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Pythagorean Theorem. a² + b² = c². Click here to see the animation. Pythagorean Theorem. a² + b² = c². c is ALWAYS the “hypotenuse” the LONGEST side across from the right angle. c² - b² = a². a or b is a “leg”. c² - a² = b². a or b is a “leg”. Find the missing length. 5. 3. 4.

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  1. Pythagorean Theorem a² + b² = c² Click here to see the animation

  2. Pythagorean Theorem a² + b² = c² c is ALWAYS the “hypotenuse” the LONGEST side across from the right angle c² - b² = a² a or b is a “leg” c² - a² = b² a or b is a “leg”

  3. Find the missing length 5 3 4

  4. Find the missing length 10 6 8

  5. Find the missing length 15 9 12

  6. Find the missing length 25 15 20

  7. Those were nice “3, 4, 5” trianglesHere is information on “triples”

  8. Find the unknown length for the triangle shown. This is neither a triple, or a 3,4,5. SOLUTION b2 c2 a2 + Pythagorean theorem = a2 + 62 = 72 Substitute 6 for band 7 for c. a2 + 36 = 49 Simplify. a Take positive square root of each side. = 13 The side length ais ANSWER 13. Example 1 Use the Pythagorean theorem a2 = 13 Subtract 36 from each side.

  9. Guided Practice for Example 1 5² + 12² = c² ANSWER 25 + 144 = c² 169 = c² √169 = √c² 13 = c The lengths of the legs of a right triangle areand 1. a = 5 b = 12. Find c.

  10. Multiple Choice Practice Example 2 In the triangle shown, D is the midpoint of segment AC, and segment BD is perpendicular to segment AC. What is the length of segment BD? 12 cm 14 cm 23 cm 16 cm

  11. c2 a2 + b2 Pythagorean theorem = SOLUTION The path of the kicked ball is the hypotenuse of a right triangle. The length of one leg is 12 yards, and the length of the other leg is 40 yards. Example 3 Solve a real-world problem SOCCER A soccer player makes a corner kick to another player, as shown. Find the distance the player kicks the ball.

  12. Example 3 Solve a real-world problem c2 122 + 402 Substitute 12 for aand 40 for b. = c2 1744 Simplify. = Take positive square root of each side. c2 = 42 ≈ 1744 The player kicks the ball about 42 yards. ANSWER

  13. The Pythagorean Theorem can be used to check and see if the lengths will form a right triangle when connected

  14. Example 4 Determine right triangles Tell whether the triangle with the given side lengths is a right triangle. a. 8, 15, 17 ? 82 + 152 172 = ? 289 64 + 225 = 289 289 = The triangle is a right triangle. ANSWER

  15. The triangle is not a right triangle. ANSWER Example 4 Determine right triangles Tell whether the triangle with the given side lengths is a right triangle. b. 5, 8, 9 ? 52 + 82 92 = ? 25 + 64 81 = 89 81 =

  16. Guided Practice for Examples 4 and 5 Tell whether the triangle with the given side lengths is a right triangle. not a right triangle 4. 7, 11, 13 ANSWER 5. 15, 36, 39 right triangle ANSWER right triangle ANSWER 6. 15, 112, 113

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