Name: ___________________________________

Date: _______________________________ …BLM 7–1...

Great North American Trigonometry Race Map

Principles of Mathematics 10: Teacher’s Resource

BLM 7–1 Great North American Trigonometry Race Map

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–2... (page 1)

Get Ready Angle Properties 1. Find each unknown angle. a)

Pythagorean Theorem 3. Find the unknown side length in each triangle. Round your answers to the nearest tenth of a unit. a)

b) b)

c)

c)

d) d)

2. Prove that the opposite angles of a pair of intersecting lines are equal.

Principles of Mathematics 10: Teacher’s Resource

BLM 7–2 Get Ready

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–2... (page 2)

Slope 4. Find the slope of each ramp. Express your answers as fractions in lowest terms. a)

b)

b) Equivalent Ratios 5. Solve each proportion. x 4 a) = 5 7 4 3 b) = y 4 m 5 c) = 5 m x +1 2 d) = x 3

c)

Transformations 6. Identify the type of transformation in each case. a) d)

Principles of Mathematics 10: Teacher’s Resource

BLM 7–2 Get Ready

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Name: ___________________________________

Shed Drawing

Principles of Mathematics 10: Teacher’s Resource

BLM 7–4 Shed Drawing

Date: _______________________________ …BLM 7–4...

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Truss Bridge

Principles of Mathematics 10: Teacher’s Resource

BLM 7–5 Truss Bridge

Date: _______________________________ …BLM 7–5...

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________

Section 7.1 Practice Master 1. Name the similar triangles in each case. Write the letters so that equal angles appear in corresponding order. a)

…BLM 7–6... (page 1)

2. Name a pair of similar triangles in each diagram and explain why they are similar. a)

b) b)

c) c)

d)

Principles of Mathematics 10: Teacher’s Resource

BLM 7–6 Section 7.1 Practice Master

3. Name a pair of similar triangles in each and explain why they are similar. a)

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–6... (page 2)

b)

4. For each pair of similar triangles in question 3, list all the pairs of corresponding angles and corresponding sides. 5. a) Draw an isosceles triangle. b) Draw an isosceles triangle that is • congruent to the one you drew • similar to the one you drew • neither congruent nor similar to the one you drew

Principles of Mathematics 10: Teacher’s Resource

BLM 7–6 Section 7.1 Practice Master

6. Are all scalene triangles similar? Justify your answer. 7. Taha wants to enlarge a 4 in. by 6 in. photo so that the width and length are in proportion. a) Find the dimensions of the photo if the width and length are doubled. b) Find the dimensions of the photo if the width and length are tripled. c) Find the dimensions of the photo if the width and length are halved. d) How are the areas of the different sizes of photos related?

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–7...

Making a Clinometer In this activity, you will build a clinometer, a device that will allow you to measure angles and calculate the heights of various objects. Materials • • • • • •

protractor drinking straw a semi-circle of cardboard tape string weight (e.g., several paper clips)

Procedure 1. Draw a baseline along the bottom edge of the cardboard, and mark the centre. 2. Use a protractor to mark the cardboard into degrees from 90° to 0° to 90° with zero at the bottom of the curve (see diagram). Make sure that the centre of the protractor meets the centre of the cardboard. 3. Tape the straw along the straight edge (top) of the semicircle. 4. Tape the string to the centre of the straight edge of the semicircle. Attach a weight to the string.

Principles of Mathematics 10: Teacher’s Resource

BLM 7–7 Making a Clinometer

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________

Section 7.2 Practice Master 1. A right triangle has side lengths 5 cm, 12 cm, and 13 cm. a) Draw the triangle. b) A similar triangle has a hypotenuse 52 cm long. What is the scale factor? c) What are the lengths of the legs of the triangle in part b)? d) Draw the similar triangle. 2. Refer to question 1. a) Find the area of each triangle. b) How are these areas related? c) How do the areas help to confirm that the triangles are similar?

…BLM 7–8... (page 1)

c)

d)

3. The triangles in each pair are similar. Find the unknown side lengths. a)

b)

Principles of Mathematics 10: Teacher’s Resource

BLM 7–8 Section 7.2 Practice Master

4. Find x. a)

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–8... (page 2)

b)

b) UDEF ∼ UXYZ. Find the area of UDEF.

6. Use the given measures to find the width of the canal.

5. a) UMNO ∼ USTU. Find the area of UMNO.

