# 9.3 Similar Right Triangles Essential Question MAKING

9.3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS Similar Right Triangles Essential Question How are altitudes and geometric means of right triangles related? G.8.A G.8.B Writing a Conjecture Work with a partner. — a. Use dynamic geometry software to construct right △ABC, as shown. Draw CD so that it is an altitude from the right angle to the hypotenuse of △ABC. 5 A 4 D 3 2 1 B 0 C 0 1 2 3 4 5 6 7 8 −1 MAKING MATHEMATICAL ARGUMENTS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results in constructing arguments. Points A(0, 5) B(8, 0) C(0, 0) D(2.25, 3.6) Segments AB = 9.43 BC = 8 AC = 5 b. The geometric mean of two positive numbers a and b is the positive number x that satisfies a x x is the geometric mean of a and b. x b Write a proportion involving the side lengths of △CBD and △ACD so that CD is the geometric mean of two of the other side lengths. Use similar triangles to justify your steps. — = —. c. Use the proportion you wrote in part (b) to find CD. d. Generalize the proportion you wrote in part (b). Then write a conjecture about how the geometric mean is related to the altitude from the right angle to the hypotenuse of a right triangle. Comparing Geometric and Arithmetic Means Work with a partner. Use a spreadsheet to find the arithmetic mean and the geometric mean of several pairs of positive numbers. Compare the two means. What do you notice? 1 2 3 4 5 6 7 8 9 10 11 A a 3 4 6 0.5 0.4 2 1 9 10 B b C D Arithmetic Mean Geometric Mean 4 3.5 3.464 5 7 0.5 0.8 5 4 16 100 Communicate Your Answer 3. How are altitudes and geometric means of right triangles related? Section 9.3 Similar Right Triangles 481 9.3 Lesson What You Will Learn Identify similar triangles. Solve real-life problems involving similar triangles. Core Vocabul Vocabulary larry Use geometric means. geometric mean, p. 484 Previous altitude of a triangle similar figures Identifying Similar Triangles When the altitude is drawn to the hypotenuse of a right triangle, the two smaller triangles are similar to the original triangle and to each other. Theorem Theorem 9.6 Right Triangle Similarity Theorem If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. C A D △CBD ∼ △ABC, △ACD ∼ △ABC, and △CBD ∼ △ACD. C Proof Ex. 45, p. 488 A B C D D B Identifying Similar Triangles Identify the similar triangles in the diagram. S U R SOLUTION T Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. S T S T U R U R T △TSU ∼ △RTU ∼ △RST Monitoring Progress Help in English and Spanish at BigIdeasMath.com Identify the similar triangles. 1. Q 2. E H T S 482 Chapter 9 R Right Triangles and Trigonometry G F Solving Real-Life Problems Modeling with Mathematics A roof has a cross section that is a right triangle. The diagram shows the approximate dimensions of this cross section. Find the height h of the roof. Y 5.5 m Z 3.1 m h W 6.3 m X SOLUTION 1. Understand the Problem You are given the side lengths of a right triangle. You need to find the height of the roof, which is the altitude drawn to the hypotenuse. 2. Make a Plan Identify any similar triangles. Then use the similar triangles to write a proportion involving the height and solve for h. 3. Solve the Problem Identify the similar triangles and sketch them. Z Z Y COMMON ERROR Notice that if you tried to write a proportion using △XYW and △YZW, then there would be two unknowns, so you would not be able to solve for h. 3.1 m 6.3 m 5.5 m 5.5 m h X W h Y W X 3.1 m Y △XYW ∼ △YZW ∼ △XZY Because △XYW ∼ △XZY, you can write a proportion. YW XY ZY XZ h 3.1 —=— 5.5 6.3 h ≈ 2.7 —=— Corresponding side lengths of similar triangles are proportional. Substitute. Multiply each side by 5.5. The height of the roof is about 2.7 meters. 4. Look Back Because the height of the roof is a leg of right △YZW and right △XYW, it should be shorter than each of their hypotenuses. The lengths of the two hypotenuses are YZ = 5.5 and XY = 3.1. Because 2.7 < 3.1, the answer seems reasonable. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the value of x. 3. 