Name · 2020. 4. 6. · Name PearsonRealize.com 8-1 Additional Practice Right Triangles and the Pythagorean Theorem For Exercises 1–9, find the value of x.Write your answers in

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8-1 Reteach to Build UnderstandingRight Triangles and the Pythagorean Theorem

1. Review the theorems in the lesson. Two of the theorems are the Pythagorean Theorem and its converse. Together, they state that sum of the squares of the lengths of the two shortest sides of a triangle equals the square of the length of the longest side if and only if the triangle is a right triangle. The Pythagorean Theorem can be used to verify the side length properties of 30°-60°-90° and 45°-45°-90° right triangles. Draw a line to match each hypothesis with a conclusion.

Hypothesis:

A C

B

a

b

c

a 2 + b 2 = c 2

A C

B

a

b

c

C B

A

s45°

45°

C B

A

s

30°

60°

Conclusion: AB = s √ __

2 △ABC is a right triangle.

a 2 + b 2 = c 2 AC = s √ __

3 ;

AB = 2s

2. Tonya and Terrence both attempted to solve for y, but they disagreed on the answer. Check their work. Who is incorrect? Explain.

Tonya Terrance

2x = 20; x = 10 2x = 20; x = 10

y = 2x = 20 y = 2 ( 10 ___ √

__ 3 ) =

20 ___ √

__ 3 = 20 √

__ 3 _____

3

3. Find x and y in the triangle.

The triangle shown is a right triangle.

Use the relationship between x and 16 to solve for x.

⋅ x = 16

x =

Use the relationship between x and y to solve for y.

y = ⋅ x

y =

xy

60°60°

30°

20

y

60°16

x30°-60°-90°

28

√ __

3 8 √

__ 3

Sample: Tonya’s last line is incorrect, since the side length x is across from the 60° angle, not the 30° angle.

enVision™ Geometry • Teaching Resources


8-1 Additional PracticeRight Triangles and the Pythagorean Theorem

For Exercises 1–9, find the value of x. Write your answers in simplest radical form.

1.

9

12 x

2.

5

x

60° 3.

9

6

x

4.

6x

5.

410

x

6.

8 x

60°

7.

8

88 x

A

CB

8.

45°

10

4

x

9.

30°

20 x

10. Simon and Micah both made notes for their test on right triangles. They noticed that their notes were different. Who is correct? Explain.

x

x

45°

45°-45°-90° 45°-45°-90°30°-60°-90° 30°-60°-90°

Simon Micah

45°

x√2

x

2x

60°

30°

x√3

45°

45°

x

√2x

√2x x

60°

30°

2x

2x√3

Both are correct. They have used different variables, but the ratios of the lengths of legs and hypotenuse are the same.

11. A rectangular lot is 165 feet long and 90 feet wide. How many feet of fencing are needed to make a diagonal fence for the lot? Round to the nearest foot. 188 ft

15 5 √ __

3 3 √ __

5

3 √ __

2 2 √

___ 21 4 √

__ 3

4 √ __

3 4 √ __

2

10



8-2 Reteach to Build UnderstandingTrigonometric Ratios

1. In a right triangle, the sine ratio of an acute angle is length of opposite leg __________________ length of hypotenuse

.

The cosine ratio is length of adjacent leg _________________ length of hypotenuse

. The tangent ratio is length of opposite leg __________________ length of adjacent leg

.

Draw a line to match each trigonometric ratio with the correct ratio of sides.

Trigonometric ratio:

sin A cos A tan A

Ratio of sides: a __ c a __ b b __ c

2. What is Yuson’s error in solving for the value for x? What is the correct value of x? Round to the nearest tenth.

cos 29° = opposite _________ hypotenuse

cos 29° = x ___ 13

x = 13 (cos 29°)

x ≈ 11.4

3. Find x and y in the triangle. Round to the nearest tenth.

Solve for x. Solve for y.

cos = leg adjacent __________ hypotenuse

sin = leg opposite __________ hypotenuse

cos = x __ 8 sin = y __

8

x = ∙ cos y = ∙ sin ≈ ≈

x

29°

13

x

y20°

70°8

2.78 8

70° 70°70°

70°

7.570°

70°

She should have used the sine ratio, not the cosine ratio; x ≈ 6.3.

A C

B

a

b

chypotenuse

leg adjacentto ∠A

leg opposite∠A



8-2 Additional PracticeTrigonometric Ratios

For Exercises 1–3, find sin A, cos A, and tan A.

1.

9 6

A

C

B

2.

4

A

C

B

45°

3.

4

C

A B

60°

For Exercises 4–9, find the value of x. Round to the nearest tenth.

