Unit 6: Triangle Geometry

Transcription

1 Unit 6: Triangle Geometry Student Tracking Sheet Math 9 Principles Name: lock: What I can do for this unit: fter Practice fter Review How I id 6-1 I can recognize similar triangles using the ngle Test, the Side Ratio Test, and the Side-ngle-Side test. 6-2 I can solve for missing lengths in similar triangles using proportion equations. 6-3 I can use the Pythagorean Theorem to solve for missing sides in right angled triangles. 6-4 I can evaluate the sine, cosine, and tangent ratio in right angled triangle and use this to find missing sides and angles. (Remember to identify the opposite, adjacent, and hypotenuse) 6-5 I can determine the area of a triangle using the formula = sin. ode Value escription N Not Yet Meeting pectations I just don t get it. MM Minimally Meeting pectations arely got it, I need some prompting to help solve the question. M Meeting pectations Got it, I understand the concept without help or prompting. ceeding pectations Wow, nailed it! I can use this concept to solve problems I may have not seen in practice. I also get little details that may not be directly related to this target correct.

2 Unit 6: Triangle Geometry ay 1 Math 9 Principles 6-1 I can recognize similar triangles using the ngle Test, the Side Ratio Test, and the Side- ngle-side test. Make the correct statement of similarity in each, unless they are not similar. F Y X Z 21 G In Δ, 38, 42. In Δ, 42, 102. In Δ, 53, 19. In Δ, 108, 53.

3 ompute the corresponding side ratios and make the statement of similarity or state not similar. orresponding Side Ratios 22.5 F Similarity: 64 F 76.8 orresponding Side Ratios 32 Similarity: In, = 11, = 16.5, = 5.5 In F, = 12, F = 24, F = 36 orresponding Side Ratios Similarity: In, = 9.5, = 18.4, = 14.2 In F, = 12, F = 24, F = 36 orresponding Side Ratios Similarity:

4 orresponding Side Ratios Similarity: F 6.9 F orresponding Side Ratios Similarity: orresponding Side Ratios F Similarity: In, = 9.4, = 4.8, = 18.2 In F, = 31.85, F = 16.45, F = 8.4 orresponding Side Ratios Similarity:

5 etermine if similarity eists. If so, write the similarity statement. F Similarity: orresponding Side Ratios F 9 orresponding Side Ratios Similarity: orresponding Side Ratios F Similarity: F orresponding Side Ratios Similarity:

6 Unit 6: Triangle Geometry ay 2 Math 9 Principles 6-2 I can solve for missing lengths in similar triangles using proportion equations. 1) For each problem, write the statement of similarity, the corresponding side ratios, and solve for each unknown using a proportion equation. Show all work. 8 4 F l Similarity: Proportion quation: (Solve for l) 12 2) 6.5 l 11 Similarity: Proportion quation: (Solve for l) 13 3) Similarity: 12 l Proportion quation: 9 (Solve for l) 12 4) l Similarity: Proportion quation: (Solve for l) 16

7 5) 6 14 Similarity: Proportion q n for l 1 : l 2 8 l 1 18 Proportion q n for l 2 : 6) G Similarity 1: ~ Prop q n l 1 : 24 l 2 l 1 Similarity 2: ~ Prop q n l 2 : F ) To find the width of the river River l Similarity: Prop q n for l: 8) is similar to PQR. Find : 11 R P 4 6 Q 9) TN is similar to QS. Find : Q 10 T 12 7 S N

8 10) Find the value of and y in each. State each similarity y F 11) P 10 y Q 20 R 3 S 24 T 12) y

9 Unit 6: Geometry ay 3 Math 9 Principles 6-3 I can use the Pythagorean Theorem to solve for missing sides in right angled triangles. Solve for the indicated missing side lengths. Use Sum of Squares or ifference of Squares. Round to one decimal place where necessary. Show your work. 1) 2) 3) ) 5) 6) ) 8) 9)

10 Solve for the indicated side length using the 2 version of the Pythagorean formula. 10) 11) 12) etermine whether or not these are right angled triangles. (show all work) 13) 14) 63 15) Solve using the Pythagorean Theorem. Sketch each equation yourself, label the dimensions, and solve. 16) screen s size is usually stated in terms of its diagonal length. ssuming the screen is square, find its size if it is stated as 180 cm. 17) From point, travel 10 km east then travel 3 km south, turn west and travel 12 km. How far are you from point? 18) If the distance between bases in baseball is 90 ft, how far is it from home plate directly to second base?

