right triangles & trigonometry chapter...

Geometry – Rt. Triangle Trig ~1~ NJCTL.org

Right Triangles & Trigonometry Chapter Questions

1. What is the Pythagorean Theorem? What is the Pythagorean Theorem Converse?

2. How do you use the Pythagorean Theorem Converse to classify triangles by their angles?

3. Why does the altitude of a right triangle divide the triangle into two smaller triangles that are similar to the original triangle and to each other?

4. What are the similarities and differences between a geometric mean and an arithmetic

mean?

5. Explain the difference of the side lengths of a 45°-45°-90° triangle and a 30°-60°-90° triangle?

6. When do you use the sine, cosine or tangent trigonometric ratios to solve right triangles?

7. When do you use the inverse sine, cosine or tangent trigonometric ratios to solve right

triangles?

8. Explain why co-functions of complementary angles are equal (sine and cosine are co-functions).

9. What of types of triangles can be solved using the Law of Sines and

Law of Cosines? What information must be given to use the Law of Sines? What

information must be given to use the Law of Cosines?


Right Triangles & Trigonometry Chapter Problems

Pythagorean Theorem and its Converse Class work

1. In the triangle above, the hypotenuse is side ______, the legs are sides _____ and ______.

Refer the diagram below to answer questions 2-7. Write the answer in simplest radical form.

2. If AB = 9 and BC = 12, then AC = _________.

3. If AB = 11 and AC = 61, then BC = _________.

4. If BC = 45 and AC = 53, then AB = __________.

5. If AC = 9 and AB = 3, then BC = _________.

6. If BC = 14 and AC = 8√7, then AB = __________.

7. If AB = 2√3 and BC = 6√5, then AC = _________.

Classify the sides as lengths of an acute, right, obtuse, or not a triangle.

8. 8, 11, 5

9. 20, 29, 21

10. 6, 2, 2√2

11. 9, 11√2, 7√5

12. Find the perimeter and area of the rectangle.


Homework

13. In the triangle above the hypotenuse is side ______, the legs are sides _____ and ______.

Refer the diagram below to answer questions 14-19. Write the answer in simplest radical form.

14. If AB = 33 and BC = 56, then AC = _________.

15. If AB = 12 and AC = 37, then BC = _________.


17. If AC = 14 and AB = 5√2, then BC = _________.


19. If AB = 5√2 and BC = 7√6, then AC = _________.

Classify the sides as lengths of an acute, right, obtuse, or not a triangle.

20. 19, 7, 15

21. 9, 4, 3

22. 14, 14, 14√2

23. 3√11, 4√7, 5√6

24. Find the perimeter and area of the triangle.

29cm

31cm


Similarity in Right Triangles Classwork 25. Find the geometric mean of the pairs. Write the answer is simplest radical form.

a. 8 and 32 b. 2 and 24 c. 10 and 15 d. 14 and 18

26. GE is the altitude of triangle DEF. What are the three similar triangles?

Solve for x. 27.

28.

29.


30.

31.

32.

Homework 33. Find the geometric mean of the pairs. Write the answer is simplest radical form.

a. 9 and 25 b. 6 and 54 c. 12 and 20 d. 10 and 28

34. SQ is the altitude of triangle PQR. What are the three similar triangles?


Solve for x. 35.

36.

37.

38.

39.


40.

Special Right Triangles

Class work

41. The length of each leg of an isosceles right triangle is 13 cm. What is the length of the hypotenuse? 42. The length of the hypotenuse of a 45o-45o-90o triangle is 16 cm. What is the length of each leg? 43. The length of the shorter leg of a 30o-60o-90o triangle is 5 cm. What is the length of the longer leg? What is the length of the hypotenuse? 44. The length of the longer leg of a 30o-60o-90o triangle is 12 cm. What is the length of the hypotenuse? What is the length of the shorter leg? 45. The length of the hypotenuse of a 30o-60o-90o triangle is 12 cm. What is the length of the longer leg? What is the length of the shorter leg? Find the lengths of the missing sides.

46.

47.

y

x

18cm

60

y

x18cm60


48.

49.

50.

51. Find the area of the triangle. Round to the nearest hundredth.

52. A skateboarder constructs a ramp using plywood. The height of the ramp is 4 feet. If the ramp falls at a 60o angle, what is the length of the plywood?

