9.1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.6.D G.9.B The Pythagorean Theorem Essential Question How can you prove the Pythagorean Theorem? Proving the Pythagorean Theorem without Words Work with a partner. a b a. Draw and cut out a right triangle with legs a and b, and hypotenuse c. a c b b. Make three copies of your right triangle. Arrange all four triangles to form a large square, as shown. c c c. Find the area of the large square in terms of a, b, and c by summing the areas of the triangles and the small square. b d. Copy the large square. Divide it into two smaller squares and two equally-sized rectangles, as shown. a e. Find the area of the large square in terms of a and b by summing the areas of the rectangles and the smaller squares. f. Compare your answers to parts (c) and (e). Explain how this proves the Pythagorean Theorem. b c a a b b b a a a b Proving the Pythagorean Theorem Work with a partner. a. Draw a right triangle with legs a and b, and hypotenuse c, as shown. Draw —. Label the lengths, as shown. the altitude from C to AB C REASONING To be proficient in math, you need to know and flexibly use different properties of operations and objects. b h c−d A a d c D B b. Explain why △ABC, △ACD, and △CBD are similar. c. Write a two-column proof using the similar triangles in part (b) to prove that a2 + b2 = c2. Communicate Your Answer 3. How can you prove the Pythagorean Theorem? 4. Use the Internet or some other resource to find a way to prove the Pythagorean Theorem that is different from Explorations 1 and 2. Section 9.1 The Pythagorean Theorem 467 9.1 Lesson What You Will Learn Use the Pythagorean Theorem. Use the Converse of the Pythagorean Theorem. Core Vocabul Vocabulary larry Classify triangles. Pythagorean triple, p. 468 Using the Pythagorean Theorem Previous right triangle legs of a right triangle hypotenuse One of the most famous theorems in mathematics is the Pythagorean Theorem, named for the ancient Greek mathematician Pythagoras. This theorem describes the relationship between the side lengths of a right triangle. Theorem Theorem 9.1 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. c a b c 2 = a2 + b2 Proof Explorations 1 and 2, p. 467; Ex. 39, p. 488 A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c2 = a2 + b2. STUDY TIP You may find it helpful to memorize the basic Pythagorean triples, shown in bold, for standardized tests. Core Concept Common Pythagorean Triples and Some of Their Multiples 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 6, 8, 10 10, 24, 26 16, 30, 34 14, 48, 50 9, 12, 15 15, 36, 39 24, 45, 51 21, 72, 75 3x, 4x, 5x 5x, 12x, 13x 8x, 15x, 17x 7x, 24x, 25x The most common Pythagorean triples are in bold. The other triples are the result of multiplying each integer in a bold-faced triple by the same factor. Using the Pythagorean Theorem Find the value of x. Then tell whether the side lengths form a Pythagorean triple. 5 SOLUTION c2 = a2 + b2 Pythagorean Theorem x2 = 52 + 122 Substitute. x2 = 25 + 144 Multiply. x2 = 169 Add. x = 13 12 x Find the positive square root. The value of x is 13. Because the side lengths 5, 12, and 13 are integers that satisfy the equation c2 = a2 + b2, they form a Pythagorean triple. 468 Chapter 9 Right Triangles and Trigonometry Using the Pythagorean Theorem Find the value of x. Then tell whether the side lengths form a Pythagorean triple. x 7 SOLUTION c2 14 = a2 + b2 Pythagorean Theorem 142 = 72 + x2 Substitute. 196 = 49 + x2 Multiply. 147 = x2 Subtract 49 from each side. — √ 147 = x — Find the positive square root. — √49 • √3 = x Product Property of Radicals — 7√ 3 = x Simplify. — — The value of x is 7 √ 3 . Because 7 √3 is not an integer, the side lengths do not form a Pythagorean triple. Solving a Real-Life Problem The skyscrapers shown are connected by a skywalk with support beams. Use the Pythagorean Theorem to approximate the length of each support beam. 23.26 m 47.57 m 47.57 m x x support beams SOLUTION Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. x2 = (23.26)2 + (47.57)2 —— Pythagorean Theorem x = √ (23.26)2 + (47.57)2 Find the positive square root. x ≈ 52.95 Use a calculator to approximate. The length of each support beam is about 52.95 meters. