8-1 The 8-1 and Its Converse 8-1 1. Plan

What You’ll Learn Check Skills You’ll Need GO for Help Skills Handbook, p. 753 Objectives 1 To use the Pythagorean • To use the Pythagorean the lengths of the sides of each . What do you notice? Theorem Theorem 1.A 2 ± 2 ≠ 2 2. A 2 To use the Converse of the • To use the Converse of the 1. 3 4 5 13 in. Pythagorean Theorem Pythagorean Theorem 3 m 5 m 2. 52 ± 122 ≠ 132 5 in. Examples . . . And Why B B C 4 m C 12 in. 1 Pythagorean Triples To find the distance between 2 Using Simplest Radical Form two docks on a lake, as in 3.A 4. A Example 3 3 Real-World Connection 10 yd 4 Is It a ? 6 yd ͙ 4 m 4 2 m 5 Classifying as Acute, B Obtuse, or Right C 8 yd CB4 m 62 ± 82 ≠ 102 42 ± 42 ≠ (42 )2 New Vocabulary " • Math Background

Some mathematical ideas assumed to be true have yet to 1 be proved, such as Goldbach’s 1 The Pythagorean Theorem conjecture: Every even number greater than 2 can be expressed The well-known right triangle relationship called the Pythagorean Theorem is as the sum of two prime numbers. named for Pythagoras, a Greek mathematician who lived in the sixth century B.C. Although several ancient cultures We now know that the Babylonians, Egyptians, and Chinese were aware of this postulated the Pythagorean relationship before its discovery by Pythagoras. Theorem and used it to measure distances, the first proof of it There are many proofs of the Pythagorean Theorem. You will see one proof in was attributed by Euclid to Exercise 48 and others later in the book. Pythagoras. The distance formula is a coordinate form of the Key Concepts Theorem 8-1 Pythagorean Theorem Pythagorean Theorem, which is the foundation of all In a right triangle, the sum of the of . the lengths of the legs is equal to the square c a of the length of the . More Math Background: p. 414C a2 + b2 = c2 b Lesson Planning and Vocabulary Tip Resources Hypotenuse Pythagorean triple A is a set of nonzero whole numbers a, b, and c that satisfy See p. 414E for a list of the 2 + 2 = 2 the equation a b c . Here are some common Pythagorean triples. resources that support this lesson. Legs 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

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If you multiply each number in a Pythagorean triple by the same whole number, Bell Ringer Practice the three numbers that result also form a Pythagorean triple. Check Skills You’ll Need For intervention, direct students to: Lesson 8-1 The Pythagorean Theorem and Its Converse 417 Skills Handbook, p. 753

Special Needs L1 Below Level L2 As you read the Pythagorean Theorem together with Before the lesson, list the squares of whole numbers the class, point out how much easier it is when stated less than 20. Also review how to simplify a radical algebraically rather than in words. Have students expression. practice reciting “a2 + b2 = c2.” learning style: verbal learning style: verbal 417 1 EXAMPLE Pythagorean Triples 2. Teach D E B C 1 A D E B C 2 A D E B C 3 A D E B C 4 A D E # B C Find the length of the hypotenuse of ABC. Do the lengths 5 A D E B B C Test-Taking Tip Guided Instruction of the sides of #ABC form a Pythagorean triple? Memorizing the 2 + 2 = 2 21 common Pythagorean a b c Use the Pythagorean Theorem. EXAMPLE 1 EXAMPLE Teaching Tip triples, like those at 212 +202 = c2 Substitute 21 for a and 20 for b. the bottom of p. 417, + = 2 AC20 Let students know that can help you solve 441 400 c Simplify. Pythagorean triples often problems more 841 = c2 appear on standardized tests. quickly. c = 29 Take the square root.

2 EXAMPLE Error Prevention The length of the hypotenuse is 29. The lengths of the sides, 20, 21, and 29, form a Pythagorean triple because they are whole numbers that satisfy a2 + b2 = c2. Some students may assume that the legs are always the known Quick Check 1 A right triangle has a hypotenuse of length 25 and a leg of length 10. Find the quantities. Point out that c is length of the other leg. Do the lengths of the sides form a Pythagorean triple? always the hypotenuse when 5;21 no applying the formula a2 + b2 = c2 " to a right triangle. In some cases, you will write the length of a side in simplest radical form.

