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I can: use the Pythagorean Theorem and its converse to solve problems.

6-3. The Pythagorean Theorem. I can: use the Pythagorean Theorem and its converse to solve problems. Vocabulary. Pythagorean Theorem leg hypotenuse. a 2 + b 2 = c 2. 41 = c. Solve for c; c = c 2. Additional Example 1A: Find the the Length of a Hypotenuse.

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I can: use the Pythagorean Theorem and its converse to solve problems.

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  1. 6-3 The Pythagorean Theorem I can: use the Pythagorean Theorem and its converse to solve problems.

  2. Vocabulary Pythagorean Theorem leg hypotenuse

  3. a2 + b2 = c2

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

  5. Solve for c; c = c2. 225 = 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) Pythagorean Theorem a2 + b2 = c2 Substitute for a and b. 92 + 122 = c2 Simplify powers. 81 + 144 = c2 15= c

  6. 74 = c Solve for c; c = c2. Try This: Example 1A Find the length of the hypotenuse. c A. 5 7 Pythagorean Theorem a2 + b2 = c2 52 + 72 = c2 Substitute for a and b. Simplify powers. 25 + 49 = c2 8.60c

  7. y x Solve for c; c = c2. 61 = c Try This: Example 1B Find the length of the hypotenuse. B. triangle with coordinates (–2, –2), (–2, 4), and (3, –2) (–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. (3, –2) (–2, –2) 7.81c

  8. 576 = 24 Additional Example: 2 Finding the Length of a Leg in a Right Triangle Solve for the unknown side in the right triangle. Pythagorean Theorem a2 + b2 = c2 25 Substitute for a and c. 72 + b2 = 252 b Simplify powers. 49 + b2 = 625 –49 –49 b2 = 576 7 b = 24

  9. 128  11.31 Try This: Example 2 Solve for the unknown side in the right triangle. a2 + b2 = c2 Pythagorean Theorem 12 Substitute for a and c. b 42 + b2 = 122 Simplify powers. 16 + b2 = 144 –16 –16 4 b2 = 128 b 11.31

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

  11. a = 21 units ≈ 4.58 units 1 2 1 2 A = hb = (4)( 21) = 2 21 units2 9.17 units2 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 2 2 Find the square root of both sides.

  12. 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. c 10 m h 12 m 3. Find the area of the largest triangle. 6 m 9 m 60 m2 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

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