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Quiz 1. Find the perimeter of the figure. 2. Find the area of the figure. 12 ft 4 ft5 ft 3. The perimeter of the triangle 4. The perimeter of the combined figure 5. The area of the trapezoid

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Pre-Algebra 6.3 The Pythagorean Theorem

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Graph the figures with the given verticals and find the area. 1. (–1, –1), (–1, 3), (6, –1) 2. (2, 1), (8, 1), (6, –3) 3. (3, –2), (15, –2), (14, 6), (4, 6) 14 units 2 12 units 2 88 units 2 Warm Up

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

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Pythagorean Theorem leg hypotenuse Vocabulary

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a 2 + b 2 = c 2

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4 5 c 6.40 c A. Pythagorean Theorem Substitute for a and b. a 2 + b 2 = c 2 4 2 + 5 2 = c 2 16 + 25 = c 2 41 = c Simplify powers. Solve for c; c = c 2. Find the length of the hypotenuse. 41 = c 2 Example: Find the the Length of a Hypotenuse

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15 = c B. Pythagorean Theorem Substitute for a and b. a 2 + b 2 = c 2 9 2 + 12 2 = c 2 81 + 141 = c 2 225 = c Simplify powers. Solve for c; c = c 2. Find the length of the hypotenuse. triangle with coordinates (1, –2), (1, 7), and (13, –2) Example: Find the the Length of a Hypotenuse

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5 7 c A. Find the length of the hypotenuse. 8.60 c Pythagorean Theorem Substitute for a and b. a 2 + b 2 = c 2 5 2 + 7 2 = c 2 25 + 49 = c 2 74 = c Simplify powers. Solve for c; c = c 2. Try This

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B. triangle with coordinates (–2, –2), (–2, 4), and (3, –2) x y The points form a right triangle. (–2, –2) (–2, 4) (3, –2) Find the length of the hypotenuse. 7.81 c Pythagorean Theorem a 2 + b 2 = c 2 6 2 + 5 2 = c 2 36 + 25 = c 2 61 = c Simplify powers. Solve for c; c = c 2. Substitute for a and b. Try This

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25 7 b 576 = 24 b = 24 a 2 + b 2 = c 2 7 2 + b 2 = 25 2 49 + b 2 = 625 –49 b 2 = 576 Solve for the unknown side in the right triangle. Pythagorean Theorem Substitute for a and c. Simplify powers. Example: Finding the Length of a Leg in a Right Triangle

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b 11.31 12 4 b a 2 + b 2 = c 2 4 2 + b 2 = 12 2 16 + b 2 = 144 –16 b 2 = 128 128 11.31 Solve for the unknown side in the right triangle. Pythagorean Theorem Substitute for a and c. Simplify powers. Try This

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a 66 44 a 2 + b 2 = c 2 a 2 + 4 2 = 6 2 a 2 + 16 = 36 a 2 = 20 a = 20 units ≈ 4.47 units Find the square root of both sides. Substitute for b and c. Pythagorean Theorem A = hb = (8)( 20) = 4 20 units 2 17.89 units 2 1212 1212 Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle. Example: Using the Pythagorean Theorem to Find Area

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a 2 + b 2 = c 2 a 2 + 2 2 = 5 2 a 2 + 4 = 25 a 2 = 21 a = 21 units ≈ 4.58 units Find the square root of both sides. Substitute for b and c. Pythagorean Theorem A = hb = (4)( 21) = 2 21 units 2 4.58 units 2 1212 1212 Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle. a 55 22 Try This

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1. Find the height of the triangle. 2. Find the length of side c to the nearest meter. 3. Find the area of the largest triangle. 4. One leg of a right triangle is 48 units long, and the hypotenuse is 50 units long. How long is the other leg? 8m 12m 60m 2 14 units h c 10 m 6 m9 m Use the figure for Problems 1-3. Lesson Quiz

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