4. E 13 H 5 3 G K x x J 4 12 L 5 M F Section 9.3 Similar Right Triangles 483 Using a Geometric Mean Core Concept Geometric Mean The geometric mean of two positive numbers a and b is the positive number x — a x that satisfies — = —. So, x2 = ab and x = √ ab . x b Finding a Geometric Mean Find the geometric mean of 24 and 48. SOLUTION x2 = ab Definition of geometric mean ⋅ x2 = 24 48 Substitute 24 for a and 48 for b. ⋅ x = √24 ⋅ 24 ⋅ 2 — x = √24 48 — — x = 24√2 Take the positive square root of each side. Factor. Simplify. — The geometric mean of 24 and 48 is 24√2 ≈ 33.9. — is drawn to the hypotenuse, forming two smaller right In right △ABC, altitude CD triangles that are similar to △ABC. From the Right Triangle Similarity Theorem, you know that △CBD ∼ △ACD ∼ △ABC. Because the triangles are similar, you can write and simplify the following proportions involving geometric means. C A D B C CD AD BD CD CB DB —=— ⋅ CD 2 = AD BD A D B C D AC AD AB CB AB AC —=— —=— ⋅ CB 2 = DB AB ⋅ AC 2 = AD AB Theorems Theorem 9.7 Geometric Mean (Altitude) Theorem C In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse. A D CD2 Proof Ex. 41, p. 488 ⋅ = AD BD Theorem 9.8 Geometric Mean (Leg) Theorem C In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. Proof Ex. 42, p. 488 484 Chapter 9 Right Triangles and Trigonometry B A ⋅ ⋅ D CB 2 = DB AB AC 2 = AD AB B Using a Geometric Mean Find the value of each variable. COMMON ERROR In Example 4(b), the Geometric Mean (Leg) Theorem gives y2 = 2 (5 + 2), not y2 = 5 (5 + 2), because the side with length y is adjacent to the segment with length 2. ⋅ ⋅ a. 2 b. x y 6 5 3 SOLUTION a. Apply the Geometric Mean (Altitude) Theorem. b. Apply the Geometric Mean (Leg) Theorem. ⋅ x2 = 6 3 x2 = 18 y2 — x = √ 18 — x = √9 ⋅ = 2 ⋅7 y2 = 2 (5 + 2) y2 = 14 ⋅ √2 — — y = √ 14 — — x = 3√ 2 The value of y is √ 14 . — The value of x is 3√ 2 . Using Indirect Measurement To find the cost of installing a rock wall in your schooll gymnasium, you need to find the height of the gym wall. all. all You use a cardboard square to line up the top and bottom om of the gym wall. Your friend measures the vertical distance ance from the ground to your eye and the horizontal distance ce from you to the gym wall. Approximate the height of the gym wall. w ft 8.5 ft SOLUTION 5 ft By the Geometric Mean (Altitude) Theorem, you know w that 8.5 is the geometric mean of w and 5. ⋅ 8.52 = w 5 Geometric Mean (Altitude) Theorem 72.25 = 5w Square 8.5. 14.45 = w Divide each side by 5. The height of the wall is 5 + w = 5 + 14.45 = 19.45 feet. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the geometric mean of the two numbers. 5. 12 and 27 x 4 6. 18 and 54 7. 16 and 18 8. Find the value of x in the triangle at the left. 9 9. WHAT IF? In Example 5, the vertical distance from the ground to your eye is 5.5 feet and the distance from you to the gym wall is 9 feet. Approximate the height of the gym wall. Section 9.3 Similar Right Triangles 485 Exercises 9.3 Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and __________________. 2. WRITING In your own words, explain geometric mean. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, identify the similar triangles. (See Example 1.) 3. F In Exercises 11–18, find the geometric mean of the two numbers. (See Example 3.) E H G 4. M 11. 8 and 32 12. 9 and 16 13. 14 and 20 14. 25 and 35 15. 16 and 25 16. 8 and 28 17. 17 and 36 18. 24 and 45 In Exercises 19–26, find the value of the variable. (See Example 4.) L N K 19. 20. x In Exercises 5–10, find the value of x. (See Example 2.) 5. 6. 25 X T 20 7 x 24 39 E H x 36 8. F 15 16 G 12 24. 4 x 26. x 3.5 ft 4.6 ft 23 ft 486 Chapter 9 2 16 8 27 5.8 ft x b 25. z 12.8 ft 16 x C 10 10.. 26.3 ft 6 34 30 10 y 5 B 9. R 23. D 25 18 16 A x 22. x Z S 7. 16 21. 12 8 y 4 Q W Y 5 Right Triangles and Trigonometry x ERROR ANALYSIS In Exercises 27 and 28, describe and MATHEMATICAL CONNECTIONS In Exercises 31–34, find correct the error in writing an equation for the given diagram. the value(s) of the variable(s). 