chapter 7 diagnostic test - nelson students have difficulty with the questions in the diagnostic...

Chapter 7 Diagnostic Test | 429

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Chapter 7 Diagnostic Test STUDENT BOOK PAGES 370–419

1. Express each ratio in simplest form.

a) 12 : 42 b) 18

5.4 c) 20 : –8 d) 6345

2. Solve each proportion.

a) x4

1512

= b) 32

45=

x c) x

5.2212

5.7= d)

2.156.98.4

=x

3. Determine the length of side a in each diagram. a) b)

4. As a shortcut to school, Jason walks across a rectangular field along the diagonal, instead of walking along two adjacent sides of the field. The dimensions of the field are 90 m by 120 m. How much shorter is Jason’s shortcut?

5. Determine the value of x in each diagram. a) c)

b) d)

430 | Principles of Mathematics 10: Chapter 7 Diagnostic Test Answers

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Chapter 7 Diagnostic Test Answers

1. a) 2 : 7

b) 41

c) 5 : –2

d) 75

2. a) 5 b) 30 c) 36 d) 7.6

3. a) about 8.2 m b) about 8.8 cm

4. 60 m

5. a) 30º b) 150º c) 123º d) 36º

If students have difficulty with the questions in the Diagnostic Test, it may be necessary to review the following topics: • solving proportions • applying the Pythagorean theorem to determine side lengths • using properties of angles in a triangle and angles formed by parallel lines and transversals

to determine angle measures

Lesson 7.1 Extra Practice | 431

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Lesson 7.1 Extra Practice STUDENT BOOK PAGES 374–381

1. i) For each pair of triangles, determine whether the triangles are congruent, similar, or neither.

ii) If the triangles are congruent, identify the corresponding angles and sides that are equal. If the triangles are similar, identify the corresponding angles that are equal, and calculate the scale factor that relates the smaller triangle as a reduction of the larger triangle.

a)

b)

2. +ABC is similar to+PRQ. Determine the length of QR.

3. In+PQR, PQ = 8.0 cm, PS = 5.0 cm, and PT = 4.0 cm.

a) Which triangles are similar? How do you

know? b) Determine the length of PR.

4. +ABC is similar to+DEF. Determine the scale factor that relates the larger triangle as an enlargement of the smaller triangle.

5. Determine the value of each lower-case letter. a)

b)

432 | Principles of Mathematics 10: Lesson 7.1 Extra Practice Answers

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Lesson 7.1 Extra Practice Answers 1. a) i) similar ii) ,ABCDEF ∠=∠ ,BACEDF ∠=∠

;ACBDFE ∠=∠

scale factor: 32

===BCEF

ACDF

ABDE

b) i) congruent ii) ,PRQMON ∠=∠ ,RPQOMN ∠=∠

;PQRMNO ∠=∠ MN = PQ, MO = PR, NO = QR

2. 15 cm

3. a) +PST ~+PQR since ,PQRPST ∠=∠ ,PRQPTS ∠=∠ and QPRSPT ∠=∠

b) 6.4 cm

4. 23

5. a) b = 12 cm b) x = 4.5 cm, y = 3.3 cm


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1. A section of roof rafters is shown below.

Determine the value of x to one decimal place.

2. A bridge is going to be built across the river at ED. Determine the width of the river where the bridge will be built.

3. On a sunny day, the shadow of a tower is 28.5 m long. A pole near the tower is 2.3 m high and casts a shadow that is 1.8 m long. Determine the height of the tower.

4. In a lumberjack competition, two poles of different heights are used for a pole-climbing event. The wires from the tops of the poles to the ground are parallel. The shorter pole is 6.0 m tall, and its wire is secured 8.0 m from the base. If the wire to the top of the taller pole is 16.0 m long, determine the height of the taller pole.

5. John uses a mirror to determine the height of a building. He knows that the angle of elevation is equal to the angle of reflection when a light is reflected off a mirror. What is the height of the building?

6. To determine the length of a lake, TR, the measurements shown in the diagram were made. How long is the lake?

When calculating side lengths and angle measures, round your answers to the number of decimal places in the given information when the required degree of accuracy is not stated.


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Lesson 7.2 Extra Practice Answers 1. about 5 m

2. about 2.8 m

3. about 36.4 m

4. 9.6 m

5. about 16.7 m

6. about 435.2 m

Chapter 7 Mid-Chapter Review Extra Practice | 435

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Chapter 7 Mid-Chapter Review Extra Practice STUDENT BOOK PAGES 389–390

1. If two triangles are similar, under what conditions would they also be congruent?

2. +MNO and+PQR are similar. Write a proportion for the corresponding side lengths.

