# Honors Geometry Companion Book, Volume 1

5.2.3 The Pythagorean Theorem (continued)

Example 1 Using the Pythagorean Theorem

The Pythagorean Theorem is used to find the length of two legs of a right triangle in this application example. There are two unknown legs, but the ratio of the lengths of the legs is given, which allows the length of one leg to be expressed in terms of the length of the other leg. To express the value of one leg length in terms of the other, set up the ratio a is to b as 4 is to 1 and cross multiply. The lengths a , b , and c can now be expressed as 12, b , and 4 b . Substitute the values of the lengths of the sides of the triangle into a 2 + b 2 = c 2 . Solve the equation for b . The solution yields b ≈ 2 feet 11 inches. This is the value for the recommended distance of the foot of the ladder from the wall. The Pythagorean Theorem is used to find the length of a leg of a right triangle in this example. The length of one leg and the hypotenuse are given. Substitute the values of the length of the leg and the hypotenuse of the triangle into the Pythagorean formula. Remember that the side opposite the right angle is the hypotenuse, or c in a 2 + b 2 = c 2 . Solve the equation for x , the length of the unknown leg. The Pythagorean Theorem is used to find an unknown value related to the lengths of the sides of a right triangle in this example. The length of one leg and the hypotenuse are given as algebraic expressions involving one unknown, x . