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The Pythagorean Theorem Section 8-1

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Use the Pythagorean Theorem.

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Key Vocabulary Leg Hypotenuse Pythagorean Theorem Pythagorean Triple

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Parts of a Right Triangle Longest side is the hypotenuse, side c (opposite the 90 o angle). The other two sides are the legs, sides a and b. Pythagoras developed a formula for finding the length of the sides of any right triangle.

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Theorem 4.7 - The Pythagorean Theorem The 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. Example: (hypotenuse) 2 =(leg) 2 +(leg) 2

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Find the length of the hypotenuse. ANSWER The length of the hypotenuse is 13. SOLUTION (hypotenuse) 2 = (leg) 2 + (leg) 2 Pythagorean Theorem c 2 = 5 2 + 12 2 Substitute. c 2 = 25 + 144 Multiply. c 2 = 169 Add. c = 13 Solve for c. Find the positive square root. c 2 = 169 Example 1

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Find the unknown side length. ANSWER The side length is about 12.1. SOLUTION (hypotenuse) 2 = (leg) 2 + (leg) 2 Pythagorean Theorem 14 2 = 7 2 + b 2 Substitute. 196 = 49 + b 2 Multiply. 147 = b 2 Simplify. 12.1 b Approximate with a calculator. 196 49 = 49 + b 2 49 Subtract 49 from each side. Find the positive square root. 147 = b2b2 Example 2

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ANSWER 8 8 about 10.6 Find the unknown side length. 1. 2. 3. Your Turn:

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Example 3a A. Find x. The side opposite the right angle is the hypotenuse, so c = x. a 2 + b 2 = c 2 Pythagorean Theorem 4 2 + 7 2 = c 2 a = 4 and b = 7

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Example 3a 65= c 2 Simplify. Take the positive square root of each side. Answer:

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Example 3b B. Find x. The hypotenuse is 12, so c = 12. a 2 + b 2 = c 2 Pythagorean Theorem x 2 + 8 2 = 12 2 b = 8 and c = 12

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Example 3b Take the positive square root of each side and simplify. x 2 + 64= 144Simplify. x 2 = 80Subtract 64 from each side. Answer:

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Your Turn: A. Find x. A. B. C. D.

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Your Turn: B. Find x. A. B. C. D.

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More Examples: 1) A=8, C =10, Find B 2) A=15, C=17, Find B 3) B =10, C=26, Find A 4) A=15, B=20, Find C 5) A =12, C=16, Find B 6) B =5, C=10, Find A 7) A =6, B =8, Find C 8) A=11, C=21, Find B A B C B = 6 B = 8 A = 24 C = 25 B = 10.6 A = 8.7 C = 10 B = 17.9

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Pythagorean Triples Pythagorean Triples Three whole numbers that work in the Pythagorean formulas are called Pythagorean Triples. The largest number in each triple is the length of the hypotenuse. Pythagorean triples are not the only possible side lengths for a right triangle. They give the triangles where all the lengths are whole numbers, but the side lengths could be any real numbers.

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Pythagorean Multiples If you multiply the lengths of all three sides of any right triangle by the same number, then the resulting triangle is a right triangle. In other words, if a 2 + b 2 = c 2, then (an) 2 + (bn) 2 = (cn) 2. Therefore, additional pythagorean triples can be found by multiplying each number in a known triple by the same factor.

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Pythagorean Triples Multiples

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Primitive Pythagorean Triples primitive Pythagorean triple A set of Pythagorean triples is considered a primitive Pythagorean triple if the numbers are relatively prime; that is, if they have no common factors other than 1. You need know the first 4 primitives: 3-4-5, 5-12-13, 7-24-25, 8-15-17. 3-4-55-12-137-24-258-15-17 9-40-4111-60-6112-35-3713-84-85 16-63-6520-21-2928-45-5333-56-65 36-77-8539-80-8948-55-7365-72-97

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Example 4 Use a Pythagorean triple to find x. Explain your reasoning.

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Example 4 Notice that 24 and 26 are multiples of 2 : 24 = 2 12 and 26 = 2 13. Since 5, 12, 13 is a Pythagorean triple, the missing leg length x is 2 5 or 10. Answer:x = 10 Check:24 2 + 10 2 = 26 2 Pythagorean Theorem ? 676 = 676Simplify.

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Your Turn: A.10 B.15 C.18 D.24 Use a Pythagorean triple to find x.

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More Practice Use Pythagorean Triples to find each missing side length. Primitive: 5-12-13 X=26 Primitive: 7-24-25 X=50 Primitive: 3-4-5 X=15