UNIT 7 – SIMILAR TRIANGLES AND TRIGONOMETRY

Date Lesson § TOPIC Homework

Dec. 3 7.1 7.1

Congruence and Similarity in Triangles Pg. 378 # 1, 4 – 8, 12

Dec. 4 7.2 7.2

Solving Similar Triangle Problems

Pg. 386 # 2 - 12

Dec. 5 7.3 7.3

Exploring Similar Right Triangles

Pg. 393 # 1 - 4

Dec. 6 OPT Mid- Chapter Review

QUIZ (7.1 – 7.2)

Pg. 390 # 1 - 10

Dec. 7 7.4 7.4

The Primary Trigonometric Ratios

Pg. 398 # 2, 3, 5, 7 - 13

Dec.

10 7.5 7.5

Solving Right Triangles

Pg. 403 # 1 – 4, 7, 8a, 9 – 11, 13ac, 14

Dec.

11 7.6 7.6

Solving Right Triangle Problems

Pg. 412 # 1 – 6, 10, 12, 14

Dec.

12 7.7 7.6

Solving Right Triangle Problems

Two-step Problems

QUIZ (7.4 – 7.5)

Pg. 413 # 11, 13, 15 – 17, 20

Dec.

13/14 7.8

Review for Unit 7 Test

Pg. 416 # 1 – 16

Plus Review WS 7.8

Dec.

17 7.9

TEST- UNIT 7

MPM 2D Lesson 7.1 Congruence and Similarity in Triangles

A

D

E F

B C

P

X

Y Z

Q R

Using a ruler and protractor measure each of the following very carefully. Measure sides to the nearest mm

and angles to the nearest degree.

AB = A = DE = D =

AC = B = DF = E =

BC = C = EF = F =

PQ = P = XY = X =

PR = Q = XZ = Y =

QR= R = YZ = Z =

When comparing ABC and PQR, what do you notice about the lengths of the sides and the measure

of the angles?

For ABC and DEF,

AB corresponds to side in DEF. A corresponds to in DEF.

AC corresponds to side in DEF. B corresponds to in DEF.

BC corresponds to side in DEF. C corresponds to in DEF.

For ABC and DEF, complete the following.

DE

AB A = D =

DF

AC B = E =

EF

BC C = F =

What do you notice about the ratios of the corresponding sides?

What do you notices about the corresponding angles?

Triangles when the ratios of the lengths of the corresponding sides are and the

measures of the corresponding angles are the triangles are called ________________

The scale factor or scale ratio is the measure of the amount of enlargement or reduction from one similar

triangle to the other. The scale factor is the ratio of any 2 corresponding sides of similar triangles.

Ex. 1 Explain why one of the triangle below is similar to ABC and the other is not.

X

A P

8 10 4 5 6 8

B 3 C

Y 6 Z Q 5.3 R

Properties of Similar Triangles

If ABC XYZ and the scale factor is XY

ABn then:

the length of any side or altitude of ABC = n(length of corresponding side or altitude of XYZ)

the perimeter of ABC = n(perimeter of XYZ)

the area of ABC = n2(area of XYZ)

To prove that 2 triangles are similar.

Angle-Angle Similarity (AA) –- Two corresponding pairs of angles share the same measure.

Side-Side -Side Similarity (SSS) -- Three corresponding pairs of sides have a common ratio

Side-Angle -Side Similarity (SSS) -- Two corresponding pairs of sides have a common ratio and the

contained angles share the same measure.

To prove that 2 triangles are congruent show that they are exactly the same.

Side-Side -Side Congruity (SSS) – Two triangles are congruent if all three sides have equal

measures.

Side-Angle -Side Congruity (SAS) – Two triangles are congruent if two sides and the contained angle

have the same measures.

Angle-Side-Angle Congruity (ASA) - Two triangles are congruent if two angles and the contained

side have the same measures.

Hypotenuse-Side (HS) - Two triangles are congruent if their hypotenuses and one of the other

sides have the same measure.

Hypotenuse-Angle (HA) - Two right triangles are congruent if their hypotenuses and one of the

acute angles have the same measure

Finally, if two triangles are congruent they are also similar, but two similar triangles are not

necessarily congruent.

Ex. 2 Show that the two triangles in this diagram are similar, then find the values of x and y.

Pg. 378 # 1, 4 – 8, 12

MPM 2D Lesson 7.2 Solving Similar Triangle Problems

Ex. 1 How tall is the tree below?

3 m

11 m

2 m

Ex. 2 Jay stands on level ground and looks at the mirror on the ground that is 2 m from his feet. He can see

the top of a flag pole that is 7 m from the mirror. If his eyes are 1.72 m from the ground, how tall is

the flag pole?

Pg. 386 # 2 - 12

MPM 2D Lesson 7.3 Exploring Similar Right Triangles

G

1. Use the triangles below to complete the tables on the next page.

E

C

A B D F

2. Use the triangles below to complete the tables on the next page.

W

X

T

P Q R S

Complete the tables below for each of the given similar triangles.

1- Give all answers correct to 3 decimal places.