4.

48°5

x

5.

x°11

23

6.

54°

x 8

7. 37°

x12

8. x°

6.5

10 9. x°

47

25

10. Skylar drew two triangles that share a side and labeled the two portions of the base w and x. Then he solved for w and x as shown. Are his calculations correct? Explain.

sin 45° = 5.5 ___ w

w = 5.5 ______ sin 45° = 7.8

sin 35° = 5.5 _____ w + x

w + x = 5.5 ______ sin 35° ≈ 9.6

x = 1.8

11. A wire makes a 70° angle with the ground and is attached to the top of a 50 ft antenna. How long is the wire? Round to the nearest foot. 53 ft

2 __ 3 ; √

__ 5 ___

3 ; 2 √

__ 5 ____

5 √

__ 2 ___

2 ; √

__ 2 ___

2 ; 1

1 __ 2 ; √

__ 3 ___

2 ; √

__ 3 ___

3

x ≈ 5.6

x ≈ 28.6 x ≈ 11.0

x ≈ 7.2

x ≈ 49.5

x ≈ 62.0

w x45°

5.5

35°

No; he is not correct. The tangent ratio is the correct trigonometric ratio to use, not the sine ratio. w = 5.5 and x = 5.5 _______

tan 35° − 5.5 ≈ 2.4

x

70°

50 ft



8-3 Reteach to Build UnderstandingThe Law of Sines

1. Complete the definition of the Law of Sines.

In any triangle, the ratio of the sine of an angle to the length of the opposite side is constant. In △ABC with angles A, B, and C and opposite side lengths a, b, and c, the following relationship is true.

sin A ______ a = sin B _____

b = sin C ______ c

2. What error does Teo make in finding the value of AC? What is the correct value of AC? Round to the nearest tenth.

Use the Law of Sines.

sin A _____ AC

= sin B _____ BA

sin 118° _______ AC

= sin 22° ______ 24

AC = 24 ∙ sin 118° ___________ sin 22°

≈ 56.6

3. What is QR to the nearest tenth?

Given information:

m∠P opposite QR is .

m∠Q opposite PR is .

PR is .

Use the Law of Sines.

sin 50° _______ QR

= sin 60° _______ 20

∙ sin 50° = QR ∙ sin

QR = 20 ∙ sin 50° ___________ sin 60°

≈

A B

C

c

b a

C

A

B

118°

22°

24

P R

Q

60°

50°20

20

20 60°

17.7

50°60°

He wrote the ratio of the sine of an angle to the length of a side adjacent to the angle, not the length of the side opposite the angle. The correct value of AC is 14.0.



8-3 Additional PracticeThe Law of Sines

For Exercises 1–9, use the Law of Sines to find the values of x and y. Round to the nearest tenth.

1.

x

y74° 39°

12

2.

x

y81° 31°

18

3.

x°

y

115°

2410

4.

x

y

75°

33°

15

5.

x

y48°

92° 8 6.

x

y

41°

87°

10

7.

x

y80°

40°

13

8.

x°

y

120°

16

20

9.

x

y

130°

12°

30

10. To find the measure of ∠ A, Shannon made the calculations shown. What mistake did she make? What is the correct measure of ∠ A? She substituted incorrectly and found the measure of ∠B. The correct measure is 17.1°. sin A _____ a =

sin C _____ c

sin A _____ 9 = sin 125° _______

12

m ∠ A = si n −1 ( 9 sin 125° _________

12 ) ≈ 37.9°

11. The diagram shows three streets that form the perimeter of a park. How far is it from the corner of Oak and Ridgewood to the corner of Oak and Savannah? Round to the nearest tenth of a yard. 276.0 yd

125°

9

12

A

C

B

42°38°

300 ydSavannahLn. Ridgewood

Ave.

Oak St.

x ≈ 7.9; y ≈ 11.5

x ≈ 8.6; y ≈ 15.2

x ≈ 8.5; y ≈ 11.4

x ≈ 9.4; y ≈ 16.9

x ≈ 6.9; y ≈ 10.8

x° ≈ 43.9°; y ≈ 6.4

x° ≈ 22.2°; y ≈ 18.0

x ≈ 15.2; y ≈ 12.0

x ≈ 8.1; y ≈ 24.1



8-4 Reteach to Build UnderstandingThe Law of Cosines

1. Complete the definition of the Law of Cosines.

Given a triangle, the length of a side is related to the cosine of the opposite angle and the lengths of the two other sides. For any △ABC with angles A, B, and C and opposite side lengths a, b, and c, you have the following relationships.

a 2 = − 2bc cos

b 2 = − 2ac cos

c 2 = − 2ab cos

2. William attempts to find the value of AB by applying the Law of Cosines. What is his error? Find the correct value of AB. Round to the nearest whole number.