11 19) 20 foot ramp rises to a doorway that is 3 feet off the ground. How far away from the building is the ramp? 20) Find the side length of the largest square that can fit inside a circle of diameter ) From point, travel 12 km south and 10 km west. Then travel 5 km north. How far are you from point? 22) rectangle is 40 units long and 15 units wide. Find the length of its diagonal. 23) Find the side length of the largest square you can fit inside a circle with radius ) How high up a wall does a 16 ft. ladder reach if the bottom is 6 ft. from the base of the wall? 25) ship travels 30 km west, then turns south and travels 15 km. How far is it from its original position? (Measured in a straight line) 26) If the hypotenuse is 34 and one other side of a right triangle is 16, find the length of the third side of the triangle.

12 Unit 6: Triangle Geometry ay 4 Math 9 Principles 6-4 I can evaluate the sine, cosine, and tangent ratio in right angled triangle and use this to find missing sides and angles. (Remember to identify the opposite, adjacent, and hypotenuse) omplete the following using the letters provided. The side opposite angle is: The side opposite angle is: z The side adjacent angle is: The side adjacent angle is: y tan = tan = Show the Tangent ratio for each, then find angle. tan = ngle = 4 9 tan = ngle = tan = ngle = tan = ngle =

13 Show the Tangent ratio of each angle and find its measure in degrees. tan = ngle = 6 tan = ngle = 17 tan = ngle = tan = ngle = tan = ngle = 41 9 tan = ngle = 40 tan = ngle = 18 8 tan = ngle = tan = ngle = tan = ngle = (Find side first) tan = ngle = 20 tan = ngle = 14 tan = ngle = 80 tan = ngle = (Find side first) 82

14 Use the Tangent ratio to calculate each indicated side length

15 Sketch a diagram and solve using the tangent ratio or arctangent as appropriate. (Remember to use arctangent when solving for an angle). ramp rises to a doorway 4 ft. off the ground. The bottom of the ramp is 18 ft. from the base of the building. Find the angle of elevation (bottom corner) of the ramp.

16 The top of a playground slide is 3.2 m high. The bottom of the slide is 4 m from the base of the ladder. Find the angle of elevation (bottom corner) of the slide. supporting wire, fastened 40 m from the base of a communications tower, makes an angle of 60 with the ground. How high up the tower does the wire reach? communications tower, on the sea coast, is 450 m high. From a ship at sea, the angle of elevation is 4. How far is the ship offshore? telephone pole is supported by a steel cable connected to the pole 9 metres up. The cable is fied into the ground, 5 metres from the base of the pole. Find the angle of elevation of the cable.

17 Unit 6: Triangle Geometry ay 5 Math 9 Principles 6-4 I can evaluate the sine, cosine, and tangent ratio in right angled triangle and use this to find missing sides and angles. Find angle in each triangle. (Round to the nearest degree) 1) 2) ) 4) ) 6)

18 Using the Sin ratio, calculate the missing length () in each. learly show your equation for each question. 7) 8) ) 10) ) 12)

19 13) 14) ) 9 47 reate a labeled diagram for each question and solve. 16) In order to create a coaster with a 70 incline that has a maimum height of 30 m, what length of track is necessary? ssume a straight track. 17) warehouse conveyer belt is 3.2 m long. If it can incline at a maimum angle of 38, what height above the ground can the top of the belt reach?

20 18) surveyor measures the angle of elevation between two points to be 8. If the distance, measured straight between those two points, is 1200 m, what is the change of elevation between those points? 19) The sun s rays create a shadow of a tall tree. The length of the shadow is 12 m. The angle of elevation of the sun is 78. alculate the height of the tree. 20) If a road with a 6 incline or angle of elevation rises 300 metres, how long is the road? 21) 2000 m stretch of road has a change of elevation of 500 m. What is the angle of elevation of the road? 22) conveyer belt is 4.8 m long. If it can incline at a maimum angle of 32, what height above the ground can it reach?

