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Shelby County Schools

Extended Learning Packet

Geometry

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Chapter 8 18 Glencoe Geometry

Study Guide and InterventionSpecial Right Triangles

Properties of 45°-45°-90° Triangles The sides of a 45°-45°-90° right triangle have a special relationship.

If the leg of a 45°-45°-90° right triangle is x units, show that the hypotenuse is x √ � 2 units.

x √⎯

x

x 245°

45°

Using the Pythagorean Theorem with a = b = x, then c2 = a2 + b2

= x2 + x2

= 2x2

c = √ �� 2x2 = x √ � 2

In a 45°-45°-90° right triangle the hypotenuse is √ � 2 times the leg. If the hypotenuse is 6 units, find the length of each leg.

The hypotenuse is √ � 2 times the leg, so divide the length of the hypotenuse by √ � 2 .a = 6 −

√ � 2

= 6 − √ � 2

. √ � 2

− √ � 2

= 6 √ � 2 −

2

= 3 √ � 2 units

ExercisesFind x.

1. x

8

45°

45°

2.

x

45°3√⎯2 3.

45°

4

x

4.

x x

18 5.

45°

16

x

x

6.

45°

24

x

√2

7. If a 45°-45°-90° triangle has a hypotenuse length of 12, find the leg length.

8. Determine the length of the leg of 45°-45°-90° triangle with a hypotenuse length of 25 inches.

9. Find the length of the hypotenuse of a 45°-45°-90° triangle with a leg length of 14 centimeters.

8-3

Example 1 Example 2

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Study Guide and Intervention (continued)

Special Right Triangles

Properties of 30°-60°-90° Triangles The sides of a 30°-60°-90° right triangle also have a special relationship.

In a 30°-60°-90° right triangle, show that the hypotenuse is twice the shorter leg and the longer leg is √ � 3 times the shorter leg.

� MNQ is a 30°-60°-90° right triangle, and the length of the hypotenuse

−−− MN is two times the length of the shorter side

−−− NQ .

Using the Pythagorean Theorem, a2 = (2x)2 - x2

= 4x2 - x2

= 3x2

a = √ �� 3x2 = x √ � 3

In a 30°-60°-90° right triangle, the hypotenuse is 5 centimeters. Find the lengths of the other two sides of the triangle.

If the hypotenuse of a 30°-60°-90° right triangle is 5 centimeters, then the length of theshorter leg is one-half of 5, or 2.5 centimeters. The length of the longer leg is √ � 3 times the length of the shorter leg, or (2.5)( √ � 3 ) centimeters.

ExercisesFind x and y.

1. x

y30°

60°12

2.

x

y

60°

8

3. x

y

11

30°

4. x

y

30°

9 √⎯3

5. x

y60°

12

6.

30°

60°

x

20y

7. An equilateral triangle has an altitude length of 36 feet. Determine the length of a side of the triangle.

8. Find the length of the side of an equilateral triangle that has an altitude length of 45 centimeters.

8-3

30°

60°x

2x

aExample 1

Example 2


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8-3 Skills PracticeSpecial Right Triangles

Find x.

1.

45°

25

x

2.

45°

17x

3. 45°

48

x

4.

45°100

x 5.

45°

100

x

6.

45°

88 x

7. Determine the length of the leg of 45°-45°-90° triangle with a hypotenuse length of 26.


Find x and y.

9.

30°

x11

y

10.

60°x

8

y

√3 11. 30°

x

5

y

√3

12.

60°

x

30

y

13. 60°

x

21y

√3

14. 60°

x

52 y√3


16. Find the length of the side of an equilateral triangle that has an altitude length of 11 √ � 3 feet.


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Find x.

1.

45°

14

x

2.

45°

45x

3. 45°

22

x

4.

45°210

x

5.

45°

88

x

6.

45°

x5 √2

Find x and y.

7.

30°

x9

y

8.

60°x

4

y

√3

9.

30°

x

20

y

10.

60°

x

98

y

11. Determine the length of the leg of 45°-45°-90° triangle with a hypotenuse length of 38.



14. BOTANICAL GARDENS One of the displays at a botanical garden is an herb garden planted in the shape of a square. The square measures 6 yards on each side. Visitors can view the herbs from a diagonal pathway through the garden. How long is the pathway?