3 EXAMPLE Technology Tip 2 EXAMPLE Using Simplest Radical Form Students may wonder why they are asked to use a calculator in Algebra Find the value of x. Leave your some exercises but not in other answer in simplest radical form (page 390). 20 8 similar exercises. Tell them that a2 + b2 = c2 Pythagorean Theorem real-world applications typically 2 + 2 = 2 require decimal answers. Point out 8 x 20 Substitute. x that radicals are exact, so they are 64 + x2 = 400 Simplify. preferred when exercises are of x2 = 336 Subtract 64 from each side. a purely mathematical nature. x = 336 Take the square root. ! x = 16(21) Simplify. PowerPoint ! x = 4 21 Additional Examples ! Quick Check 2 The hypotenuse of a right triangle has length 12. One leg has length 6. Find the 1 A right triangle has legs of length of the other leg. Leave your answer in simplest radical form. length 16 and 30. Find the length 6 3 of the hypotenuse. Do the lengths " of the sides form a Pythagorean 3 EXAMPLE Real-World Connection triple? 34; yes Gridded Response The Parks Department rents paddle boats at docks near each entrance to the park. To the 2 Find the value of x. Leave your 430 // nearest meter, how far is it to paddle from one dock to answer in simplest radical form. .... the other? 000 1111 12 a2 + b2 = c2 Pythagorean Theorem x 2222 3333 2 + 2 = 2 350 m 4 444 250 350 c Substitute. 5555 185,000 = c2 Simplify. 10 6666 7777 = 185,000 8888 c Take the square root. 2 11 9999 ! " c = 430.11626 Use a calculator. 250 m 3 A baseball diamond is a square It is 430 m from one dock to the other. with 90-ft sides. Home plate and second base are at opposite Quick Check 3 Critical Thinking When you want to know how far you have to paddle a boat, why vertices of the square. About how is an approximate answer more useful than an answer in simplest radical form? far is home plate from second You want to know the nearest whole number value, which may not be base? about 127 ft apparent in a radical expression. 418 Chapter 8 Right Triangles and

Advanced Learners L4 English Language Learners ELL Have students describe how a triangle whose sides Review the term converse, using the Pythagorean form a Pythagorean triple and a triangle whose sides Theorem and its converse as an example. Then have are a multiple of that triple are related. Students students write the Pythagorean Theorem as a should recognize that they are similar triangles. biconditional statement. 418 learning style: verbal learning style: verbal Guided Instruction 12 The Converse of the Pythagorean Theorem Technology Tip Have students use You can use the Converse of the Pythagorean Theorem to determine whether a software to explore and triangle is a right triangle. You will prove Theorem 8-2 in Exercise 58. demonstrate the theorems If c2 a2 + b2, the triangle is 2 2 + 2 Key Concepts Theorem 8-2 Converse of the Pythagorean Theorem obtuse and If c a b , the triangle is acute. Direct students If the square of the length of one side of a triangle is equal to the sum of the to keep a and b constant while squares of the lengths of the other two sides, then the triangle is a right triangle. manipulating c by altering the opposite c.

5 EXAMPLE Error Prevention 4 EXAMPLE Is It a Right Triangle? Remind students that c must be Is this triangle a right triangle? 85 13 the longest side of the triangle 2 c2 0 a2 + b2 84 for the comparison of c and a2 + b2 to give a valid triangle 852 0 132 + 842 Substitute the greatest length for c. classification. Also, students 7225 0 169 + 7056 Simplify. For: Pythagorean Activity should use the Triangle Inequality Use: Interactive Textbook, 8-1 7225 = 7225 ✓ Theorem to check that a + b c so that the side lengths form c2 = a2 + b2, so the triangle is a right triangle. a triangle. Quick Check 4 A triangle has sides of lengths 16, 48, and 50. Is the triangle a right triangle? no PowerPoint Additional Examples You can also use the squares of the lengths of the sides of a triangle to find whether the triangle is acute or obtuse. The following two theorems tell how. 4 Is this triangle a right triangle?