21 Unit 6: Triangle Geometry ay 6 Math 9 Principles 6-4 I can evaluate the sine, cosine, and tangent ratio in right angled triangle and use this to find missing sides and angles. 1) Find angle in each triangle using the arccosine function. (Round to the nearest degree) 2) ) 44 4) 36 Use the cosine ratio to find the indicated length () in each. 5) ) 32 7)

22 8) ) reate a labeled diagram for each question and solve using the cosine ratio. 10) ladder is 12 m long. If safety rgulations prohibit the ladder to be inclined against a building at an angle greater that 70, at least how far away should the ladder be from the base of the building? 11) Upon takeoff, an airplane maintains a constant angle climb of 18. If the flight travelled is 10km, what ground distance has been travelled? 12) Seen below is the side veiw of a roof truss. For the type of roofing that is to be installed, the angle of inclination for the roof should be 20. If the width of the house is 8m, calculate the length of the roof slope. 8 20

23 Identify the three trigonometric ratios. o not calculate angles. 13) Sin = Sin = os = os = Tan = Tan = Identify the trigonometric ratio. omplete with either Sin, os, Tan, or None. 15) 14) = = 17) 16) = ) 18) = = =

24 20) Use either arcsine, arccosine, or arctangent to determine the angle measured in each. (Round to the nearest degree) 21) ) 23) ) 13 25) Use either sin, cos, or tan to determine the indicated side in each. (Round to the nearest tenth) 26) 27) 30 8m 7m 50

25 28) 29) m 12cm 30) 31) 30 15m 40 4cm Sketch a right angled triangle for each, label, and solve using the appropriate trigonometric ratio. 32) ladder is 6m long. To what height of a building will the ladder reach when its angle of elevation is 75? 33) From a ship, the angle of elevation to the top of the communication tower on the shore is 16. If the tower is 150m high, how far offshore is the ship?

26 34) The width of a cabin is 8m. If the roof truss needs and angle of elevation of 60, calculate the length of the roof incline. 35) rectangle is 12 by 30. Its diagonal cuts each corner right angle into two angles. What are the angles? 36) n isosceles triangle has side lengths 9, 12, and 12. Find the measure of the base angles. 37) fter travelling 2300m along a road with an incline, the gain in elevation is 150m. Find the angle of elevation of the road. 38) n airplane takes off from a runway which is 5000m from the base of a hill which rises 800m almost straight up. If the plane is going to just clear the top of the hill, what should its minimum angle of elevation be?

27 Unit 6: Triangle Geometry ay 7 Math 9 Principles 6-5 I can determine the area of a triangle using the formula. Find the area for each triangle. 1) 2) 3) ) ) ) ) ) )

28 Solve each question. Round all answers to the nearest tenth. 10) Find the area of quadrilateral. Hint: ivide it into 2 triangles ) Find the area of an equilateral triangle with side lengths of ) 3. Find the area of a parallelogram with sides of length 32 and 15 and one corner angle of 115. HINT: Sketch it and divide it into 2 triangles. 13) n isosceles triangle has a base length of 15. Its other two sides are 24 units long. Its base angles are 72. Find the area.

29 14) Find the area of quadrilateral Find the length of the indicated side in each pair of similar triangles. 15) 16) 9 y y G 17) ) Y 16 X F

30 . Use the Pythagorean Theorem to calculate the length of the missing side. Round your answer to the nearest tenth. 19) 20) X 6 X 21) X 9 22) Find the length of the legs of the right, isosceles triangles shown. X 30 X 23) From point, travel 6 km east, turn south and travel 4 km, and then turn east and travel 10 km to point. Find the shortest distance between and. 24) From point, travel 10 km south, turn east and travel 14 km, and then turn north and travel 3 km to point. Find the shortest distance between and.

31 Unit 6: Triangle Geometry Review Math 9 Principles 1. In the following diagram, which is considered the adjacent side to angle? a c b 2. Using the Pythagorean Theorem, what is the length of side c? Using the Pythagorean Theorem, what is the length of side c? Given the two triangles below, write a similarity statement F 40 52

32 5. Is angle equal to 90? Is angle equal to 90? Given the two triangles below, write the similarity statement F 8. In the diagram below, write the tan ratio of angle. z y 9. In the diagram in #8, write the sin ratio of angle. 10. In the diagram in #8, write the cos ratio of angle. 11. In the diagram below, the degree measure of angle is approimately 27 23

33 12. In the diagram below, the degree measure of angle is approimately alculate the area of the following triangle: Use the Pythagorean Theorem to find the lengths. Round answer to one decimal place where necessary Find the length of the side of a square that fits inside a circle of radius 12 cm.

