10.5 – The Pythagorean Theorem. leg legleg hypotenuse hypotenuse leg legleg.

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Presentation transcript:

10.5 – The Pythagorean Theorem

leg legleg

hypotenuse hypotenuse leg legleg

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.

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

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

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

Ex. 1 Find the length of the missing side. a 25 a 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 = 625 a = 625

Ex. 1 Find the length of the missing side. a 25 a 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 = 625 a = 625 a 2 = 525 a 2 = 525

Ex. 1 Find the length of the missing side. a 25 a 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 = 625 a = 625 a 2 = 525 a 2 = 525 a ≈ a ≈ 22.91

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

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

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

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

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) = = 841

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) = = = = 841

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) = = = = 841, so they form a rt. triangle!, so they form a rt. triangle!