Chapter 7 Diagnostic Test
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1 Chapter 7 Diagnostic Test STUDENT BOOK PAGES Epress each ratio in simplest form : c) 20 : 8 d) Solve each proportion = = 3 c) = d) = Determine the length of side a in each diagram. 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 in each diagram. c) d) Chapter 7 Diagnostic Test 429
2 Chapter 7 Diagnostic Test Answers 1. 2 : c) 5 : 2 d) c) 36 d) about 8.2 m about 8.8 cm m 5. 30º 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 430 Principles of Mathematics 10: Chapter 7 Diagnostic Test Answers
3 Lesson 7.1 Etra Practice STUDENT BOOK PAGES 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. 3. In+PQR, PQ = 8.0 cm, PS = 5.0 cm, and PT = 4.0 cm. Which triangles are similar? How do you know? 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. 2. +ABC is similar to+prq. Determine the length of QR. Lesson 7.1 Etra Practice 431
4 Lesson 7.1 Etra Practice Answers 1. i) similar ii) DEF = ABC, EDF = BAC, DFE = ACB; DE DF EF 2 scale factor: = = = AB AC BC 3 i) congruent ii) MON = PRQ, OMN = RPQ, MNO = PQR; MN = PQ, MO = PR, NO = QR cm 3. +PST ~+PQR since PST = PQR, PTS = PRQ, and SPT = QPR 6.4 cm b = 12 cm = 4.5 cm, y = 3.3 cm 432 Principles of Mathematics 10: Lesson 7.1 Etra Practice Answers
5 Lesson 7.2 Etra Practice STUDENT BOOK PAGES 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. 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? 1. A section of roof rafters is shown below. Determine the value of 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. 6. To determine the length of a lake, TR, the measurements shown in the diagram were made. How long is the lake? 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. Lesson 7.2 Etra Practice 433
6 Lesson 7.2 Etra Practice Answers 1. about 5 m 2. about 2.8 m 3. about 36.4 m m 5. about 16.7 m 6. about m 434 Principles of Mathematics 10: Lesson 7.2 Etra Practice Answers
7 Chapter 7 Mid-Chapter Review Etra Practice STUDENT BOOK PAGES 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. 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 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. 4. Determine the value of each lower-case letter. 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? c) Chapter 7 Mid-Chapter Review Etra Practice 435
8 Chapter 7 Mid-Chapter Review Etra 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., MN MO NO = = PQ PR QR a = 4.5 cm, b = 4.0 cm 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 m m, 9.6 m 436 Principles of Mathematics 10: Chapter 7 Mid-Chapter Review Etra Practice Answers
9 Lesson 7.4 Etra Practice STUDENT BOOK PAGES Determine each ratio to four decimal places. 4. Determine the measure of θ to one decimal place. sin P d) cos Q cos P e) tan P c) tan Q f) sin Q 2. Solve for, and epress your answer to one decimal place. sin 40º = 12 d) cos 60º = cos 47º = 35 c) tan 65º = e) tan 15º = f) sin 60º = c) 3. Determine the value of. Then write the primary trigonometric ratios for θ. 5. Determine the measure of θ to one decimal place. 5 1 cos θ = d) cos θ = 6 4 sin θ = 11 7 e) tan θ = 1 15 c) tan θ = 7 f) sin θ = 2 1 Lesson 7.4 Etra Practice 437
10 Lesson 7.4 Etra Practice Answers sin P = = cos P = = c) tan Q = = d) cos Q = = e) tan P = = º 46.4º c) 73.7º º 39.5º c) 65.0º d) 75.5º e) 45.0º f) 30.0º 20 f) sin Q = = units 73.3 units c) 16.3 units d) 17.0 units e) 11.5 units f) 27.7 units 3. = 39 cm 15 5 sin θ = = cos θ = = tan θ = = = 40 cm 40 sin θ = 41 9 cos θ = tan θ = Principles of Mathematics 10: Lesson 7.4 Etra Practice Answers
11 Lesson 7.5 Etra Practice STUDENT BOOK PAGES For each triangle, i) state two trigonometric ratios that you could use to determine ii) determine to the nearest unit 3. Solve each triangle. Round the measure of each angle to the nearest degree. Round the length of each side to one decimal place. 2. Calculate the measures of A and B in each triangle using trigonometric ratios. Round your answers to the nearest degree. 4. In+JKL, J = 31º, K = 90º, and k = 45.3 m. Calculate j. In+RST, R = 80º, T = 90º, and s = 20.5 m. Calculate r. 5. In+ABC, B = 90º, b = 35.9 cm, and c = 16.2 cm. Calculate C. 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. Lesson 7.5 Etra Practice 439
12 Lesson 7.5 Etra Practice Answers 1. i) sin 60º = 15, cos 30º = 15 ii) = 13 cm 28 i) tan 42º =, tan 48º = 28 ii) = 25 cm 2. A = 52º, B = 38º A = 37º, B = 53º 3. B = 31º, a = 38.3 cm, c = 44.7 cm D = 58º, F = 32º, e = 40.8 cm 4. about 23.3 m about m 5. about 27º about 55º 6. about 76º 440 Principles of Mathematics 10: Lesson 7.5 Etra Practice Answers
13 Lesson 7.6 Etra Practice STUDENT BOOK PAGES 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. 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 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? Determine the angle between the line y = and the -ais, 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? 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. Lesson 7.6 Etra Practice 441
14 Lesson 7.6 Etra Practice Answers 1. about 78 m 2. about m 3. about 57º cm m 6. about 79 m 7. about 75.6 m 8. about 363 cm 2 9. about m º º 442 Principles of Mathematics 10: Lesson 7.6 Etra Practice Answers
15 Chapter 7 Review Etra Practice STUDENT BOOK PAGES 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. 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 2. Determine the value of each lower-case letter. 6. Determine the value of to one decimal place sin 25º = tan 65º = In+ABC, B = 90º, b = 12 cm, and c = 8 cm. Determine the measure of A. In+DEF, E = 90º, F = 35º, and d = 17.6 cm. Determine the length of side f. 8. Solve this triangle. 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? 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. Chapter 7 Review Etra Practice 443
16 Chapter 7 Review Etra Practice Answers 1. +ABC ~+EDC; A = E, B = D; 1 scale factor: 6 2. = 24 cm, y = 12 cm = 6.6 cm, y = m m 5. i) 15 cm ii) sin A =, cos A =, tan A = iii) A = 62 i) 25 cm ii) sin A =, cos A =, tan A = iii) A = about 48º about 12.3 cm 8. θ = 18º, a = 3.9 cm, b = 12.6 cm m º 444 Principles of Mathematics 10: Chapter 7 Review Etra Practice Answers
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