34 Triangle Similarity. In #18 - #20, find the indicated side () by matching up corresponding sides of the similar triangles Use the appropriate trigonometric ratios to determine the side marked () in each of the following questions. e sure to show all work and round answers to the nearest tenth

35 Use the appropriate trigonometric ratio to determine the angle labelled in each of the following diagrams. Round answers to the nearest degree Use the area formula to determine the area of the following triangle ladder is 6m in length and makes an angle of 25 with a house. How far from the base of the house is the ladder? Round your answer to 1 decimal place.

36 29. tree casts a shadow that is 8.5m long. If the angle of inclination to the sun is 35, how tall is the tree? 30. n airplane takes off at an angle of elevation of 12. If it continues at this angle until it reaches an altitude of 4000m, how far has the plane travelled in ground distance? nswer to the nearest km. 31. Find the area of a parallelogram with side lengths 12 and 25, and one angle of hill is a 1550m walk to the top. What is the height of the hill, assuming it has a constant angle of inclination of 8?

37 Unit 6: Triangle Geometry Key Math 9 Principles 6-1: 1) ~ GF 2) not similar 3) ~ 4) ~ 5) not similar 6) ~ F 7)not similar 8) ~ 9) ~ F 10)not similar 11) Not similar 12)not similar 13) ~ F 14) ~ F 15)not similar 16) ~ F 17) ~ F 18)not similar 19) ~ F 20)not similar 6-2: 1) ~ F, l = 6 2) ~, l = 5.5 3) ~, l = 16 4) ~, l = ) ~, l 1 = 7.2, l 2 = 15 6) ~ FG, ~ FG, l 1 = 6.4, l 2 = ) ~, 30 8) = ) ) ~ F, = 4.8, y = ) PQR~ PST, = 15, y = 2 12) ~, = 7.2, y = 7 6-3: 1) = 15.7 (7 5) 2) = 12.7 ( 161) 3) = 12.8 (2 41) 4) = 26.0 ( 674) 5) = 11.6 (3 15) 6) = 15.6 (2 61) 7) = 27.6 ( ) 8) = 13.4 (6 5) 9) = 29.7 ( 881) 10) = 14.8 ( 220.5) 11) = 25.5 (18 2) 12) = 33.9 (24 2) 13)no 14) yes 15) no 16) cm 17) 3.6 km 18) ft 19)19.8 ft 20) )12.2 km 22)42.7 units 23) )14.8 ft 25) 33.5 km 26) 30 27) ) ) 77.8 ft 30) 47.7 ft 31) ) ) ) ) ) : 1)y 2) 3) 4)y 5) y 6) y 10) = ) tan =, = 70.6, tan = )tan = )tan = ) tan = 14 20, = 60.3, tan = , = 24.0, tan = 8 20, = 35.0, tan = 14 7) = )11 = ) = , = 19.4, = )tan = 9 40, = ) tan = 20, = )tan = 18, = 12.7, tan = , = 67.4, tan = 48 18, = 77.3, tan = 80, = 77.3, = 22.6, = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) ) ) = 69.3 m 30) = m 31) : 1) 46 2) 26 3) 14 4) 25 5) 13 6)63 7)18.9 8)26.7 9) ) ) ) ) ) ) )31.9 m 17)2 m 18) 12.7 km 19) m 20) 7.5 ft 21)56.5 m 22)2870 m 23) ) 2.5 m 6-6: 1) 63 2) 66 3) 35 4) ) ) )12.3 8)16.8 9) )4.1 11)9.5 12)4.3 13) sin = , sin =, cos =, cos =, tan =, tan = 14)tan ) none 17) none 18) sin 19) cos 20)62 21) 25 22)31 23)46 24) 58 25)54 26)4.0 m 27) 8.3 m 28)19.9 cm 29) 70.7 m 30) 8.0 cm 31)19.6 m 32) 5.8 m 33)523.1 m 34)8 m 35)68, 22 36) 45 37)3.7 38) )sin

38 6-7: 1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = 4. 4, y = ) ) ) ) ) 16.5 km 24)15.7 km Unit 6 Review: 1)b 2)13 3)10.8 4) ~ F 5)yes 6)no 7) ~ F 8)tan = y 9)sin = 10) cos = y 11) = ) = ) ) ) 29.7 z z 16) ) )55 19) ) 5 21) )9.2 23) )42 25)39 26) 39 27) ) ) ) m 31) ) m