Principles of Mathematics 10: Teacher’s Resource

BLM 7–8 Section 7.2 Practice Master

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________

Section 7.3 Practice Master 1. Find the tangent of the angle indicated, to four decimal places. a)

b)

c)

d)

…BLM 7–10... (page 1)

4. Find the measure of each angle, to the nearest degree. a) tan θ = 0.7145 b) tan C = 0.4163 c) tan D = 2.7143 d) tan M = 1.7500 9 e) tan θ = 14 10 f) tan L = 7 5. Find the measures of both acute angles in each triangle, to the nearest degree. a)

b)

6. Find the length of the unknown side, to the nearest tenth of a unit. a) 2. Refer to question 1. Find the tangent of the other acute angle, to four decimal places. 3. Evaluate with a calculator. Round your answers to four decimal places. a) tan 38° b) tan 23° c) tan 6° d) tan 30° e) tan 57.4° f) tan 82.7°

Principles of Mathematics 10: Teacher’s Resource

BLM 7–10 Section 7.3 Practice Master

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–10... (page 2)

b)

7. Find the value of x, to the nearest tenth of a metre. a)

Principles of Mathematics 10: Teacher’s Resource

BLM 7–10 Section 7.3 Practice Master

b)

8. In order to measure the height of a tree, Dan calculated that its shadow is 12 m long and that the line joining the top of the tree to the tip of the shadow forms an angle of 52° with the flat ground. a) Draw a diagram to illustrate this problem. b) Find the height of the tree, to the nearest tenth of a metre.

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Overlapping Triangles

Principles of Mathematics 10: Teacher’s Resource

BLM 7–11 Overlapping Triangles

Date: _______________________________ …BLM 7–11...

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________

Section 7.4 Practice Master 1. Find sin θ, cos θ, and tan θ for each triangle, expressed as fractions in lowest terms. a)

…BLM 7–12... (page 1)

2. Find the three primary trigonometric ratios for ∠A, to four decimal places. a)

b) b)

c) 3. Evaluate with a calculator. Round your answers to four decimal places. a) sin 72° b) sin 16° c) sin 64° d) sin 23° d)

Principles of Mathematics 10: Teacher’s Resource

BLM 7–12 Section 7.4 Practice Master

4. Evaluate with a calculator. Round your answers to four decimal places. a) cos 42° b) cos 85° c) cos 14° d) cos 36°

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–12... (page 2)

5. Find the measure of each angle, to the nearest degree. a) sin θ = 0.5189 b) sin Q = 0.8476 3 c) sin θ = 8 9 d) sin R = 11 6. Find the measure of each angle, to the nearest degree. a) cos θ = 0.7258 b) cos W = 0.3194 5 c) cos θ = 9 9 d) cos B = 10

b) Find the value of x, to the nearest tenth of a centimetre, by applying the cosine ratio.

8. Solve each triangle. Round side lengths to the nearest tenth of a metre. a)

7. a) Find the value of x, to the nearest tenth of a metre, by applying the sine ratio.

b) In UPQR, ∠P = 34°, ∠Q = 90°, and q = 20 m.

Principles of Mathematics 10: Teacher’s Resource

BLM 7–12 Section 7.4 Practice Master

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________

Section 7.5 Practice Master 1. A flagpole is secured with a guy wire, as shown. The guy wire makes an angle of 65° with the ground and is secured 4 m from the bottom of the flagpole. a) Find the height of the flagpole, to the nearest tenth of a metre. b) Find the length of the guy wire, to the nearest tenth of a metre.

…BLM 7–14... (page 1)

3. Zidane and Shemique are looking up at their school from the playing field at the back of the school. From Zidane’s point of view, the top of the school is at an angle of elevation of 43°. From Shemique’s point of view, directly closer to the school, it is 62°. The school is 27 m high. How far apart are Zidane and Shemique? Round your answer to the nearest tenth of a metre. 4. At the top of a hiking trail, there are two vertical posts. One is 5 m tall, and the other is 7 m tall. The ground between the posts is level, and the bases of the posts are 4 m apart. The posts are connected by two straight wires.

2. Aimee and Russell are facing each other on opposite sides of an 8-m telephone pole. From Aimee’s point of view, the top of the telephone pole is at an angle of elevation of 52°. From Russell’s point of view, the top of the telephone pole is at an angle of elevation of 38°. How far apart are Aimee and Russell? Round your answer to the nearest tenth of a metre.