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the value of x. Then tell whether the side lengths form a Pythagorean triple. 1. 2. x 4 x 3 5 6 3. An anemometer is a device used to measure wind speed. The anemometer shown is attached to the top of a pole. Support wires are attached to the pole 5 feet above the ground. Each support wire is 6 feet long. How far from the base of the pole is each wire attached to the ground? 6 ft 5 ft d Section 9.1 The Pythagorean Theorem 469 Using the Converse of the Pythagorean Theorem The converse of the Pythagorean Theorem is also true. You can use it to determine whether a triangle with given side lengths is a right triangle. Theorem Theorem 9.2 Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. B c a C If c2 = a2 + b2, then △ABC is a right triangle. b A Proof Ex. 39, p. 474 Verifying Right Triangles Tell whether each triangle is a right triangle. a. 8 4 95 15 36 113 SELECTING TOOLS Use a calculator to determine that — √113 ≈ 10.630 is the length of the longest side in part (a). b. 7 SOLUTION Let c represent the length of the longest side of the triangle. Check to see whether the side lengths satisfy the equation c2 = a2 + b2. — 2 ? a. ( √ 113 ) = 72 + 82 ? 113 = 49 + 64 113 = 113 ✓ The triangle is a right triangle. 2 ? ( 4√— 95 ) = 152 + 362 b. 42 ⋅ ( √95 ) = 15 + 36 ? 16 ⋅ 95 = 225 + 1296 — 2 ? 2 1520 ≠ 1521 2 ✗ The triangle is not a right triangle. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the triangle is a right triangle. 4. 9 3 34 15 470 Chapter 9 Right Triangles and Trigonometry 5. 22 14 26 Classifying Triangles The Converse of the Pythagorean Theorem is used to determine whether a triangle is a right triangle. You can use the theorem below to determine whether a triangle is acute or obtuse. Theorem Theorem 9.3 Pythagorean Inequalities Theorem For any △ABC, where c is the length of the longest side, the following statements are true. If c2 < a2 + b2, then △ABC is acute. If c2 > a2 + b2, then △ABC is obtuse. A A c b a C c2 < a2 c b + a C B c2 b2 > a2 + B b2 Proof Exs. 42 and 43, p. 474 REMEMBER The Triangle Inequality Theorem (Theorem 6.11) on page 343 states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Classifying Triangles Verify that segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle. Is the triangle acute, right, or obtuse? SOLUTION Step 1 Use the Triangle Inequality Theorem (Theorem 6.11) to verify that the segments form a triangle. ? 4.3 + 5.2 > 6.1 9.5 > 6.1 ? ✓ 4.3 + 6.1 > 5.2 10.4 > 5.2 ? ✓ 5.2 + 6.1 > 4.3 11.3 > 4.3 ✓ The segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle. Step 2 Classify the triangle by comparing the square of the length of the longest side with the sum of the squares of the lengths of the other two sides. c2 6.12 37.21 a2 + b2 Compare c2 with a2 + b2. 4.32 + 5.22 Substitute. 18.49 + 27.04 Simplify. c 2 is less than a2 + b2. 37.21 < 45.53 The segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form an acute triangle. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 6. Verify that segments with lengths of 3, 4, and 6 form a triangle. Is the triangle acute, right, or obtuse? 7. Verify that segments with lengths of 2.1, 2.8, and 3.5 form a triangle. Is the triangle acute, right, or obtuse? Section 9.1 The Pythagorean Theorem 471 Exercises 9.1 Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. VOCABULARY What is a Pythagorean triple? 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. Find the length of the longest side. Find the length of the hypotenuse. 3 Find the length of the longest leg. 4 Find the length of the side opposite the right angle. Monitoring Progress and Modeling with Mathematics In Exercises 3–6, find the value of x. Then tell whether the side lengths form a Pythagorean triple. (See Example 1.) 3. 30 16 x 10. 50 7 x 48 4. 7 9. 9 x x ERROR ANALYSIS In Exercises 11 and 12, describe and 11 5. 6. correct the error in using the Pythagorean Theorem (Theorem 9.1). 6 x 11. 40 4 x ✗ x 7 9 c2 = a2 + b2 x 2 = 72 + 242 x 2 = (7 + 24)2 x 2 = 312 x = 31 24 In Exercises 7–10, find the value of x. Then tell whether the side lengths form a Pythagorean triple. (See Example 2.) 7. 8 472 x 17 Chapter 9 8. 24 12. 