3. The triangles below are similar. Determine the scale factor that relates the smaller triangle as a reduction of the larger triangle.


b)

c)

5. +ABC ~+DEF, BC = 5.2 cm, AC = 3.4 cm, DE = 8.2 cm, and EF = 10.6 cm. Determine the lengths of AB and DF.

6. The shadow of an apartment building is 106.0 m long. A 1.0 m pole near the building casts a shadow that is 1.8 m long. Determine the height of the building.

7. Determine the length of the lake, DE.

8. A 5 m ladder rests against a vertical wall, with its base 3 m from the wall. Another ladder, 12 m long, is resting against the wall, parallel to the first ladder. How far up the wall does each ladder reach, to one decimal place?

436 | Principles of Mathematics 10: Chapter 7 Mid-Chapter Review Extra Practice Answers

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Chapter 7 Mid-Chapter Review Extra Practice Answers 1. If two triangles are similar and two corresponding

sides are equal, then the triangles are congruent.

2. Answers may vary, e.g.,

QRNO

PRMO

PQMN

==

3. 31

4. a) a = 4.5 cm, b = 4.0 cm b) c = 3.0 cm c) d = 5.0 cm, e = 7.5 cm

5. AB = 4.0 cm, DF = 6.9 cm

6. about 58.9 m

7. 72 m

8. 4.0 m, 9.6 m


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1. Determine each ratio to four decimal places.

a) sin P d) cos Q

b) cos P e) tan P c) tan Q f) sin Q

2. Solve for x, and express your answer to one decimal place.

a) sin 40º = 12x d) cos 60º =

34x

b) cos 47º = x

50 e) tan 15º = 43x

c) tan 65º = x

35 f) sin 60º = x

24

3. Determine the value of x. Then write the primary trigonometric ratios for θ. a)

b)

4. Determine the measure of θ to one decimal place. a)

b)

c)

5. Determine the measure of θ to one decimal place.

a) cos θ = 65 d) cos θ =

41

b) sin θ = 117 e) tan θ = 1

c) tan θ = 7

15 f) sin θ = 21


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Lesson 7.4 Extra Practice Answers

1. a) sin P = 5248 = 0.9231

b) cos P = 5220 = 0.3846

c) tan Q = 4820 = 0.4167

d) cos Q = 5248 = 0.9231

e) tan P = 2048 = 2.4000

f) sin Q = 5220 = 0.3846

2. a) 7.7 units b) 73.3 units c) 16.3 units d) 17.0 units e) 11.5 units f) 27.7 units

3. a) x = 39 cm

sin θ = 135

3915

=

cos θ = 1312

3936

=

tan θ = 125

3615

=

b) x = 40 cm

sin θ = 4140

cos θ = 419

tan θ = 940

4. a) 36.9º b) 46.4º c) 73.7º

5. a) 33.6º b) 39.5º c) 65.0º d) 75.5º e) 45.0º f) 30.0º


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1. For each triangle, i) state two trigonometric ratios that you could

use to determine x ii) determine x to the nearest unit a)

b)

2. Calculate the measures of A∠ and B∠ in each triangle using trigonometric ratios. Round your answers to the nearest degree. a)

b)

3. Solve each triangle. Round the measure of each angle to the nearest degree. Round the length of each side to one decimal place.

a)

b)

4. a) In+JKL, =∠J 31º, =∠K 90º, and k = 45.3 m. Calculate j.

b) In+RST, =∠R 80º, =∠T 90º, and s = 20.5 m. Calculate r.

5. a) In+ABC, =∠B 90º, b = 35.9 cm, and c = 16.2 cm. Calculate .C∠

b) In+DEF, =∠F 90º, d = 9.3 cm, and e = 13.2 cm. Calculate .E∠

6. A 10.5 m ladder is leaning against a wall. The foot of the ladder is 2.6 m from the wall. Determine the angle between the ladder and the ground.


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Lesson 7.5 Extra Practice Answers

1. a) i) sin 60º = 15x , cos 30º =

15x

ii) x = 13 cm

b) i) tan 42º = 28x , tan 48º =

x28

ii) x = 25 cm

2. a) =∠A 52º, =∠B 38º b) =∠A 37º, =∠B 53º

3. a) =∠B 31º, a = 38.3 cm, c = 44.7 cm b) =∠D 58º, =∠F 32º, e = 40.8 cm

4. a) about 23.3 m b) about 116.3 m

5. a) about 27º b) about 55º

6. about 76º


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1. A runner estimates that the slope of a steep hill makes an angle of 50º with the ground. If the hill is 60 m high, what distance will the runner have to travel to get to the top of the hill?