Principles of Mathematics 10: Teacher’s Resource

BLM 7–14 Section 7.5 Practice Master

a) What angle does each wire make with the ground? Round your answer to the nearest degree. b) What is the length of each wire? Round your answer to the nearest tenth of a metre.

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–14... (page 2)

5. From the top of a building, the angle of elevation of the top of a nearby building is 28° and the angle of depression of the bottom of the nearby building is 48°. The distance between the two buildings is 50 m. What is the height of the taller building? Round your answer to the nearest metre.

Principles of Mathematics 10: Teacher’s Resource

BLM 7–14 Section 7.5 Practice Master

6. A square-based pyramid has a height of 182 m and a base length of 280 m. Find the angle, to the nearest degree, that one of the edges of the pyramid makes with the base. Round your answer to the nearest degree.

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–15... (page 1)

Chapter 7 Review 7.1 Investigate Properties of Similar Triangles 1. a) Draw two triangles that are similar. b) Draw two hexagons that are congruent. 2. Name the two similar triangles and explain why they are similar. a)

7.2 Use Similar Triangles to Solve Problems 3. The pairs of triangles are similar. Find the unknown side lengths. a)

b)

b)

4. The tips of the shadows of a flagpole and a 1.5-m fence post meet at the point S. The following lengths are measured: ST = 2.7 m and QT = 7.4 m. Use this information to find the height of the flagpole. Round your answer to the nearest tenth of a metre.

5. Nimo has constructed a deck in the shape of an equilateral triangle with each side length equal to 2 m. If she enlarges her deck to a similar shape whose side lengths are doubled, what will the area of the new deck be?

Principles of Mathematics 10: Teacher’s Resource

BLM 7–15 Chapter 7 Review

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–15... (page 2)

7.3 The Tangent Ratio

b)

6. Find the tangent of ∠A, to four decimal places. a)

b)

9. The angle of elevation of a ramp is 4°. The horizontal length of the ramp is 18 m. What is the vertical height of the ramp, to the nearest tenth of a metre? 7.4 The Sine and Cosine Ratios 10. Find sin θ, cos θ, and tan θ for each triangle, expressed as fractions in lowest terms. a)

7. Find the measure of each angle, to the nearest degree. a) tan θ = 0.8173 b) tan E = 1.5413 13 c) tan θ = 18 23 d) tan B = 12 8. Find x, to the nearest tenth of a metre. a)

Principles of Mathematics 10: Teacher’s Resource

BLM 7–15 Chapter 7 Review

b)

11. Find the measure of each angle, to the nearest degree. a) sin θ = 0.4152 b) sin T = 0.8731 11 c) cos θ = 15 3 d) cos S = 8

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Date: _______________________________ …BLM 7–15... (page 3)

12. Find x, to the nearest tenth of a metre. a)

b)

13. Solve UPQR. Round angles to the nearest degree.

15. Find the length of x, to the nearest tenth of a centimetre.

16. The Carziz Tunnel cuts through Mount Mainet. At the start of the tunnel, the angle of elevation of the top of Mount Mainet is 38°. At the end of the tunnel, the angle of elevation of the top of Mount Mainet is 42°. The height of Mount Mainet above the tunnel passage is 584 m. How long is the Carziz tunnel through Mount Mainet? Round your answer to the nearest metre.

7.5 Solve Problems Involving Right Triangles 14. Solve each triangle. Round side lengths to the nearest tenth of a unit. a)

b)

Principles of Mathematics 10: Teacher’s Resource

BLM 7–15 Chapter 7 Review

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Chapter 7 Practice Test 1. sin 35°, to four decimal places, is A 0.8192 B 0.7002 C 0.5736 D −0.4282 2. If tan θ = 0.5512, then θ, to the nearest degree, is A 33° B 57° C 55° D 29°

Date: _______________________________ …BLM 7–17...

5. Sketch a pair of similar figures for each. a) triangle b) kite 6. Sketch a pair of congruent figures for each. a) rectangle b) parallelogram 7. In UPQR, find x, to the nearest tenth of a metre.

3. In UABC, x, to the nearest tenth of a metre, equals

8. Solve UPQR. Round side lengths to the nearest tenth of a kilometre. A B C D

10.1 cm 14.9 cm 12.1 cm 10.6 cm

4. In UDEF, ∠D, to the nearest degree, is

9. From a point 10 m from the base of a building, the angle of elevation of the top of the building is 54°. Find the height of the building, to the nearest metre.

