1. Find the ratios , and . Round answers to the nearest hundredth.
The Tangent Ratio
May 9, 2003
Lesson 8-3
Check Skills You’ll Need
(For help, go to Lessons 7-1 and 8-2.)
Find the ratios , and Round answers to the nearest hundredth. BC AB AC 1. 2. 3. 4. 5. 7. 6. x 3 = 4 7 6 11 9 8 15 5 12
Check Skills You’ll Need
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2. The Tangent Ratio
May 9, 2003
Lesson 8-3
Check Skills You’ll Need
1. Use the Pythagorean Theorem to find AB: a2 + b2 = c (12)2 + (16)2 = AB = AB2 AB2 = AB = = 20; substitute 20 for AB, 16 for AC, and 12 for BC; = = 0.60; = = 0.80; = = 0.75
Solutions BC AB 12 20 AC AB 16 20 BC AC 12 16
2. BC = 10; use the Pythagorean Theorem to find AB: a2 + b2 = c2 (10)2 + (10)2 = AB = AB2 AB2 = AB = = ; substitute for AB, 10 for AC, and 10 for BC; = ; = ; = = 1 BC AB 10
10 2 AC AB 10
10 2 BC AC 10
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 = , and AB = 2AC = 2(4) = 8; By substitution, = = ;
= = = 0.5; = = BC AB
4 3 8 3 2 AC AB 4 8 1 2 BC AC
4 3 4
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3. 4. Multiply both sides of = by 3: x = • = or 1
The Tangent Ratio
May 9, 2003
Lesson 8-3
Check Skills You’ll Need
4. Multiply both sides of = by 3: x = • = or 1 x 3 4 7 1 12 5
5. Multiply both sides of = by 9: x = • = or 4 6 11 9 54 10
= ; (8)x = (4)(15)(Cross Products); simplify: 8x = 60; divide by 8: x = = or 7 8 15 60 2
= ; (5)(12) = (7)x (Cross Products); simplify: 60 = 7x; divide by 7: x = or 8
Solutions (continued)
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4. Special Right Triangles
May 9, 2003
Special Right Triangles
Lesson 8-2
Lesson Quiz
Use ABC for Exercises 1–3.
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.
AC = 18; AB =
AC = ; AB = 36
AC = ; AB =
, or about 2.8 cm
, or about 20.8 mm
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5. Textbook

6. The Tangent Ratio
May 9, 2003
Lesson 8-3
Notes
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7. AA Similarity Postulate
The Tangent Ratio
May 9, 2003
Lesson 8-3
Notes
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.
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8. The Tangent Ratio
May 9, 2003
Lesson 8-3
Notes
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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9. Writing Tangent Ratios
The Tangent Ratio
May 9, 2003
Lesson 8-3
Additional Examples
Writing Tangent Ratios
Write the tangent ratios for A and B.
tan A 20 21 =
opposite
adjacent BC AC
tan B 21 20 =
opposite
adjacent AC BC
Quick Check
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10. Real-World Connection
The Tangent Ratio
May 9, 2003
Lesson 8-3
Additional Examples
Real-World Connection
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° =
height
125
Use the tangent ratio.
height = 125 (tan 32°)
Solve for height.
Use a calculator.
The tree is about 78 ft tall.
Quick Check
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11. Using the Inverse of Tangent
The Tangent Ratio
May 9, 2003
Lesson 8-3
Additional Examples
Using the Inverse of Tangent
Find m R to the nearest degree.
tan R = 47 41
Find the tangent ratio.
m R tan–1
Use the inverse of the tangent. 47 41
Use a calculator. 47 41
So m R
Quick Check
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12. Use the figure for Exercises 1–3.
The Tangent Ratio
May 9, 2003
Lesson 8-3
Lesson Quiz
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. 15 8 8 15 28
Find x to the nearest whole number. 4. 5. 52 29
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