2. Using sonar, a trawler captain detects a school of fish at a depth of 65.5 m. The angle of depression of the sounding is 18º. How far will the trawler have to travel to be directly above the school of fish?

3. A 25 m cellphone tower is supported by wires on opposite sides. The wires are anchored to the ground at a distance of 16 m from the foot of the tower. What is the angle of inclination that each wire makes with the ground?

4. The support for a shelf makes an angle of 48º with the wall. If the shelf is 32 cm wide, what is the length of the support? Round your answer to the nearest tenth of a centimetre.

5. A tree casts a shadow that is 20.0 m long when the angle of elevation of the Sun is 34º. Determine the height of the tree to one decimal place.

6. From the top of a building that is 55 m tall, the angle of depression to a car on the road is 35º. How far is the car from the base of the building?

7. An observer, who is 1.8 m tall, estimates that the angle of elevation of a cliff is 60º. The observer is 42.6 m from the base of the cliff. Determine the height of the cliff.

8. An equilateral triangle has side lengths of 29 cm. Determine the area of the triangle.

9. A satellite dish is mounted on the top of a building that is 100.0 m tall. The angle of elevation from the satellite dish to the top of a second building is 43º. The angle of depression to the base of the second building is 54º. How tall is the second building?

10. Determine the angle between the line 154

+= xy

and the x-axis, to the nearest degree.

11. A truck drives 15.9 km up a road, until it has gone 2.1 km vertically. If the road has a steady incline, what is the angle of elevation of the road to the nearest tenth of a degree?


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Lesson 7.6 Extra Practice Answers 1. about 78 m

2. about 201.6 m

3. about 57º

4. 43.1 cm

5. 13.5 m

6. about 79 m

7. about 75.6 m

8. about 363 cm2

9. about 167.8 m

10. 39º

11. 7.6º

Chapter 7 Review Extra Practice | 443

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Chapter 7 Review Extra Practice STUDENT BOOK PAGES 415–417

1. Determine whether the triangles in the diagram are similar. If they are similar, determine the scale factor that relates the smaller triangle as a reduction of the larger triangle.


b)

3. A 4.2 m ladder leans against a vertical wall, with its foot 1.2 m from the wall. A 5.6 m ladder leans against the wall, parallel to the first ladder. How far is the base of the second ladder from the wall?

4. On a sunny day, a tree casts a shadow that is 28.0 m long. Andrew, who is 1.9 m tall, is standing near the tree and casts a shadow that is 3.5 m long. What is the height of the tree?

5. In each triangle, i) determine the unknown side length ii) state the three primary trigonometric ratios

for A∠ iii) determine the measure of A∠ a) b)

6. Determine the value of x to one decimal place.

a) sin 25º = 6.15

x b) tan 65º = x

7.41

7. a) In+ABC, =∠B 90º, b = 12 cm, and c = 8 cm. Determine the measure of A∠ .

b) In+DEF, =∠E 90º, =∠F 35º, and d = 17.6 cm. Determine the length of side f.

8. Solve this triangle.

9. A tree casts a shadow that is 25.0 m long when the angle of elevation of the Sun is 36º. Determine the height of the tree to one decimal place.

10. A truck driver estimates that a country road rises 40 cm every 6 m along the road. What is the angle of elevation of the road? Round your answer to the nearest tenth of a degree.

444 | Principles of Mathematics 10: Chapter 7 Review Extra Practice Answers

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Chapter 7 Review Extra Practice Answers 1. +ABC ~+EDC; ,EA ∠=∠ ;DB ∠=∠

scale factor: 61

2. a) x = 24 cm, y = 12 cm b) x = 6.6 cm, y = 85˚

3. 1.6 m

4. 15.2 m

5. a) i) 15 cm

ii) sin A = 1715 , cos A =

178 , tan A =

815

iii) =∠A 62˚ b) i) 25 cm

ii) sin A = 2524 , cos A =

257 , tan A =

724

iii) =∠A 74˚

6. a) 6.6 b) 19.4

7. a) about 48º b) about 12.3 cm

8. θ = 18º, a = 3.9 cm, b = 12.6 cm

9. 18.2 m

10. 3.8º

chapter 7 diagnostic test - nelson students have difficulty with the questions in the diagnostic...

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