A B C D

44° 46° 35° 55°

Principles of Mathematics 10: Teacher’s Resource

BLM 7–17 Chapter 3 Practice Test

Copyright © 2007 McGraw-Hill Ryerson Limited

Name: ___________________________________

Chapter 7 Test 1. The value of cos 35°, to four decimal places, is A 0.8192 B 0.5736 C 1.4281 D 0.2212 2. If tan θ = 0.6385, then θ, to the nearest degree, is A 33° B 40° C 50° D 64°

Date: _______________________________ …BLM 7–18...

5. Sketch a pair of similar figures for each. a) parallelogram b) square 6. Sketch a pair of congruent figures for each. a) triangle b) pentagon 7. In USTU, find x, to the nearest tenth of a metre.

3. In UDEF, x, to the nearest tenth of a metre, is

8. Solve UXYZ. Round side lengths to the nearest tenth of a metre. A B C D

14.4 cm 19.1 cm 17.3 cm 16.6 cm

4. In UXYZ, ∠X, to the nearest degree, is

A B C D

58° 32° 38° 52°

Principles of Mathematics 10: Teacher’s Resource

BLM 7–18 Chapter 7 Test

9. From a point 25 m from the bottom of a maple tree, the angle of elevation to the top of the maple tree is 21°. Find the height of the maple tree, to the nearest metre.

Copyright © 2007 McGraw-Hill Ryerson Limited

…BLM 7–20... (page 1)

BLM Answers Get Ready 1. a) a = 55°, b = 70°, c = 70°, d = 110° b) m = 90°, n = 45°, p = 27° c) m = 115°, x = 45°, y = 65°, z = 70°, w = 70° d) d = 60°, e = 70°, f = 50° 2. Answers may vary. For example:

4.

5. 6. 7.

3. 4. 5. 6.

Since a and b make up a straight angle, a + b = 180° c Since b and c make up a straight angle, b + c = 180° d Subtract equation d from equation c to get a − c = 0, or a = c. Since c and d make up a straight angle, c + d = 180° e Subtract equation e from equation d to get b − d = 0, or b = d. Since a = c and b = d, opposite angles are equal. a) 6.7 m b) 8.1 cm c) 8.4 mm d) 7.1 km 5 1 a) b) 12 9 20 16 a) x = b) y = 7 3 c) m = −5 or 5 d) x = −3 or 2 a) translation b) rotation c) reflection d) dilatation

Section 7.1 Practice Master 1. Order of vertices may vary. For example: a) UDEF ∼ UPQR b) UJKL ∼ USTU c) USRQ ∼ USTU d) UEAB ∼ UECD 2. Answers may vary. For example: a) UMNQ ∼ UOPQ because ∠NMQ = ∠POQ = 90° and ∠MQN = ∠OQP = 90° (common angles). b) UPQR ∼ UTSR because ∠PQR = ∠TSR (alternate angles) and ∠QPR = ∠STR (alternate angles). c) UACE ∼ UBCD because ∠AEC = ∠BDC and ∠ACE = ∠BCD (common angles). 3. Answers may vary. For example: a) UABC ∼ UDEF because the ratios of corresponding sides are all equal to 2:1. Principles of Mathematics 10: Teacher’s Resource

Chapter 7 Practice Masters Answers

b) Using the Pythagorean theorem, MC ∼ 18.0 cm. Then, UMCD ∼ UKMC because the ratios of two pairs of corresponding sides are equal to 2:1. a) ∠BAC = ∠EDF, ∠ACB = ∠DFE, ∠CBA = ∠FED; BA:ED = AC:DF = CB:FE b) ∠MCD = ∠KMC, ∠CDM = ∠MCK, ∠DMC = ∠CKM; MC:KM = CD:MC = DM:CK Answers will vary. Answers may vary. For example: No. The three angles in a scalene triangle may not equal the three angles in another scalene triangle. a) width 8 in. length 12 in. b) width 12 in. length 18 in. c) width 2 in. length 3 in. d) Answers may vary. For example: The area of each enlarged or reduced photo equals the area of the original photo multiplied by the square of the scale factor.