Triangle hypotenuse

oppositesin

hypotenuse

adjacentcos

adjacent

oppositetan

ABC

AC

BC

AC

AB

AB

BC

ADE

AE

DE

AE

AD

AD

DE

AFG

AG

FG

AG

AF

AF

FG

____________sin

____________cos ____________tan

2- Give all answers correct to 3 decimal places.

Triangle hypotenuse

oppositesin

hypotenuse

adjacentcos

adjacent

oppositetan

PQT

PT

QT

PT

PQ

PQ

QT

PRX

PX

RX

PX

PR

PR

RX

PSW

PW

SW

PW

PS

PS

SW

____________sin

____________cos ____________tan

The Primary Trigonometric Ratios can only be used for right triangles.

Trig ratios are simply the ratio of the sides of a right angled triangle.

Each trig ratio represents the ratio of two different sides.

For the triangle below and , For the triangle below and ,

O

p

p

o hypotenuse hypotenuse adjacent

s

i

t

e

adjacent opposite

The side that is labelled opposite and the side labelled adjacent depends on which angle is being used.

Unless told otherwise always find the length of a side to 1 decimal place and the measure of an angle

to the nearest degree.

TOAadjacent

oppositeTANGENT

CAHhypotenuse

adjacentCOSINE

SOHhypotenuse

oppositeSINE

tan:

cos:

sin:

Ex. 1 Find the value of x, correct to 1 decimal place. Ex. 2 Find A to the nearest degree.

A A

12

x 21 5

B C

32

B C

Ex. 3 Solve each of the following triangles. (ie: find all missing sides and angles)

a) P b) 4

Y Z

46

12

X

Q R

5

Pg. 393 # 1 - 4

15

8.4 cm

x

x

8.2 km

42

12.1

cm

8.2

cm

C

BA

MPM 2D Lesson 7.4 The Primary Trigonometric Ratios

Ex. 1 Determine length x in each triangle. Round your answer to

one decimal place.

a)

b)

Ex. 2 Determine the measure of A , to the nearest degree.

Ex. 3 A hot-air balloon on the end of a taut 95 m rope rises from its platform. Sam, who is in the basket, estimates

that the angle of depression to the rope is about 50 .

a) How far, to the nearest metre, did the balloon drift horizontally?

b) How high, to the nearest metre, is the balloon above ground?

c) Viewed from the platform, what is the angle of elevation, to nearest degree,

Ex. 4 A wheelchair ramp is safe to use if it has a minimum angle of 4.8 and a maximum angleof 11.3.

What are the minimum and maximum slopes of such a ramp? Round your answers to 2 decimal places.

Pg. 398 # 2, 3, 5, 7 - 13

MPM 2D Lesson 7.5 Solving Right Triangles

Ex. 1 Solve the following triangles.

a) b)

Ex. 2 During its approach to Earth, the space shuttle’s glide angle changes. When the shuttle’s altitude is

about 15.7 miles, its horizontal distance to the runway is about 59 miles.

a) What is its glide angle? Round your answer to the nearest tenth of a degree.

When you are asked to solve a triangle, you are being asked to

find all of the unknown angle measures and side lengths.

b) When the space shuttle is 5 miles from the runway, its glide angle is about 19. Find the shuttle’s

altitude at this point in its descent. Round your answer to the nearest tenth.

Ex. 3 During a flight, a hot air balloon is observed by two persons standing at points A and B as illustrated in

the diagram. The angle of elevation of point A is 28. Point A is 1.8 miles from the balloon as measured

along the ground. Round answers to the nearest tenth.

a) What is the height, h, of the balloon?

h

B A

b) Point B is 2.4 miles from point A. Find the angle of elevation of point B.

Pg. 403 # 1 – 4, 7, 8a, 9 – 11,

13ac, 14

MPM 2D Lesson 7.6 Solving Right Triangle Problems

Ex. 1 A carpenter leans a 4.3 m ladder up against a wall. If it reaches 3.8 m up the wall,

determine, to the nearest degree, the angle the ladder makes with the wall.

Ex. 2 A missile is launched at an angle of elevation of 80°. If it travels in a straight line, what is its altitude,

correct to 1 decimal place, when it hits the training drone 15 km down range?

Ex. 3 Catalina’s parents have a house with a triangular front lawn as shown. They want to cover the lawn with

sod. How much would it cost to put sod in, if it costs $13.75 per square metre?

Pg. 412 # 1 – 6, 10, 12, 14

MPM 2D Lesson 7.7 Solving Right Triangle Problems – Two-Step Problems

Ex. 1 Jon is standing on a 40 m high seaside cliff flying a kite. The angle of depression of the kite string is 38.

If the kite string is 320.0 m long, how far above the water is the kite?

Ex. 2 Find the value of x.

25.0 m

49 33

x

Ex. 3 From the bridge of The Maid of the Mist on the Niagara River, the angle of elevation to the top of

Niagara Falls is 64. The angle of depression to the bottom of the falls is 6. If the bridge of the boat

is 2.8 m above the water, calculate the height of the falls, correct to one decimal place.

Ex. 4 Find the value of h, correct to one decimal place.

50.0 m

x y

h

50 70

Pg. 413 # 11, 13, 15 – 17, 20