27. ✗ x ✗ 6 a+5 z y 33. v z2 = w 18 34. x z ⋅(w + v) y 16 12 y 32 24 x z e g d h 35. REASONING Use the diagram. Decide which f proportions are true. Select all that apply. C D ⋅ d2 = f h A MODELING WITH MATHEMATICS In Exercises 29 and 30, use the diagram. (See Example 5.) B A —=— ○ DC DB DB DA B —=— ○ CB BD CA BA D —=— ○ BC BA C —=— ○ BA CA BA CB DB DA 36. ANALYZING RELATIONSHIPS You are designing 7.2 ft 5.5 ft Ex. 29 b+3 8 32. 12 w 28. 31. 6 ft 9.5 ft Ex. 30 a diamond-shaped kite. You know that AD = 44.8 centimeters, DC = 72 centimeters, and AC = 84.8 centimeters. You want to use a straight —. About how long should it be? Explain crossbar BD your reasoning. 29. You want to determine the height of a monument at D a local park. You use a cardboard square to line up the top and bottom of the monument, as shown at the above left. Your friend measures the vertical distance from the ground to your eye and the horizontal distance from you to the monument. Approximate the height of the monument. 30. Your classmate is standing on the other side of the monument. She has a piece of rope staked at the base of the monument. She extends the rope to the cardboard square she is holding lined up to the top and bottom of the monument. Use the information in the diagram above to approximate the height of the monument. Do you get the same answer as in Exercise 29? Explain your reasoning. A B C 37. ANALYZING RELATIONSHIPS Use the Geometric Mean Theorems (Theorems 9.7 and 9.8) to find AC and BD. B 20 A Section 9.3 15 D C Similar Right Triangles 487 38. HOW DO YOU SEE IT? In which of the following 40. MAKING AN ARGUMENT Your friend claims the triangles does the Geometric Mean (Altitude) Theorem (Theorem 9.7) apply? A ○ geometric mean of 4 and 9 is 6, and then labels the triangle, as shown. Is your friend correct? Explain 9 4 6 your reasoning. B ○ In Exercises 41 and 42, use the given statements to prove the theorem. C ○ Given △ABC is a right triangle. — is drawn to hypotenuse AB —. Altitude CD D ○ 41. PROVING A THEOREM Prove the Geometric Mean (Altitude) Theorem (Theorem 9.7) by showing that CD2 = AD BD. ⋅ 42. PROVING A THEOREM Prove the Geometric Mean 39. PROVING A THEOREM Use the diagram of △ABC. (Leg) Theorem (Theorem 9.8) by showing that CB2 = DB AB and AC2 = AD AB. ⋅ Copy and complete the proof of the Pythagorean Theorem (Theorem 9.1). 43. CRITICAL THINKING Draw a right isosceles triangle Given In △ABC, ∠BCA is a right angle. A f Prove c2 = a2 + b2 b D 1. In △ABC, ∠BCA is a and label the two leg lengths x. Then draw the altitude to the hypotenuse and label its length y. Now, use the Right Triangle Similarity Theorem (Theorem 9.6) to draw the three similar triangles from the image and label any side length that is equal to either x or y. What can you conclude about the relationship between the two smaller triangles? Explain your reasoning. c e a C STATEMENTS B REASONS 1. ________________ right angle. 2. Draw a perpendicular segment (altitude) —. from C to AB 44. THOUGHT PROVOKING The arithmetic mean and 2. Perpendicular geometric mean of two nonnegative numbers x and y are shown. x+y arithmetic mean = — 2 — geometric mean = √ xy Postulate (Postulate 3.2) 3. ce = a2 and cf = b2 3. ________________ 4. ce + b2 = ____ + b2 4. Addition Property Write an inequality that relates these two means. Justify your answer. of Equality 5. ce + cf = a2 + b2 5. ________________ 6. c(e + f ) = a2 + b2 6. ________________ 7. e + f = ____ 7. Segment Addition 8. c 9. ⋅ c2 45. PROVING A THEOREM Prove the Right Triangle Similarity Theorem (Theorem 9.6) by proving three similarity statements. Postulate (Postulate 1.2) c = a2 + b2 = a2 + b2 ⋅ Given △ABC is a right triangle. — is drawn to hypotenuse AB —. Altitude CD 8. ________________ Prove △CBD ∼ △ABC, △ACD ∼ △ABC, △CBD ∼ △ACD 9. Simplify. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Solve the equation for x. (Skills Review Handbook) x 5 46. 13 = — 488 Chapter 9 x 4 47. 29 = — 78 x 48. 9 = — Right Triangles and Trigonometry 115 x 49. 30 = —

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