Key Concepts Theorem 8-3 7 m B 4 m If the square of the length of the longest side of a triangle is greater than the sum of the squares of the c a 6 m lengths of the other two sides, the triangle is obtuse. no 2 2 2 A If c . a + b , the triangle is obtuse. C b 5 The numbers represent the Theorem 8-4 lengths of the sides of a triangle. B Classify each triangle as acute, If the square of the length of the longest side of a obtuse, or right. triangle is less than the sum of the squares of the a c a. 15, 20, 25 right lengths of the other two sides, the triangle is acute. b. 10, 15, 20 obtuse A If c2 , a2 + b2, the triangle is acute. C b Resources • Daily Notetaking Guide 8-1 L3 • Daily Notetaking Guide 8-1— EXAMPLE Classifying Triangles as Acute, Obtuse, or Right 5 Adapted Instruction L1 Classify the triangle whose side lengths are 6, 11, and 14 as acute, obtuse, or right. Real-World Connection 142 0 62 + 112 Compare c2 to a2 ± b2. Substitute the greatest length for c. The length to the brace along 0 + each leg is 36 in. The brace is 196 36 121 Closure 26 in. long to guarantee that 196 . 157 the triangle is acute. The of ABC is 20 ft2. Find Since c2 . a2 + b2, the triangle is obtuse. AC and BC. Leave your answer in simplest radical form. Quick Check 5 A triangle has sides of lengths 7, 8, and 9. Classify the triangle by its . acute C Lesson 8-1 The Pythagorean Theorem and Its Converse 419 A 2 ft 8 ft B AC ≠ 2 5 ft; BC ≠ 4 5 ft " "

419 3. Practice EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving Assignment Guide 2 A Practice by Example x Algebra Find the value of x. 1 AB1-17, 27-29, 32, 34-39, Example 1 1. 8 10 2. 25 7 3. 34 48-53 x x (page 418) 16 for 6 2 AB 18-26, 30, 31, 33, GO Help x 24 40-47 30 12 65 8 C Challenge 54-58 4. 5. 6. 15 20 x 97 72 Test Prep 59-64 x Mixed Review 65-73 16 x 17

Homework Quick Check Does each set of numbers form a Pythagorean triple? Explain. To check students’ understanding 7. 4, 5, 6 8. 10, 24, 26 9. 15, 20, 25 of key skills and concepts, go over no; 42 ± 52 u 62 yes; 102 ± 242 ≠ 262 yes; 152 ± 202 ≠ 252 Exercises 16, 18, 30, 36, 50. Example 2 x 2 Algebra Find the value of x. Leave your answer in simplest radical form. Exercises 14, 15 These exercises (page 418) 33 10.41 11." x 12. 15 anticipate the special right triangle " x 4 3 11 relationships in Lesson 8-2. Ask: 4 " x What is the ratio a : b : c in each 7 18 triangle? 1:1: 2 5 " 13.2 89 14.3 2 15. 5 2 Exercises 21–26 In only some " " " x of the exercises do the first two 10 x 6 x lengths represent a and b. Remind 5 students to compare the sum of 16 the squares of the two smaller x 5 lengths with the square of the greatest length. Example 3 16. Home Maintenance A painter leans a 15-ft ladder against a house. The base (page 418) of the ladder is 5 ft from the house. 14.1 ft a. To the nearest tenth of a foot, how high on the house does the ladder reach? b. The ladder in part (a) reaches too high on the house. By how much should the painter move the ladder’s base away from the house to lower the top by 1 ft? about 2.3 ft 17. A walkway forms the diagonal of a square playground. The walkway is 24 m long. To the nearest tenth of a meter, how long is a side of the playground? 17.0 m

GPS Guided Problem Solving L3 Example 4 Is each triangle a right triangle? Explain. (page 419) Enrichment L4 18. 19.25 20. 65 28 8 33 Reteaching L2 20 Adapted Practice L1 24 2 ± 2 u 2 56 PracticeName Class Date L3 no; 8 24 25 19 yes; 332 ± 562 ≠ 652 Practice 8-1 Ratios and Proportions no; 192 ± 202 u 282 1. The Washington Monument in Washington, D.C., is about 556 ft tall. A three-dimensional puzzle of the Washington Monument is 24 in. tall. What is the ratio of the height of the puzzle to the height of the real monument? Example 5 The lengths of the sides of a triangle are given. Classify each triangle as acute,

