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Exercise Solve x2 = 4. x=±2 Exercise Solve x2 = –4. no real solution Exercise Solve √ x = 4. x = 16 Exercise Solve √ x = –4. no real solution Exercise Solve √ – x = 4. x = –16 leg leg 25 square units 4 16 square units 5 3 9 square units 25 square units + 16 square units 4 5 3 = 9 square units 2 4 16 + 2 39 = 2 525 The Pythagorean Theorem If the hypotenuse of a right triangle has length c, and the legs have lengths a and b, then a2 + b2 = c2. Example 1 Find the hypotenuse of a right triangle with legs of 8 and 15. c2 = a2 + b2 c2 = 289 c2 = 82 + 152 √ c2 = √ 289 c2 = 64 + 225 c = 17 Example 2 Find the hypotenuse of a right triangle with legs of 6 and 7. c2 = a2 + b2 c2 = 85 c2 = 62 + 72 √ c2 = √ 85 c2 = 36 + 49 c = √ 85 ≈ 9.2 Example Find the hypotenuse of a right triangle with legs of 9 and 12. 15 Example Find the hypotenuse of a 1 right triangle with legs of 2 and √ 3 . 2 1 Example Find the hypotenuse of a right triangle with legs of 1 and 1. √2 Example 3 Find the leg of a right triangle whose hypotenuse is 16 and other leg is 7. a2 + 72 = 162 a2 + 49 = 256 a2 + 49 – 49 = 256 – 49 a2 = 207 a = √ 207 ≈ 14.4 Example Find the length of a leg of a right triangle whose hypotenuse is 39 and whose other leg is 15. 36 Example Find the length of a leg of a right triangle whose hypotenuse is 20 and whose other leg is 10. √ 300 ≈ 17.3 Converse The converse is the statement resulting when the “if” part and the “then” part of a conditional statement are switched. Converse of the Pythagorean Theorem If a triangle has sides a, b, and c, such that a2 + b2 = c2, then the triangle is a right triangle. Example 4 Determine whether a triangle with sides of 12, 35, and 37 is a right triangle. a2 + b2 = c2 yes 2 2 2 12 + 35 = 37 144 + 1,225 = 1,369 1,369 = 1,369 Example 5 Determine whether a triangle with sides of 8, 12, and 14 is a right triangle. a2 + b2 = c2 no 2 2 2 8 + 12 = 14 64 + 144 = 196 208 ≠ 196 Example Determine whether a triangle with sides of 15, 18, and 22 is a right triangle. no; 152 + 182 ≠ 222 Example Determine whether a triangle with sides of 16, 30, and 34 is a right triangle. yes; 162 + 302 = 1,156 = 342 Exercise A 16 ft. ladder leans up against the side of a building. If the base of the ladder is 4 ft. from the base of the building, how high up the side of the building will the ladder reach? 15.5 ft. Exercise A 200 ft. tower is braced to the ground by a cable, from a point 150 ft. above the ground to a point 87 ft. from the base of the tower. How long is the cable? 173.4 ft. Exercise The distance between bases on a baseball diamond is 90 ft. How far is it from home plate to second base? 127.3 ft. Exercise An opening for a window is 23” wide, 54” tall, and 60” diagonally. Is the opening “square”; that is, do the height and width form a right angle? no