The Pythagorean Theorem Tutorial 12d

The Pythagorean Theorem In a right triangle, the side opposite the right angle is the longest side. It is the hypotenuse. The other two sides are the legs of a right triangle. Side c is the hypotenuse, and is the longest side. c a Sides a and b are the legs of the triangle. b

a2 + b2 = c2 32 + 42 = 52 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2 c =5 a =3 Example: 32 + 42 = 52 b =4

Example #1 c c c 1. 2. 3. 100m 70m 18in 35ft 68ft 34in a2 + b2 = c2

Example #2 a2 + b2 = c2 x2 + 42 = 62 x2 + 16 = 36 -16 -16 x2 = 20 Find the value of the variable. Round your answers to the nearest tenth. a2 + b2 = c2 x2 + 42 = 62 x2 + 16 = 36 -16 -16 x2 = 20 a2 + b2 = c2 102 + y2 = 142 100 + y2 = 196 -100 -100 y2 = 96 a2 + b2 = c2 92 + p2 = 102 81+ p2 = 100 -81 -81 p2 = 19 x  4.5 y  9.8 p  4.4

Pythagorean Theorem When the lengths of the sides of a right triangle are integers, the integers form a Pythagorean triple. Example: 32 + 42 = 52 9 + 16 = 25 5 3 25 = 25 4 The numbers 3, 4, & 5 are called Pythagorean Triples

The Converse of the Pythagorean Theorem If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. Is this triangle a right triangle? 62 + 82 = 102 10 6 36 + 64 = 100 100 = 100 8 Yes it is a right triangle

Are the following triangles right triangles? More Examples Are the following triangles right triangles? 1. 2. 3. 7.52 + 42 ? 8.52 152 + 72 ? 172 122 + 182 ? 212 56.25 +16 ? 72.25 225 +49 ? 289 144 +324 ? 441 72.25 ? 72.25 = 274 ? 289  468 ? 441  Yes, it is a right triangle No, it is not a right triangle No, it is not a right triangle

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