Closet Find the actual dimensions of each room. Bath (page 419) right, or obtuse. Room Playroom 2. playroom

3. library Master 4. master bedroom Bedroom Library 21. 4, 5, 6 acute 22. 0.3, 0.4, 0.6 obtuse 23. 11, 12, 15 acute 5. bathroom ft 24. 3,2,3 obtuse 25. 30, 40, 50 right 26. 11,7 , 4 acute 16 ؍ .closet Scale: 1 in .6 Algebra If x ≠ 5, which of the following must be true? y 8 ! ! ! y = = = 8 7. 8x 5y 8. 5x 8y 9. x 5 y y x 1 y x = x = = 13 10. 5 8 11. 8 5 12. y 8 x = 10 x = 5 x = 5 13. y 16 14. 2y 4 15. x 2 y 3

Algebra Solve each proportion for x. 420 Chapter 8 Right Triangles and Trigonometry x = 9 6 = x 6 = 2 16. 4 3 17. 11 22 18. x 11 7 = x 2 = x 3 = 8 19. 5 3 20. x 32 21. 11 x x = 3 x 1 1 = 7 5 = 3 22. x 1 2 4 23. x 5 24. x x 1 1

For each rectangle, find the ratio of the longer side to the shorter side.

25. 26. 27. 25 ft 21 in. 18 cm 70 ft 3 ft 12 cm © Pearson Education, Inc. All rights reserved. Complete each of the following. = x = ? a = b a = ? 28. If 3x 8y, then y ?. 29. If 7 13, then b ?.

420 Error Prevention! B Apply Your Skills x 2 Algebra Find the value of x. Leave your answer in simplest radical form. Exercise 28 Students may think 27.26 26 28. 29. the triangle with side lengths x, x x 3 4͙ළ5 45 , and 20 is a right triangle. x " 48 2 Point out that there is no right 10 4 16 angle symbol in the large triangle. 3 8 5 Students must use the Pythagorean " 2 2 " Theorem twice, first to find the 30. Answers may vary. 30. Writing Each year in an ancient land, a large river overflowed its banks, side of the smallest triangle, and Sample: Have three often destroying boundary markers. The royal surveyors used a rope with then to find the hypotenuse of people hold the rope knots at 12 equal intervals to help reconstruct boundaries. Explain how the triangle with base 16. 3 units, 4 units, and a surveyor could use this rope to form a right angle. (Hint: Use the 5 units apart in the Pythagorean triple 3, 4, 5.) Exercise 31 Show students how shape of a triangle. to use Pythagorean triples to 31. Multiple Choice Which triangle is not a right triangle? B check for right triangles. For answer choice A, they can 8͙5 multiply each side by 10 to get 0.6 0.8 4͙5 sides of 6, 8 and 10. They should recognize this as a multiple of a 3, 1.0 7͙5 4, 5 triangle. Similarly, by dividing each side in answer choice B by 5, students can recognize that 45 5͙2 " 36 5 the triangle cannot be a right triangle. 27 GO for Help 5 Exercise 44 Some students may be unfamiliar with the terms 32. Embroidery You want to embroider a square design. x For a guide to solving embroider and embroidery You have an embroidery hoop with a 6 in. diameter. Exercise 32, see p. 424. hoop. Ask a volunteer to bring Find the largest value of x so that the entire square x embroidery materials and an will fit in the hoop. Round to the nearest tenth. 4.2 in. embroidery hoop to class 33. In parallelogram RSTW, RS = 7, ST = 24, and and demonstrate how to use RT = 25. Is RSTW a rectangle? Explain. the hoop. Yes; 72 ± 242 ≠ 252, so lRST is a rt. l. Proof 34. Coordinate Geometry You can use the y Pythagorean Theorem to prove the Distance Q(x2,y2) Formula. Let points P(x1, y1) and Q(x2, y2) be the endpoints of the hypotenuse of a right triangle. a. Write an algebraic expression to complete P(x1,y1) R(x2,y1) 2 ≠ – 2 ± each of the following: 34b. PQ (x2 x1) PR = j and QR = j.»x 2 x …; »y 2 y … O x (y – y )2 2 1 2 1 2 1 b. By the Pythagorean Theorem, 2 = 2 + 2 PQ PR QR . Rewrite this statement substituting the algebraic 35. Answers may vary. expressions you found for PR and QR in part (a). Sample: Using 2 c. Complete the proof by taking the square root of each side of the equation segments of length 1, GO nline that you wrote in part (b). PQ ≠ (x 2 x )2 1 (y 2 y )2 Homework Help " 2 1 2 1 construct the hyp. of 35. Constructions Explain how to construct a ͙3 the right k formed by Visit: PHSchool.com segment of length n , where n is any positive these segments. Using Web Code: aue-0801 ! ͙2 integer, and you are given a segment of length 1. 1 the hyp. found as one 1 (Hint: See the diagram.) See margin. leg and a segment of 1 length 1 as the other leg, construct the hyp. of the Find a third whole number so that the three numbers form a Pythagorean triple. k formed by those legs. Continue this process 36. 20, 21 29 37. 14, 48 50 38. 13, 85 84 39. 12, 37 35 until constructing a hypotenuse of length n. Lesson 8-1 The Pythagorean Theorem and Its Converse 421 "

