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

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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 8-3

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 8-3

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) 8-3

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 8-3

Textbook

The Tangent Ratio May 9, 2003 Lesson 8-3 Notes 8-3

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. 8-3

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. 8-3

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 8-3

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 8-3

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 8-3

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 8-3