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Lesson 8-3

(For help, go to Lessons 7-1 and 8-2.)

Find the ratios , and . Round answers to the nearest hundredth.BCAB

ACAB

BCAC

The Tangent RatioThe Tangent Ratio

1. 2. 3.

4. 5. 7.6.x3 =

47

6 11

x9=

8 15 =

4x

5x =

7 12

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8-3

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Lesson 8-3

The Tangent RatioThe Tangent Ratio

Solutions

10 10 2

32

4 38

1. Use the Pythagorean Theorem to find AB: a2 + b2 = c2

(12)2 + (16)2 = AB2 144 + 256 = AB2 AB2 = 400 AB = 400 = 20;

substitute 20 for AB, 16 for AC, and 12 for BC; = = 0.60;

= = 0.80; = = 0.75

BCAB

1220AC

AB1620

BCAC

1216

BCAB

ACAB

BCAC

1010

BCAB

ACAB

48

12

BCAC

4 34

10 10 2

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2. BC = 10; use the Pythagorean Theorem to find AB: a2 + b2 = c2

(10)2 + (10)2 = AB2 100 + 100 = AB2 AB2 = 200 AB = 200 = 10 2;

substitute 10 2 for AB, 10 for AC, and 10 for BC;

= 0.71; = 0.71; = = 1

3. AC = 4; The triangle is a 30-60°-90° triangle, so the sides opposite the

angles in order are in the ratio 1 : 3 : 2. BC is AC 3 = 4 3, and

AB = 2AC = 2(4) = 8; By substitution, = = 0.87;

= = = 0.5; = = 3 1.73

8-3

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4. Multiply both sides of = by 3: x = • = or 1x3

47

47

31

127

57

5. Multiply both sides of = by 9: x = • = or 4 6 11

x9

6 11

91

5411

1011

6. = ; (8)x = (4)(15)(Cross Products); simplify: 8x = 60;

divide by 8: x = = or 7

8 15

4x 60

8152

12

7. = ; (5)(12) = (7)x (Cross Products); simplify: 60 = 7x;

divide by 7: x = or 8

5x

7 12

607

47

Solutions (continued)

Lesson 8-3

The Tangent RatioThe Tangent Ratio

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8-3

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AC = 6 3; AB = 12 3

AC = 18 3; AB = 36

AC = 18; AB = 18 2

Lesson 8-2

Special Right TrianglesSpecial Right Triangles

Use ABC for Exercises 1–3.

Lesson Quiz

2 2 , or about 2.8 cm

12 3, or about 20.8 mm

1. If m A = 45, find AC and AB.

2. If m A = 30, find AC and AB.

3. If m A = 60, find AC and AB.

4. Find the side length of a 45°-45°-90° triangle with a 4-cm hypotenuse. 5. Two 12-mm sides of a triangle form a 120° angle. Find the length of the third side.

8-3

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Lesson 8-3

The Tangent RatioThe Tangent Ratio

Notes

8-3

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Lesson 8-3

The Tangent RatioThe Tangent Ratio

Notes

8-3

The tangent ratio for an acute angle does not depend on leg lengths of a right triangle. To see why this is so, consider the congruent angles, T and T’, in the two right triangles shown here.

TOW ~ T’O’W’ AA Similarity Postulate

Corresponding sides of ~ triangles are proportional.

tan T = tan T’ Substitute.

' '

' '

OW O W

TO T O

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Lesson 8-3

The Tangent RatioThe Tangent Ratio

Notes

8-3

If you know leg lengths for a right triangle, you can find the tangent ratio for each acute angle. Conversely, if you know the tangent ratio for an angle, you can use inverse of tangent, tan-1, to find the measure of the angle.

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tan A2021

= = =oppositeadjacent

BCAC

tan B2120

= = =oppositeadjacent

ACBC

Write the tangent ratios for A and B.

Lesson 8-3

The Tangent RatioThe Tangent Ratio

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Additional Examples

8-3

Writing Tangent Ratios

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To measure the height of a tree, Alma walked 125 ft from the tree and measured a 32° angle from the ground to the top of the tree. Estimate the height of the tree.

The tree forms a right angle with the ground, so you can use the tangent ratio to estimate the height of the tree.

tan 32° = height125 Use the tangent ratio.

height = 125 (tan 32°) Solve for height.

125 32 78.108669 Use a calculator.

The tree is about 78 ft tall.

Lesson 8-3

The Tangent RatioThe Tangent Ratio

Quick Check

Additional Examples

8-3

Real-World Connection

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Find m R to the nearest degree.

tan R =4741 Find the tangent ratio.

So m R 49.

Lesson 8-3

The Tangent RatioThe Tangent Ratio

m R tan–1 Use the inverse of the tangent.4741

Use a calculator.48.9004944741

Quick Check

Additional Examples

8-3

Using the Inverse of Tangent

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Use the figure for Exercises 1–3.

1. Write the tangent ratio for K.

2. Write the tangent ratio for M.

3. Find m M to the nearest degree.

Find x to the nearest whole number.

4. 5.

28

158

8 15

52 29

Lesson 8-3

The Tangent RatioThe Tangent Ratio

Lesson Quiz

8-3