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messersmith_power_intermediate_algebra_1e_ch4_7_10

3 Solve Problems Using the Pythagorean Theorem A right triangle is a triangle that contains a 90 (right) angle. We can label a right triangle as follows. The side opposite the 90 angle is the longest side of hypotenuse c 90�� the triangle and is called the hypotenuse. The other two b (leg) 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 one which is used in many different ways. a (leg) Definition Pythagorean Theorem Given a right triangle with legs of length a and b and hypotenuse of length c, c b a the Pythagorean theorem states that a2 b2 c2 or (leg)2 (leg)2 (hypotenuse)2. 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. 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 ✓ Notice that the negative solution does not make sense because it represents the side length of a triangle. a 13 12 www.mhhe.com/messersmith SECTION 7.5 Applications of Quadratic Equations 405


messersmith_power_intermediate_algebra_1e_ch4_7_10
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