sides are called the legs. The Pythagorean theorem states a relationship between the lengths of the sides of a right triangle. This is a very important relationship in mathematics and is used in many different ways. Definition Pythagorean Theorem Given a right triangle with legs of length a and b and hypotenuse of length c, the Pythagorean theorem states that a2 b2 c2 Cor (leg)2 (leg)2 (hypotenuse)2D. c b a The Pythagorean theorem is true only for right triangles. EXAMPLE 3 Find the length of the missing side. 13 12 Solution Since this is a right triangle, we can use the Pythagorean theorem to fi nd the length of the side. Let a represent its length, and label the triangle. a 13 12 The length of the hypotenuse is 13, so c 13. a and 12 are legs. Let b 12. a2 b2 c2 Pythagorean theorem a2 (12)2 (13)2 Substitute values. a2 144 169 a2 25 0 Write the equation in standard form. (a 5)(a 5) 0 Factor. b R a 5 0 or a 5 0 Set each factor equal to 0. a 5 or a 5 Solve. a 5 does not make sense as an answer because the length of a side of a triangle cannot be negative. Therefore, a 5. Check: 52 (12)2 (13)2 25 144 169 ✓ In-Class Example 3 Find the length of the missing side. 3 5 Answer: 4 YOU TRY 3 Find the length of the missing side. 4 5 www.mhhe.com/messersmith SECTION 7.6 Applications of Quadratic Equations 435
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