Pythagorean Theorem 6 3 The Pythagorean Theorem Warm

Pythagorean Theorem 6 -3 The Pythagorean Theorem Warm Up Problem of the Day Lesson Presentation Pre-Algebra

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

6 -3 The Pythagorean Theorem Problem of the Day A side of a square A is 5 times the length of square B. How many times as great is the area of square A than the area of square B? 25 Pre-Algebra

6 -3 The Pythagorean Theorem Learn to use the Pythagorean Theorem and its converse to solve problems. Pre-Algebra

6 -3 The Pythagorean Theorem Vocabulary Pythagorean Theorem leg hypotenuse Pre-Algebra

6 -3 The Pythagorean Theorem a 2 + b 2 = c 2 Pre-Algebra

6 -3 The Pythagorean Theorem Additional Example 1 A: Find the Length of a Hypotenuse Find the length of the hypotenuse. c A. 4 5 a 2 + b 2 = c 2 42 + 5 2 = c 2 16 + 25 = c 2 41 = c 6. 40 c Pre-Algebra Pythagorean Theorem Substitute for a and b. Simplify powers. Solve for c; c = c 2.

6 -3 The Pythagorean Theorem Additional Example 1 B: Find the Length of a Hypotenuse Find the length of the hypotenuse. B. triangle with coordinates (1, – 2), (1, 7), and (13, – 2) a 2 + b 2 92 + 122 81 + 141 225 15 Pre-Algebra = = = c 2 c 2 c c Pythagorean Theorem Substitute for a and b. Simplify powers. Solve for c; c = c 2.

6 -3 The Pythagorean Theorem Try This: Example 1 A Find the length of the hypotenuse. A. c 5 7 a 2 + b 2 = c 2 52 + 7 2 = c 2 25 + 49 = c 2 74 = c 8. 60 c Pre-Algebra Pythagorean Theorem Substitute for a and b. Simplify powers. Solve for c; c = c 2.

6 -3 The Pythagorean Theorem Try This: Example 1 B Find the length of the hypotenuse. B. triangle with coordinates (– 2, – 2), (– 2, 4), and (3, – 2) (– 2, 4) y The points form a right triangle. x (– 2, – 2) Pre-Algebra (3, – 2) a 2 + b 2 = c 2 62 + 5 2 = c 2 36 + 25 = c 2 61 = c 7. 81 c Pythagorean Theorem Substitute for a and b. Simplify powers. Solve for c; c = c 2.

6 -3 The Pythagorean Theorem Additional Example: 2 Finding the Length of a Leg in a Right Triangle Solve for the unknown side in the right triangle. 25 7 a 2 + b 2 = c 2 b 72 + b 2 = 252 49 + b 2 = 625 – 49 b 2 = 576 b = 24 Pre-Algebra Pythagorean Theorem Substitute for a and c. Simplify powers. 576 = 24

6 -3 The Pythagorean Theorem Try This: Example 2 Solve for the unknown side in the right triangle. 12 b 4 a 2 + b 2 = c 2 42 + b 2 = 122 16 + b 2 = 144 – 16 b 2 = 128 b 11. 31 Pre-Algebra Pythagorean Theorem Substitute for a and c. Simplify powers. 128 11. 31

6 -3 The Pythagorean Theorem Additional Example 3: Using the Pythagorean Theorem to Find Area Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle. 6 a 4 6 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 4 1 A = 1 hb = (8)( 2 2 Pre-Algebra Pythagorean Theorem Substitute for b and c. Find the square root of both sides. 20) = 4 20 units 2 17. 89 units 2

6 -3 The Pythagorean Theorem Try This: Example 3 Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle. 5 a 2 + b 2 = c 2 a 2 + 2 2 = 5 2 a 2 + 4 = 25 2 1 A = 1 hb = (4)( 2 2 Pre-Algebra Pythagorean Theorem Substitute for b and c. a 2 = 21 a = 21 units ≈ 4. 58 units Find the square root of both sides. 21) = 2 21 units 2 4. 58 units 2

6 -3 The Pythagorean Theorem Lesson Quiz Use the figure for Problems 1 -3. 1. Find the height of the triangle. 8 m 2. Find the length of side c to the nearest meter. 12 m 3. Find the area of the largest triangle. 60 m 2 c 10 m h 6 m 9 m 4. One leg of a right triangle is 48 units long, and the hypotenuse is 50 units long. How long is the other leg? 14 units Pre-Algebra