Properties all parallelograms have

If a quadrilateral is a parallelogram the following properties always exist in the figure

1) Opposite sides are parallel

2) Opposite sides are congruent

3) Opposite angles are congruent

4) Diagonals bisect each other

5) Consecutive angles are supplementary

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Properties of Special parallelograms

All of the special parallelograms have all of the properties normal parallelograms have but they

also have additional properties as well.

Rhombus -All sides are congruent

- diagonals are perpendicular - diagonals bisect the opposite angles

Rectangle -All angles are congruent

- diagonals bisect each other

Square -All properties of rhombus, rectangle and

parallelogram

A rhombus has diagonals of length 12cm and 7cm. Find the larger angle of the rhombus.

1

6tan

3.5

tan 1.714

tan (1.714)

59.7

Rhombus angle 2(59.7) 119.4

o

o

The diagonals of a rhombus bisect each

other, are perpendicular with each other, and they

bisect the angles of a rhombus

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2 2

1

) 12 4 12.6

3) tan

12.6

3tan

12.6

13.4o

a DF cm

b

Angles of Elevation and

Depression

An angle of elevation is an angle measured above the horizontal and an angle of

depression is an angle measured below the horizontal

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Angles of Elevation and Depression

tan 56.553

53tan 56.5

80.1

Full height includes your eye level above the ground

Full height 80.1 5.5 85.6

o

o

h

ft

h

h ft

ft

Assignment pg. 106 #3

pg. 108 #1-6

pg. 109 #2,4

pg. 112 #4ab,6

pg. 117 #2,4,6

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Thursday 1/25

Warm-up

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Warm-up

cos6432

32cos64 14.0

o x

x

1

132sin

229

132sin 35.2

229

o

o

x

x

90 54 36

17tan 54

1712.4

tan 54

17sin 54

1721.0

sin 54

o

o

o

x

x

y

y

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Law of Sines

In any triangle you can use the law of sines to find a side length or an angle measure if you know:

1) 2 side lengths and a non included angle 2) 1 side length and 2 angle measures

Law of Sines

12

sin110 sin 40

12

0.9397 0.6428

12(0.9397)17.5

0.6428

o o

x

x

x

1

18 12

sin sin 38

18 12

sin 0.6157

sin .92355

sin .92355 67.5

12

sin 74.5 sin 38

12

0.9636 0.6157

12(0.9636)18.8

0.6157

o o

o

o

o

o o

x

x

x

x

y

y

y

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P

QR

25o

60o

15km

x

15

sin 25 sin 60

15

0.4226 0.8660

15(0.4226)7.32

0.8660

o o

x

x

x km

Law of Cosines

In any triangle you can use the law of cosines to find a side length or an angle measure if you know:

1) 2 side lengths and an included angle 2) 3 side lengths

a2 = b2 + c2 – 2(b)(c)(cos A) b2 = a2 + c2 – 2(a)(c)(cos B) c2 = a2 + b2 – 2(a)(b)(cos C)

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The Cosine Rule

2 2 2

2

2

27 23 2(27)(23)cos36

729 529 1004.799

253.2

15.9

ox

x

x

x

Y

X

Z30km

25km

18km

2 2 2

1

25 18 30ˆcos2(25)(18)

49ˆcos900

49ˆ cos 86.9900

o

Z

Z

Z

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Area of a Triangle with Trig

If you know the length of 2 sides and the measure of the included angle you can use the following formula to find the area of a triangle

Area of the triangle = ½(a)(b)sin C

Area of the triangle = ½(8)(4.5)sin 99o

Area of a Triangle with Trig

Find the area of each triangle

A = ½(8)(3)sin 50o A = 12(.7660) A = 9.2 cm2

A = ½(20)(9)sin 65o A = 90(.9063) A = 81.6 ft2

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2

2

2

2

1) ( )( )sin

2

1( )sin

2

1)4 ( )sin 30

2

8 ( )(.5)

16

4

o

a Area x x B

Area x B

b x

x

x

x cm

Assignment pg. 121 #1ab,3,5ab,8

pg. 123 #1ab,2ab,4,8,9

pg. 125 #1,4

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