421 r 5 a q 5 b Find integers j and k so that (a) the two given integers and j represent the lengths 4. Assess & Reteach 48. a c and b c. So 2 2 of the sides of an acute triangle and (b) the two given integers and k represent the a = rc and b = qc. 40–47. Answers may vary. Samples 2 2 lengths of the sides of an obtuse triangle. PowerPoint a + b = rc + qc = are given. (r + q)c = c2 40. 4, 5 6; 7 41. 2, 4 4; 5 42. 6, 9 8; 11 43. 5, 10 11; 12 Lesson Quiz 44. 6, 7 8; 10 45. 9, 12 14; 16 46. 8, 17 18; 19 47. 9, 40 39; 42 1. Find the value of x. Proof 48. Prove the Pythagorean Theorem. C Given: #ABC is a right triangle x b a 9 Prove: a2 + b2 = c2 q r AB (Hint: Begin with proportions suggested D 12 by Theorem 7-3 or its corollaries.) c 15 49. Astronomy The Hubble Space Telescope 2. Find the value of x. Leave your x is orbiting Earth 600 km above Earth’s 600 km answer in simplest radical form. surface. Earth’s radius is about 6370 km. Use the Pythagorean Theorem to find 6370 km the distance x from the telescope to Earth’s 14 horizon. Round your answer to the nearest 8 ten kilometers. 2830 km not to scale x The figures below are drawn on centimeter grid paper. 2 33 " Find the perimeter of each shaded figure to the nearest tenth. 3. The town of Elena is 24 mi 17.9 cm north and 8 mi west of Real-World Connection 50. 12 cm 51. 52. Holberg. A train runs on GPS Research by Edwin Hubble 12.5 cm a straight track between the (1889–1953), here guiding a two towns. How many miles telescope in 1923, led to the does it cover? Round your Big Bang Theory of the formation of the universe. answer to the nearest whole number. 25 mi 2 - 53a. Answers may vary. 53. a. The ancient Greek philosopher Plato used the expressions 2n, n 1, and 2 4. The lengths of the sides of a Sample: n ≠ 6; 12, 35, 37 n + 1 to produce Pythagorean triples. Choose any integer greater than 1. triangle are 5 cm, 8 cm, and 10 Substitute for n and evaluate the three expressions. 122 ± 352 ≠ 372 cm. Is it acute, right, or obtuse? b. Verify that your answers to part (a) form a Pythagorean triple. obtuse C Challenge c. Show that, in general,(2n)2 1 (n2 2 1)2 5 (n2 1 1)2 for any n.

