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

Pre-Algebra 6-3 The Pythagorean Theorem Warm Up Graph the figures with the given vertices 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 units2 12 units2 88 units2

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

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

Vocabulary Pythagorean Theorem leg hypotenuse

a2 + b2 = c2

Additional Example 1A: Find the the Length of a Hypotenuse Find the length of the hypotenuse. c A. 4 5 a2 + b2 = c2 Pythagorean Theorem 42 + 52 = c2 Substitute for a and b. 16 + 25 = c2 Simplify powers. 41 = c2 41 = c Solve for c; c = c2. 6.40  c

Additional Example 1B: Find the the Length of a Hypotenuse Find the length of the hypotenuse. triangle with coordinates B. (1, –2), (1, 7), and (13, –2) a2 + b2 = c2 Pythagorean Theorem 92 + 122 = c2 Substitute for a and b. 81 + 144 = c2 Simplify powers. 225 = c Solve for c; c = c2. 15 = c

Find the length of the hypotenuse. Try This: Example 1A Find the length of the hypotenuse. c A. 5 7 a2 + b2 = c2 Pythagorean Theorem 52 + 72 = c2 Substitute for a and b. 25 + 49 = c2 Simplify powers. 74 = c Solve for c; c = c2. 8.60  c

Find the length of the hypotenuse. Try This: Example 1B Find the length of the hypotenuse. B. triangle with coordinates (–2, –2), (–2, 4), and (3, –2) x y (–2, 4) The points form a right triangle. a2 + b2 = c2 Pythagorean Theorem 62 + 52 = c2 Substitute for a and b. 36 + 25 = c2 Simplify powers. (–2, –2) (3, –2) 61 = c Solve for c; c = c2. 7.81  c

Additional Example: 2 Finding the Length of a Leg in a Right Triangle Solve for the unknown side in the right triangle. a2 + b2 = c2 Pythagorean Theorem 25 b 72 + b2 = 252 Substitute for a and c. 49 + b2 = 625 Simplify powers. –49 –49 b2 = 576 7 b = 24 576 = 24

Try This: Example 2 Solve for the unknown side in the right triangle. a2 + b2 = c2 Pythagorean Theorem 12 b 42 + b2 = 122 Substitute for a and c. 16 + b2 = 144 Simplify powers. –16 –16 4 b2 = 128 b  11.31 128  11.31

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. a2 + b2 = c2 Pythagorean Theorem a2 + 42 = 62 Substitute for b and c. a2 + 16 = 36 6 6 a a2 = 20 a = 20 units ≈ 4.47 units 4 4 Find the square root of both sides. A = hb = (8)( 20) = 4 20 units2  17.89 units2 1 2

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. a2 + b2 = c2 Pythagorean Theorem a2 + 22 = 52 Substitute for b and c. 5 5 a2 + 4 = 25 a a2 = 21 a = 21 units ≈ 4.58 units 2 2 Find the square root of both sides. A = hb = (4)( 21) = 2 21 units2  9.17 units2 1 2

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. 10 m c h 12 m 3. Find the area of the largest triangle. 60 m2 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