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10.5 – The Pythagorean Theorem

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leg legleg

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hypotenuse hypotenuse leg legleg

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hypotenuse hypotenuse leg legleg Pythagorean Theorem: For a right triangle with legs a and b and hypotenuse c, then a 2 + b 2 = c 2, where a is the smaller of the two legs.

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Ex. 1 Find the length of the missing side. a 25 a 25 10 10

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Ex. 1 Find the length of the missing side. a 25 a 25 10 10 a 2 + b 2 = c 2 a 2 + b 2 = c 2

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Ex. 1 Find the length of the missing side. a 25 a 25 10 10 a 2 + b 2 = c 2 a 2 + b 2 = c 2 a 2 + (10) 2 = (25) 2 a 2 + (10) 2 = (25) 2

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Ex. 1 Find the length of the missing side. a 25 a 25 10 10 a 2 + b 2 = c 2 a 2 + b 2 = c 2 a 2 + (10) 2 = (25) 2 a 2 + (10) 2 = (25) 2 a 2 + 100 = 625 a 2 + 100 = 625

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Ex. 1 Find the length of the missing side. a 25 a 25 10 10 a 2 + b 2 = c 2 a 2 + b 2 = c 2 a 2 + (10) 2 = (25) 2 a 2 + (10) 2 = (25) 2 a 2 + 100 = 625 a 2 + 100 = 625 a 2 = 525 a 2 = 525

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Ex. 1 Find the length of the missing side. a 25 a 25 10 10 a 2 + b 2 = c 2 a 2 + b 2 = c 2 a 2 + (10) 2 = (25) 2 a 2 + (10) 2 = (25) 2 a 2 + 100 = 625 a 2 + 100 = 625 a 2 = 525 a 2 = 525 a ≈ 22.91 a ≈ 22.91

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Ex. 2 Determine whether the following side measures form right triangles. a. 20, 21, 29

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Ex. 2 Determine whether the following side measures form right triangles. a. 20, 21, 29 Shorter lengths must be the legs.

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Ex. 2 Determine whether the following side measures form right triangles. a. 20, 21, 29 Shorter lengths must be the legs. So, a 2 + b 2 = c 2

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Ex. 2 Determine whether the following side measures form right triangles. a. 20, 21, 29 Shorter lengths must be the legs. So, a 2 + b 2 = c 2 (20) 2 + (21) 2 = (29) 2 (20) 2 + (21) 2 = (29) 2

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Ex. 2 Determine whether the following side measures form right triangles. a. 20, 21, 29 Shorter lengths must be the legs. So, a 2 + b 2 = c 2 (20) 2 + (21) 2 = (29) 2 (20) 2 + (21) 2 = (29) 2 400 + 441 = 841 400 + 441 = 841

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Ex. 2 Determine whether the following side measures form right triangles. a. 20, 21, 29 Shorter lengths must be the legs. So, a 2 + b 2 = c 2 (20) 2 + (21) 2 = (29) 2 (20) 2 + (21) 2 = (29) 2 400 + 441 = 841 400 + 441 = 841 841 = 841 841 = 841

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Ex. 2 Determine whether the following side measures form right triangles. a. 20, 21, 29 Shorter lengths must be the legs. So, a 2 + b 2 = c 2 (20) 2 + (21) 2 = (29) 2 (20) 2 + (21) 2 = (29) 2 400 + 441 = 841 400 + 441 = 841 841 = 841 841 = 841, so they form a rt. triangle!, so they form a rt. triangle!

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