6 yd 6 yd

6 yd

6 yd

PracticeSpecial Right Triangles

8-3


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Trigonometric Ratios The ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan, respectively.

sin R = leg opposite ∠R

− hypotenuse

cos R = leg adjacent to ∠R

−− hypotenuse

tan R = leg opposite ∠R

−− leg adjacent to ∠R

= r − t = s − t = r − s

Find sin A, cos A, and tan A. Express each ratio as a fraction and a decimal to the nearest hundredth.

sin A = opposite leg

− hypotenuse

cos A = adjacent leg

− hypotenuse

tan A = opposite leg

− adjacent leg

= BC − BA

= AC − AB

= BC − AC

= 5 − 13

= 12 − 13

= 5 − 12

≈ 0.39 ≈ 0.92 ≈ 0.42

ExercisesFind sin J, cos J, tan J, sin L, cos L, and tan L. Express each ratio as a fraction and as a decimal to the nearest hundredth.

1. 2. 3.

12

135

C

B

A

Study Guide and InterventionTrigonometry

8-4

s

tr

T

S

R

Example

20

12

16

40

24

32

36

12 √324 √3


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Study Guide and Intervention (continued)

Trigonometry

Use Inverse Trigonometric Ratios You can use a calculator and the sine, cosine, or tangent to find the measure of the angle, called the inverse of the trigonometric ratio.

Use a calculator to find the measure of ∠T to the nearest tenth.

The measures given are those of the leg opposite ∠T and the hypotenuse, so write an equation using the sine ratio.

sin T = opp

− hyp

sin T = 29 − 34

If sin T = 29 − 34

, then sin-1 29 − 34

= m∠T.

Use a calculator. So, m∠T ≈ 58.5.

ExercisesUse a calculator to find the measure of ∠T to the nearest tenth.

1.

3414 √3

2. 7

18

3.

87

34

4.

10132

5.

67

10 √3

6.

39

14 √2

8-4

Example

34

29

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Find sin R, cos R, tan R, sin S, cos S, and tan S. Express each ratio as a fraction and as a decimal to the nearest hundredth.

1. r = 16, s = 30, t = 34 2. r = 10, s = 24, t = 26

Use a special right triangle to express each trigonometric ratio as a fraction and as a decimal to the nearest hundredth.

3. sin 30° 4. tan 45° 5. cos 60°

6. sin 60° 7. tan 30° 8. cos 45°

Find x.

9.

7 x

36°

10. 11.

Use a calculator to find the measure of ∠B to the nearest tenth.

12. 13. 14.

sR

S

T

rt

Skills PracticeTrigonometry

8-4

12x

15°

6

x°

18

15

x°

12

22

x°

19

12 x

63°


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Find sin L, cos L, tan L, sin M, cos M, and tan M. Express each ratio as a fraction and as a decimal to the nearest hundredth.

1. � = 15, m = 36, n= 39 2. � = 12, m = 12 √ � 3 , n = 24

Find x.

3.

1164°

x

4. 5.

41°x

32

Use a calculator to find the measure of∠B to the nearest tenth.

6.

8

14

7.

30

25

8.

9. GEOGRAPHY Diego used a theodolite to map a region of land for his class in geomorphology. To determine the elevation of a vertical rock formation, he measured the distance from the base of the formation to his position and the angle between the ground and the line of sight to the top of the formation. The distance was 43 meters and the angle was 36°. What is the height of the formation to the nearest meter?

29

29°x

36°43m

ML

N

m

n

PracticeTrigonometry

8-4

739

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Geometry Educational Websites and Web Resources

Title of Resource Web Address Description Student Access

eMathInstruction https://www.youtube.com/watch?v=ZH7cXKnLCJY Students will be able to view a

video that corresponds to the

worksheets provided.

Students will need to agree to

the terms outlined by the

website for free access

Khan Academy https://www.khanacademy.org Students will be able to get

additional practice with skills in

various subjects and test prep.

Students will need to sign up for

a free account if they do not

already have an account, however, the videos are

accessible.

https://www.youtube.com/watch?v=ZH7cXKnLCJY

http://www.khanacademy.org/

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