Section 7.2 Practice Master 1. a)

b) 4 d)

c) 20 cm, 48 cm

2. a) area of first triangle 30 cm2; area of second triangle 480 cm2 b) The area of the second triangle equals the area of the first triangle times 16. c) The area factor is equal to the square of the scale factor. 3. a) d = 12 cm, e = 10 cm b) x = 9 mm, z = 15 mm c) j = 12 m, n = 15 m d) p = 10 km, r = 12 km 4. a) 6 cm b) 4.5 m 5. a) 81 cm2 b) 100 m2 6. 21 m

Copyright © 2007 McGraw-Hill Ryerson Limited

…BLM 7–20... (page 2)

Section 7.3 Practice Master

Chapter 7 Review

1. a) 0.4286 b) 0.5714 c) 0.8125 d) 0.8947 2. a) 2.3333 b) 1.7500 c) 1.2308 d) 1.1176 3. a) 0.7813 b) 0.4245 c) 0.1051 d) 0.5774 e) 1.5637 f) 7.8062 4. a) θ = 36° b) ∠C = 23° c) ∠D = 70° d) ∠M = 60° e) θ = 33° f) ∠L = 55° 5. a) ∠P = 52°, ∠Q = 38° b) ∠T = 58°, ∠U = 32° 6. a) 2.9 cm b) 9.1 mm 7. a) 7.3 m b) 20.9 m 8. a)

1. Answers will vary. 2. a) UHFG ∼ UHKJ because ∠HFG = ∠HKJ (alternate angles) and ∠HGF = ∠HJK (alternate angles). b) URQP ∼ URST because the ratios of corresponding sides are all equal to 2:1. 3. a) x = 8 cm, c = 10 cm b) f = 7 cm, g = 16 cm 4. 5.6 m 5. 4 3 m2 6. a) 0.6364 b) 1.0435 7. a) θ = 39° b) ∠E = 57° c) θ = 36° d) ∠B = 62° 8. a) 19.6 m b) 15.7 cm 9. 1.3 m 7 24 7 , cos θ = , tan θ = 10. a) sin θ = 25 25 24 3 4 3 b) sin θ = , cos θ = , tan θ = 5 5 4 11. a) θ = 25° b) ∠T = 61° c) θ = 43° d) ∠S = 68° 12. a) 19.7 cm b) 8.0 m 13. p = 7.5 m, ∠Q = 37°, ∠R = 53° 14. a) ∠B = 63°, a = 14.5 cm, b = 28.5 cm b) ∠P = 48°, y = 25.2 m, t = 37.7 m 15. 7.2 m 16. 1396 m

b) 15.4 m

Section 7.4 Practice Master 3 4 3 , cos θ = , tan θ = 5 5 4 5 12 5 b) sin θ = , cos θ = , tan θ = 13 13 12 15 25 3 c) sin θ = , cos θ = , tan θ = 29 29 5 25 5 5 d) sin θ = , cos θ = , tan θ = 32 8 4 a) sin A = 0.7657, cos A = 0.6400, tan A = 1.1964 b) sin A = 0.5000, cos A = 0. 8662, tan A = 0.5772 a) 0.9511 b) 0.2756 c) 0.8988 d) 0.3907 a) 0.7431 b) 0.0872 c) 0.9703 d) 0.8090 a) θ = 31° b) ∠Q = 58° c) θ = 22° d) ∠R = 55° a) θ = 43° b) ∠W = 71° c) θ = 56° d) ∠B = 26° a) 10.3 m b) 11.0 cm a) ∠A = 55°, a = 12.3 m, b = 8.6 m b) ∠R = 56°, p = 11.2 m, r = 16.6 m

1. a) sin θ =

2. 3. 4. 5. 6. 7. 8.

Section 7.5 Practice Master 1. 2. 3. 4. 5. 6.

a) 8.6 m 16.5 m 14.6 m a) 51°, 60° 82 m 43°

b) 9.5 m

Chapter 3 Practice Test 1. 2. 3. 4. 5. 6. 7. 8. 9.

C D A C Answers will vary. Answers will vary. 17.3 m ∠P = 50°, p = 16.7 km, r = 21.8 km 14 m

Chapter 7 Test 1. 2. 3. 4. 5. 6. 7. 8. 9.

A A D A Answers will vary. Answers will vary. 28.4 m ∠Y = 40°, y = 15.1 m, z = 23.5 m 10 m

b) 6.4 m, 8.1 m

Principles of Mathematics 10: Teacher’s Resource

Chapter 7 Practice Masters Answers

Copyright © 2007 McGraw-Hill Ryerson Limited