9 x Right Triangles and Trigonometry ✗ 26 x 10 c2= a2 + b2 x 2 = 102 + 262 x 2 = 100 + 676 x 2 = 776 — x = √776 x ≈ 27.9 13. MODELING WITH MATHEMATICS The fire escape In Exercises 21–28, verify that the segment lengths form a triangle. Is the triangle acute, right, or obtuse? (See Example 5.) forms a right triangle, as shown. Use the Pythagorean Theorem (Theorem 9.1) to approximate the distance between the two platforms. (See Example 3.) 16.7 ft x 21. 10, 11, and 14 22. 6, 8, and 10 23. 12, 16, and 20 24. 15, 20, and 36 25. 5.3, 6.7, and 7.8 26. 4.1, 8.2, and 12.2 — 27. 24, 30, and 6√ 43 — 28. 10, 15, and 5√ 13 29. MODELING WITH MATHEMATICS In baseball, the lengths of the paths between consecutive bases are 90 feet, and the paths form right angles. The player on first base tries to steal second base. How far does the ball need to travel from home plate to second base to get the player out? 8.9 ft 14. MODELING WITH MATHEMATICS The backboard of the basketball hoop forms a right triangle with the supporting rods, as shown. Use the Pythagorean Theorem (Theorem 9.1) to approximate the distance between the rods where they meet the backboard. x 30. REASONING You are making a canvas frame for a painting using stretcher bars. The rectangular painting will be 10 inches long and 8 inches wide. Using a ruler, how can you be certain that the corners of the frame are 90°? 13.4 in. 9.8 in. In Exercises 15 –20, tell whether the triangle is a right triangle. (See Example 4.) 15. In Exercises 31–34, find the area of the isosceles triangle. 16. 21.2 97 65 11.4 31. 32. 17 m 23 h 20 ft 19. 2 26 1 5 10 6 16 m 18. 4 19 14 20 ft 32 ft 72 17. h 17 m 34. 33. 10 cm 10 cm h 12 cm 20. 50 m h 50 m 89 3 5 80 28 m 39 Section 9.1 The Pythagorean Theorem 473 35. ANALYZING RELATIONSHIPS Justify the Distance 41. MAKING AN ARGUMENT Your friend claims 72 and Formula using the Pythagorean Theorem (Thm. 9.1). 75 cannot be part of a Pythagorean triple because 722 + 752 does not equal a positive integer squared. Is your friend correct? Explain your reasoning. 36. HOW DO YOU SEE IT? How do you know ∠C is a right angle without using the Pythagorean Theorem (Theorem 9.1)? 42. PROVING A THEOREM Copy and complete the proof of the Pythagorean Inequalities Theorem (Theorem 9.3) when c2 < a2 + b2. C Given In △ABC, c2 < a2 + b2, where c is the length of the longest side. △PQR has side lengths a, b, and x, where x is the length of the hypotenuse, and ∠R is a right angle. 8 6 A B 10 Prove △ABC is an acute triangle. 37. PROBLEM SOLVING You are making ng a kite and need to figure out how much binding to buy. You need the binding for 12 in. the perimeter of the kite. The binding comes in packages of two yards. How many packages should you buy? P A 15 in. c 12 in. B x b a C Q b a R 20 in. STATEMENTS REASONS 1. In △ABC, c2 < a2 + b2, where 1. _______________ c is the length of the longest side. △PQR has side lengths a, b, and x, where x is the length of the hypotenuse, and ∠R is a right angle. 38. PROVING A THEOREM Use the Pythagorean Theorem (Theorem 9.1) to prove the Hypotenuse-Leg (HL) Congruence Theorem (Theorem 5.9). 2. a2 + b2 = x2 3. c2 < x2 2. _______________ 4. c < x 4. Take the positive 3. _______________ square root of each side. 39. PROVING A THEOREM Prove the Converse of the Pythagorean Theorem (Theorem 9.2). (Hint: Draw △ABC with side lengths a, b, and c, where c is the length of the longest side. Then draw a right triangle with side lengths a, b, and x, where x is the length of the hypotenuse. Compare lengths c and x.) and n, where m > n. Do the following expressions produce a Pythagorean triple? If yes, prove your answer. If no, give a counterexample. 2mn, n2, m2 + 5. _______________ 6. m∠C < m∠R 6. Converse of the Hinge Theorem (Theorem 6.13) 40. THOUGHT PROVOKING Consider two integers m m2 − 5. m∠R = 90° 7. m∠C < 90° 7. _______________ 8. ∠C is an acute angle. 8. _______________ 9. △ABC is an acute triangle. 9. _______________ 43. PROVING A THEOREM Prove the Pythagorean Inequalities Theorem (Theorem 9.3) when c2 > a2 + b2. (Hint: Look back at Exercise 42.) n2 Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Simplify the expression by rationalizing the denominator. (Skills Review Handbook) 7 44. — — √2 474 Chapter 9 14 45. — — √3 8 46. — — √2 Right Triangles and Trigonometry 12 47. — — √3

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