53c. (2n)2 + (n2 – 1)2 54. Geometry in 3 Dimensions The box at the right is a D rectangular solid. =4n2 + n4 – 2n2 + 1 d2 2 in. Alternative Assessment a. Use #ABC to find the length d of the diagonal of = n4 + 2n2 + 1 1 B the base. 5 in. 2 2 Have students use the Pythagorean =(n + 1) # 4 in. b. Use ABD to find the length d2 of the diagonal of d1 Theorem to find the length of the the box. 29 " A 3 in. C diagonal of their notebook paper c. You can generalize the steps in parts (a) and (b). and explain in writing how the 2 + 2 = 2 ≠ 2 2 2 Use the facts that AC BC d1 and d2 BD 1 AC 1 BC Pythagorean Theorem was used. 2 + 2 = 2 " d1 BD d2 to write a one-step formula to find d2. Then have them measure the d. Use the formula you wrote to find the length of the longest fishing pole you diagonal to confirm the length can pack in a box with dimensions 18 in., 24 in., and 16 in. 34 in. found using the Pythagorean Theorem. z Geometry in 3 Dimensions Points P(x1, y1, z1) and Q(x2, y2, z2) at the left are points in a three-dimensional coordinate system. Use the following formula to find P Q PQ. Leave your answer in simplest radical form. O y d = (x 2 x )2 1 (y 2 y )2 1 (z 2 z )2 " 2 1 2 1 2 1 x 55. P(0, 0, 0), Q(1, 2, 3) 56. P(0, 0, 0), Q(-3, 4, -6) 57. P(-1, 3, 5), Q(2, 1, 7) 14 61 17 " " " 422 Chapter 8 Right Triangles and Trigonometry

422 Test Prep Proof 58. Use the plan and write a proof of Theorem 8-2, the Converse of the Pythagorean Theorem. A sheet of blank grids is available Given: #ABC with sides of length a, b, and c B in the Test-Taking Strategies with where a2 + b2 = c2 Transparencies booklet. Give this a c Prove: #ABC is a right triangle. sheet to students for practice with filling in the grids. Plan: Draw a right triangle (not #ABC) CAb with legs of lengths a and b. Label the Resources 2 + 2 = 2 hypotenuse x. By the Pythagorean Theorem, a b x . Use substitution to For additional practice with a # compare the lengths of the sides of your triangle and ABC. Then prove the variety of test item formats: triangles congruent. See margin. • Standardized Test Prep, p. 465 • Test-Taking Strategies, p. 460 • Test-Taking Strategies with Transparencies Test Prep

Gridded Response 59. The lengths of the legs of a right triangle are 17 m and 20 m. To the nearest tenth of a meter, what is the length of the hypotenuse? 26.2 60. The hypotenuse of a right triangle is 34 ft. One leg is 16 ft. Find the length of the other leg in feet. 30 58. Draw right kFDE with 61. What whole number forms a Pythagorean triple with 40 and 41? 9 legs DE of length a and 62. The two shorter sides of an obtuse triangle are 20 and 30. What is the least EF of length b, and hyp. whole number length possible for the third side? 37 of length x. Then a2 ± b2 2 63. Each leg of an isosceles right triangle has measure 10 cm. To the nearest ≠ x by the Pythagorean tenth of a centimeter, what is the length of the hypotenuse? 14.1 Thm. We are given kABC with sides of 64. The legs of a right triangle have lengths 3 and 4. What is the length, to the length a, b, c and nearest tenth, of the to the hypotenuse? 2.7 a2 ± b2 ≠ c2. By subst., c2 ≠ x2, so c ≠ x. Since all side lengths of kABC k Mixed Review and FDE are the same, Mixed Review kABC OkFDE by SSS. lC OlE by CPCTC, so Lesson 7-5 For the figure at the right, complete the proportion. mlC ≠ 90. Therefore, for x7 10 y k k GO 10 y j y ABC is a right . Help 65. j 5 18 15 66. x 5 18 x – 7 x 15 18 67. Find the values of x and y. x = 21, y = 12

) Lesson 5-2 In the second figure,PS bisects lRPT. Solve for R each variable. Then find RS. P = + = - = 7 = 7 68. RS 2x 19, ST 7x 16; x , RS 7; 33 S 69. RS = 2(7y - 11), ST = 5y + 5; y = 7, RS = 7 3; 20 T Lesson 4-1 #PQR O#STV. Solve for each variable. 70. m&P = 4w + 5, m&S = 6w - 15 10 71. RQ = 10y - 6, VT = 5y + 9 3 72. m&T = 2x - 40, m&Q = x + 10 50 73. PR = 2z + 3, SV = 4z - 11 7

lesson quiz, PHSchool.com, Web Code: aua-0801 Lesson 8-1 The Pythagorean Theorem and Its Converse 423

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