Homework

53. The length of each leg of an isosceles right triangle is 9 cm. What is the length of the hypotenuse? 54. The length of the hypotenuse of a 45o-45o-90o triangle is 10 cm. What is the length of each leg?

yx

18cm

60

yx

6cm

45

y

x6cm45


55. The length of the shorter leg of a 30o-60o-90o triangle is 13 cm. What is the length of the longer leg? What is the length of the hypotenuse? 56. The length of the longer leg of a 30o-60o-90o triangle is 21 cm. What is the length of the hypotenuse? What is the length of the shorter leg? 57. The length of the hypotenuse of a 30o-60o-90o triangle is 21 cm. What is the length of the longer leg? What is the length of the shorter leg? Find the lengths of the missing sides.

58.

59.

60.

61.

y

x

24cm

60

y

x

24cm60

y

x

24cm

60

y

x

10cm45


62.

63. Find the area of the triangle. Round to the nearest hundredth.

64. A skateboarder constructs a ramp using plywood. The length of the plywood is 7 feet long and falls at a 40o angle. What is the height of the ramp?

Trigonometric Ratios

Class work

Refer to the triangle below to answer questions 65-66.

65. side opposite to ∠W is _______side adjacent to ∠W is _______

side opposite to∠B is _______side adjacent to ∠B is _______hypotenuse is _______.

66. sinW = ______ cosW = ______ tanW = _______.

sinB = ______ cosB = ______ tanB = _______.

y

x10cm45


Evaluate. Round the nearest ten-thousandth.

67. sin90o

68. cos52o

69. tan25o

70. tan88o

71. Find the length of AF̅̅̅̅ . Round to the nearest hundredth.

72. Find the length of SO̅̅̅̅ . Round to the nearest hundredth.

73. Find the length of KU̅̅ ̅̅ . Round to the nearest hundredth.

74. If the sin30 = .5, the cos60= ______________. Why?

If the cos25= .9063, the sin65=_____________. Why?


Homework

Refer to the triangle below to answer questions 75-76.

75. side opposite to ∠W is _______side adjacent to ∠W is _______

side opposite to ∠B is _______side adjacent to ∠B is _______hypotenuse is _______.

76. sinW = ______ cosW = ______ tanW = _______.

sinB = ______ cosB = ______ tanB = _______.

Evaluate. Round the nearest ten-thousandth.

77. sin45o

78. tan 77o

79. cos69o

80. tan33o

81. Find the length of RF̅̅̅̅ . Round to the nearest hundredth.

82. Find the length of SC̅̅ ̅. Round to the nearest hundredth.


83. Find the length of BK̅̅ ̅̅ BK. Round to the nearest hundredth.

84. If the sin60 = .8660, the cos30= ______________. Why?

If the cos65= .4226, the sin25=_____________. Why?

Solving Right Triangles

Class work

Solve the triangle. Round to the nearest hundredth.

85.

m∠C = ______ AC = ______ BC = ______

86.

m∠D = ______ m∠F = ______ DF = ______


87.

m∠G = ______ m∠I = ______ GH = ______

88.

m∠J = ______ KL = ______ JL = ______

Homework

Solve the triangle. Round to the nearest hundredth.

89.

m∠C = ______ AC = ______ BC = ______

90.

m∠D = ______ m∠F = ______ DF = ______


91.

m∠G = ______ m∠I = ______ GH = ______

92.

m∠J = ______ KL = ______ JL = ______

Angle of Elevation and Depression

Class work

93. Angela flies a kite at a 60o angle of elevation. The kite’s string is 275 feet. Angela’s arm is 4.5 feet off the ground. How high is the kite off the ground? (Round to the nearest hundredth)

94. An airplane flies 6 miles above the ground. The distance from the start of the runway to the airplane is 15 miles. What is the angle of depression the airplane must use to land? (Round to the nearest hundredth)

95. You are looking at the top of a tree. The angle of elevation is 78o. The distance from the top of the tree to your position is 125 feet. If you are 5.25 feet tall, how far are you from the base of the tree? (Round to the nearest hundredth) Homework 96. An airplane is 20 miles from the start of the runway. The angle of depression the airplane must use to land is 40o. How high is the airplane above the ground? (Round to the nearest hundredth) 97. You are standing on a mountain that is 6842 feet high. You look down at your campsite at angle of 38o. If you are 5.6 feet tall, how far is the base of the mountain from the campsite? (Round to the nearest hundredth)

98. You are looking at the top of a tree. The angle of elevation is 66o. The distance from the top of the tree to your position is 108 feet. If you are 5.75 feet tall, how tall is the tree? (Round to the nearest hundredth)


Law of Sines and Law of Cosines

Class Work

99. Solve the triangle.

m∠P = ______ m∠Q = ______ PR = ______


m∠S = ______ m∠T = ______ m∠U = ______


m∠E = ______ m∠F = ______ EF = ______


F

E

D 26

19

42°

z 65y

x

25

30



104. Show why 𝑠𝑖𝑛𝐴

𝑎=

𝑠𝑖𝑛𝐶

𝑐 by drawing an auxiliary line from a vertex perpendicular to the

opposite side.

