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3. Find the length of the hypotenuse of the right triangle. Tell whether the side lengths form a Pythagorean triple. Because the side lengths 5, 12, and 13 are integers, they form a Pythagorean triple. 12 x 5 P ROVING THE P YTHAGOREAN T HEOREM Example 1

4. A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c 2 = a 2 + b 2. For example, the integers 3, 4, and 5 form a Pythagorean triple because 5 2 = 3 2 + 4 2. U SING THE P YTHAGOREAN T HEOREM

5. The most common Pythagorean Triples are: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 REMEMBER: The Pythagorean Theorem ONLY works in Right Triangles ! P ROVING THE P YTHAGOREAN T HEOREM

6. x 14 7 Many right triangles have side lengths that do not form a Pythagorean triple. Find the length of the leg of the right triangle. S OLUTION U SING THE P YTHAGOREAN T HEOREM Example 2

7. Find x. Example 3 U SING THE P YTHAGOREAN T HEOREM

8. The Pythagorean Theorem and Ladders You're locked out of your house and the only open window is 25 feet above the ground. You need use a ladder to get in. There's a bush along the edge of the house, so you'll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the window? You need a ladder that's 27 feet long. U SING THE P YTHAGOREAN T HEOREM Example 5

9. S UPPORT B EAM These skyscrapers are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam. U SING THE P YTHAGOREAN T HEOREM Example 6 Indirect Measurement

10. The Pythagorean Theorem and Baseball You've just picked up a ground ball at first base, and you see the other team's player running towards third base. How far do you have to throw the ball to get it from first base to third base, and throw the runner out? You need to throw the ball 127.3 feet to get it from first base to third base. U SING THE P YTHAGOREAN T HEOREM Example 7

11. A ramp was constructed to load a truck. If the ramp is 9 feet long and the horizontal distance from the bottom of the ramp to the truck is 7 feet, what is the vertical height of the ramp? U SING THE P YTHAGOREAN T HEOREM Example 8

12. leg hypotenuse

13. My Uncle Bob just emailed me and said he's locked himself out of his house. He left the water running in the bathtub and needs to get back in before it overflows. Walking around the house, he noticed that a window was open on the second floor, 25 feet above the ground. There is also a bush under the window, so he'll have to place the ladder 10 feet from the house. All his neighbors have different length ladders. What length of ladder does he need to borrow?"

14. PYTHAGORAS and the Pythagorean Theorem a 2 + b 2 = c 2 c b a