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The Pythagorean Theorem and Its Converse 417 Skills Handbook, P 8-1 The Pythagorean Theorem 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 Square the lengths of the sides of each triangle. 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 Right Triangle? 6 yd ͙ 4 m 4 2 m 5 Classifying Triangles as Acute, B Obtuse, or Right C 8 yd CB4 m 62 ± 82 ≠ 102 42 ± 42 ≠ (42 )2 New Vocabulary Pythagorean triple " • 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 squares of trigonometric functions. the lengths of the legs is equal to the square c a of the length of the hypotenuse. 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 PowerPoint 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 Trigonometry 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 geometry 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 Ͼ a2 + b2, the triangle is 2 Ͻ 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 angle 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 Ͼ 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.
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