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Warm Up Estimate the value of each to the nearest tenth. 1. √30 2. √14 3. √55 4. √48 5.4 or 5.5 3.7 or 3.8 7.4 or 7.5 6.9

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**The Pythagorean Theorem**

Learn to use the Pythagorean Theorem and its converse to solve problems. Course 3

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Vocabulary Pythagorean Theorem leg hypotenuse

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**Pythagoras was born on the Aegean island of Samos**

Pythagoras was born on the Aegean island of Samos. He is best known for the Pythagorean Theorem, which relates the side lengths of a right triangle.

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**Directions: Find the leg of the hypotenuse to the nearest tenth.**

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**Example 1 a2 + b2 = c2 Pythagorean Theorem Substitute for a and b.**

Find the length of each leg first. a2 + b2 = c2 Pythagorean Theorem = c2 Substitute for a and b. = c2 Simplify powers. 225 = c Solve for c; c = c2. 15 = c

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Example 2 c 4 5 a2 + b2 = c2 Pythagorean Theorem = c2 Substitute for a and b. = c2 Simplify powers. 41 = c2 41 = c Solve for c; c = c2. 6.4 c

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**Example 3 Pythagorean Theorem a2 + b2 = c2 Substitute for a and b.**

5 7 c Pythagorean Theorem a2 + b2 = c2 Substitute for a and b. = c2 Simplify powers. = c2 Solve for c; c = c2. 74 = c 8.6 c

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Example 4 a2 + b2 = c2 Pythagorean Theorem 25 b 72 + b2 = 252 Substitute for a and c. 49 + b2 = 625 Simplify powers. – –49 Subtract 49 from each side. b2 = 576 7 Find the positive square root b = 24

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Example 5 a2 + b2 = c2 Pythagorean Theorem 12 b 42 + b2 = 122 Substitute for a and c. 16 + b2 = 144 Simplify powers. – –16 Subtract 16 from both sides. 4 b2 = 128 Find the positive square root. b 11.3

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Directions: Determine whether or not the given numbers are possible measures for the sides of a right triangle.

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Example 6 a = 3, b=4 and c = 5

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Example 7

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Lesson Summary Use the figure for Problems 1 and 2. 1. Find the height h of the triangle. 8 m 2. Find the length of side c to the nearest meter. 10 m c h 12 m 3. An escalator in a shopping mall is 40 ft long and 32 ft tall. What distance does the escalator carry shoppers? 6 m 9 m 2624 ≈ 51 ft

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