(AB) 2 = 29 2 + 2 8 2 + 2(29)(28) cos 52°

(AB) 2 ≈ 2,625

AB ≈ 51.2

3. What is BC? Round to the nearest tenth.

Given information: Use the Law of Cosines.

m∠A opposite BC is . (BC) 2 = ( ) 2 + ( ) 2 Length of ‾ AB is . − (50)(35) cos Length of ‾ AC is . (BC) 2 ≈

BC ≈

A B

C

c

b a

B

C

A

52°

29 28

A C

B

40°35

50

40° 50

32.3

352 40°

1,043.85035

William added twice the product of the given side lengths and the included angle instead of subtracting. The correct value is 25.

b 2 + c 2 ABC

a 2 + c 2 a 2 + b 2



8-4 Additional PracticeThe Law of Cosines

For Exercises 1–4, use the Law of Cosines to find the values of x and y. Round to the nearest tenth.

1.

x° y°

16

10 12

2.

y°

x

10

20

30°

3. x°

y°

5

8 6

4.

y°

x

35

40

45°

5. William calculated the measure of the largest angle for a triangle with sides 8, 11, and 13. What mistake did he make? What is the correct angle measure?

c 2 = a 2 + b 2 − 2ab cos C

8 2 = 1 3 2 + 1 1 2 − 2(13)(11) cos C

64 = 169 + 121 − 286 cos C

64 − (169 + 121) = −286 cos C

226 ____ 286

= cos C

m∠C ≈ 37.8°

6. Two planes are flying at the same altitude. One airplane is 60 miles due north of the control tower. Another airplane is located 70 miles from the tower at a heading of 80 ° east of south. To the nearest tenth of a mile, how far apart are the two airplanes?

He found the measure of the smallest angle. He needed to substitute 13 for c and 8 for a. The measure of the largest angle is 84.8°.

N

E

70 mi

60 mi

80°

tower

A

B

x ≈ 48.5; y ≈ 38.6

x ≈ 38.6; y ≈ 92.9

99.8 mi

x ≈ 12.4; y ≈ 126.2

x ≈ 29.1; y ≈ 76.6



8-5 Reteach to Build UnderstandingProblem Solving With Trigonometry

1. Label each diagram as angle of depression or angle of elevation.

line of

sight

line of sighthorizontal line

horizontal line

2. Dyani attempts to find the area of △ABC. What is her error? Find the correct area. Round to the nearest square centimeter.

area = 1 __ 2 (AB)(BC) tan C

area = 1 __ 2 (14)(12) tan 65°

≈ 180 c m 2

3. A lifeguard, seated 10 ft up, is looking down at a swimmer. Her line of sight forms a 55° angle with the horizontal line. How far is the swimmer from the base of the lifeguard stand?

Given information:

Angle of depression is .

The length of the side opposite the angle of depression is ft.

The length of the side adjacent to the angle of depression is ft.

Use the ratio.

tan = 10 ___ x

x = 10 _______ tan 55°

≈ 7

B

A

C65°

14 cm

12 cm

a

b

c

x ft55°

10 ft

55°

The swimmer is 7 feet from the base of the lifeguard stand.

55°

tangent

10

x

She should use the sine function, not the tangent function. 76 c m 2

angle of elevation angle of depression



8-5 Additional PracticeProblem Solving With Trigonometry

Find the value of x. Round to the nearest tenth.

1.

900 ft

x40°

2.

x

1,000 ft37°

3.

60 m x

18°

4.

80 ftx

48°

5. 12,000 ft

x

22° 6.

2,800 ft

x

33°

7. Rachel is attempting to find the area of the triangle shown. She remembers the formula A = 1 __

2 bh. What line

should she draw to apply the formula? What is b? What is h? What is the area? Round to the nearest tenth.the altitude; b = 9 cm; h = 6 sin 70° cm; 25.4 c m 2

8. Two office buildings are 100 ft apart, as shown. From the edge of the roof of the shorter building, the angle of elevation to the top of the taller building is 28° and the angle of depression to the bottom is 42°. How tall is each building? Round to the nearest foot.

90 ft; 143 ft

9 cm

6 cm

70°

Creativeicon

pending

100 ft

28°42°

1,072.6 ft

194.2 m

4,848.3 ft

753.6 ft

59.5 ft

1,818.3 ft


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Name · 2020. 4. 6. · Name PearsonRealize.com 8-1 Additional Practice Right Triangles and the Pythagorean Theorem For Exercises 1–9, find the value of x.Write your answers in