Homework


m∠P = ______ m∠Q = ______ QR = ______


m∠S = ______ m∠T = ______ m∠U = ______

z

40

yx

10

60

b

ac

B

C A



m∠E = ______ m∠F = ______ DE = ______



110. Show why 𝑠𝑖𝑛𝐵

𝑏=

𝑠𝑖𝑛𝐶

𝑐 by drawing an auxiliary line from a vertex perpendicular to the

opposite side.

F

E

D 10

12

105°

z 58

y

x

25

23

z

45

y

x

12

95

ac

bC

B

A


Area of an Oblique Triangle

Class Work

Find the area of the triangle for #’s 111-114

111.

112.

113.

114.

55°

13

12

55°

47°

12

8

8

14 14

70°48°

18

14


115. Derive the formula A=1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex

perpendicular to the opposite side. Given: a, b and m∠C Find: A=1/2 ab sin(C).

Homework

Find the area of the triangle #’s 116-119

116.

117.

118.

b

ac

B

C A

30°

23

18

45°

37°

10

16

8

10 10


100°

43°16

22119.

120. Derive the formula A=1/2 ac sin(B) for the area of a triangle by drawing an auxiliary line from a vertex

perpendicular to the opposite side.

Right Triangles & Trigonometry Unit Review Multiple Choice - Choose the correct answer for each question. No partial credit will be given. 1. Find the length of the missing side of the triangle. Write the answer in simplest radical form.

a. 117 b. 9√13 c. √117 d. 3√13 2. Find the length of the missing side of the triangle. Write the answer in simplest radical form.

a. 4√22 b. 16√22 c. 18.76 d. √352

3. Tell whether the lengths 7√2, 3√7, and 5 form the sides of an acute, right, obtuse, or not a triangle.

a. acute b. right c. obtuse d. not a triangle

ac

bC

B

A


4. Find the value of x. Write the answer in simplest radical form.

a. 15√2

2 b. 7.5 c. 10√3 d. 9√13


a. 4√2 b. 8√2 c. 8√2

2 d. 4


a. 2√2 b. 4√2 c. 4√2

2 d. 2

7. In the triangle below, what is the side opposite to ∠J.

a. JR b. LJ c. JL d. LR


8. What is the tanK?

a. 2 b. 1

2 c.

5

7 d.

7

5

9. Find the m∠V. Round the answer to the nearest hundredth.

a. 33.69o b. 56.31o c. 41.81o d. 48.19o

10. Find the length of DT̅̅ ̅̅ . Round the answer to the nearest hundredth.

a. 8.31 b. 6.64 c. 6.26 d. 3.99

11. Find the m∠D. Round the answer to the nearest hundredth.

a. 50.71o b. 39.29o c. 35.10o d. 54.90o


12. Find the m∠V. Round the answer to the nearest hundredth.

a. 30.18o b. 27.45o c. 32o d. 30.179o

13. Find the m∠C. Round the answer to the nearest hundredth.

a. 71.44o b. 54.70o c. 85.90o d. 39.40o 14. Find the geometric mean of 5 and 9.

a. 3√5 b. 7 c. 5√3 d. 9√3 15. Solve for x.

a. 7 b. 12 c. 5√7 d. 25

16. In the figure, sin 47 = 15/x, which of the following equations is also true?

a. sin 43 = x/15 b. cos 43 = 15/x c. cos 47 = 15/x d. tan 47 = x/15

x 9

16

47°

15x


Short Constructed Response - Write the answer for each question. No partial credit will be given.

17. Tell whether the lengths 7√3, 3√11, and 4√5 form the sides of an acute, right, obtuse or not a triangle. 18. Michael is flying a kite at an angle of 45o. The kite's string is 202 feet long and Amy arm is 4 feet off the ground. How high is the kite off the ground? 19. Find the area of the triangle.

Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given. 20. Solve the triangle and find the area. Round to the nearest hundredth.


