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FeatureLesson Geometry Lesson Main Lesson 8-3 (For help, go to Lessons 7-1 and 8-2.) Find the ratios, and. Round answers to the nearest hundredth. BC AB AC AB BC AC The Tangent Ratio x3x3 = x9x9 = 8 15 = 4x4x 5x5x = 7 12 Check Skills Youll Need 8-3

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FeatureLesson Geometry Lesson Main Lesson 8-3 The Tangent Ratio Solutions Use the Pythagorean Theorem to find AB: a 2 + b 2 = c 2 (12) 2 + (16) 2 = AB = AB 2 AB 2 = 400 AB = 400 = 20; substitute 20 for AB, 16 for AC, and 12 for BC; = = 0.60; = = 0.80; = = 0.75 BC AB AC AB BC AC BC AB AC AB BC AC 10 BC AB AC AB BC AC Check Skills Youll Need 2.BC = 10; use the Pythagorean Theorem to find AB: a 2 + b 2 = c 2 (10) 2 + (10) 2 = AB = AB 2 AB 2 = 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; = =

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FeatureLesson Geometry Lesson Main 4.Multiply both sides of = by 3: x = = or 1 x3x Multiply both sides of = by 9: x = = or x9x = ; (8)x = (4)(15)(Cross Products); simplify: 8x = 60; divide by 8: x = = or x4x = ; (5)(12) = (7)x (Cross Products); simplify: 60 = 7x; divide by 7: x = or 8 5x5x Solutions (continued) Lesson 8-3 The Tangent Ratio Check Skills Youll Need 8-3

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FeatureLesson Geometry Lesson Main AC = 6 3; AB = 12 3 AC = 18 3; AB = 36 AC = 18; AB = 18 2 Lesson 8-2 Special 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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FeatureLesson Geometry Lesson Main Textbook

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FeatureLesson Geometry Lesson Main Lesson 8-3 The Tangent Ratio Notes 8-3

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FeatureLesson Geometry Lesson Main Lesson 8-3 The 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 ~ TOW AA Similarity Postulate Corresponding sides of ~ triangles are proportional. tan T = tan TSubstitute.

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FeatureLesson Geometry Lesson Main Lesson 8-3 The 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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FeatureLesson Geometry Lesson Main tan A === opposite adjacent BC AC tan B === opposite adjacent AC BC Write the tangent ratios for A and B. Lesson 8-3 The Tangent Ratio Quick Check Additional Examples 8-3 Writing Tangent Ratios

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FeatureLesson Geometry Lesson Main 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. Lesson 8-3 The Tangent Ratio Quick Check Additional Examples 8-3 Real-World Connection

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FeatureLesson Geometry Lesson Main Find m R to the nearest degree. tan R = Find the tangent ratio. So m R 49. Lesson 8-3 The Tangent Ratio m R tan –1 Use the inverse of the tangent Use a calculator Quick Check Additional Examples 8-3 Using the Inverse of Tangent

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FeatureLesson Geometry Lesson Main 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 Lesson 8-3 The Tangent Ratio Lesson Quiz 8-3

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