Answer Key

1. hypotenuse – DL/sides - DU and UL 2. 15 3. 60 4. 28

5. 6√2

6. 6√7

7. 8√3 8. obtuse 9. right 10. Not a Triangle 11. acute 12. Perimeter = 46in/Area = 120in2 13. hypotenuse – ME/sides - MQ and QE 14. 65 15. 35 16. 39

17. √146

18. 5√7

19. 2√86 20. obtuse 21. Not a Triangle 22. right 23. acute

24. Perimeter = (62+4√30)cm≈83.9cm

/Area = 58√30cm2≈306.7 cm2

25. a. 16

b. 4√3

c. 5√6

d. 6√7

26. FEGEDGFDE

27. 4.8

28. 2√10 29. 4 or 16

30. 4√6 31. 7.2

32. 51

3

33. a. 15 b. 18

c. 4√15

d. 2√70

34. RQPQSPRSQ

35. 5.4

36. x=2

37. 4√6 38. 4 39. 30 40. 420/2

41. 13√2cm

42. 8√2cm

43.LL = 5√3cm H=8√3𝑐𝑚

44. 𝐻 = 8√3cm SL=4√3

45. LL = 6√3, SL = 6

46. x = 18√3𝑐𝑚, y = 36cm

47. x = 9𝑐𝑚, y = 9√3cm

48. 𝑥 = 6√3𝑐𝑚, y=12√3𝑐𝑚

49. x=6cm, y=6√2𝑐𝑚

50. x=3√2, y=3√2 51. 292.72ft2 52. 8ft

53. 9√2 cm

54. 5√2 cm

55. H=26cm LL=13√3 cm

56. SL=7√3 cm H=14√3 cm

57. LL=(21√3)/2cm SL=10.5cm

58. x=24√3 𝑐𝑚 y=48

59. x=12√3cm, y=12cm

60. x=16√3 𝑐𝑚, y=8√3 𝑐𝑚

61. x=10, y=10√2 cm

62. x=5√2 cm, y=5√2 cm 63. 187.06 ft2 64. 6.58ft 65. 10,24,24,10,26 66. 5/3,12/13,5/12,12/13,5/13,12/5 67. 1.0000 68. 0.6157 69. 0.4663 70. 28.6363 71. 8.07 72. 14.83 73. 16.26 74. 0.5, 0.9063 because co-functions of complementary angels are equal 75. 14, 48, 48, 14, 50 76. 7/25, 24/25, 7/24, 24/25, 7/25, 24/7 77. 0.7072 78. 4.3315 79. 0.3584 80. 0.6494 81. 22.28


82. 23.66 83. 2.20 84. 0.8660, 0.4226 because co-functions of complementary angels are equal

85. m∠C = 63o, BC = 6.62, AC = 14.59

86. m∠D = 59.74o, m∠F = 30.26o, DF =

13.89

87. m∠G = 39.52o, m∠I = 50.48o, GH = 8.49

88. m∠J = 22o, KL = 6.74, JL = 16.69

89. m∠C = 57o, BC = 5.84, AC = 10.73 90. m∠D = 60.26o, m∠F = 29.74o, DF = 16.12

91. m∠G = 53.13o, m∠I = 36.87o, GH = 9 92. m∠J = 16, KL = 5.51, JL = 19.23 93. 242.66 ft 94. 23.58o 95. 25.99ft 96. 12.86ft 97. 8764.37ft 98. 104.41ft 99. m∠P = 57.41o, m∠Q = 85.59o, PR = 8.28

100. m∠S = 126.22o, m∠T = 31.47o, m∠U = 22.31o

101. m∠E = 89o, m∠F = 49o, EF = 17.4 102. x=85o, y=13.79, z=27.48 103. x=80o, y=8.79, z=6.53 104. sin C = h/a, h=a sinC sin A = h/c, h=c sinA a sin C = c sin A (sin C)/c = (sin A)/a

105. m∠P = 90.02o, m∠Q = 57.98o, QR = 9.44

106. m∠S = 39.57o, m∠T = 121.86o, m∠U = 18.57o

107. m∠E = 53.6o, m∠F = 21.4o, DE = 4.53 108. x=99 o, y=54.26, z=63.19 109. x=40 o,y=18.60,z=13.20 110. sin C = h/b, h=b sinC sin B = h/c, h=c sinB b sin C = c sin B (sin C)/c = (sin B)/b 111. 63.89 sq units 112. 46.95 sq units 113. 53.66 sq units 114. 93.64 sq units 115. sin C = h/a h=a sin C A=(1/2)(base)(height) A=(1/2)b (a sinC) A=(1/2)ab sin C 116. 103.5 sq units 117. 79.22 sq units 118. 34 sq units 119. 173.33 sq units 120. sin B = h/c h=c sin B A=(1/2)(base)(height) A=(1/2)a (c sinB) A=(1/2)ac sin B

Right Triangles & Trig Unit Review 1. D 2. A 3. C 4. C 5. A 6. B 7. D 8. A 9. A 10. C 11. D

12. A 13. C 14. A 15. B 16. B 17. acute 18. 146.83 feet 19. 22√26 20. BC = 9.54, m∠C = 20, m∠B=126.97, A=22.87 sq units

right triangles & trigonometry chapter...

Documents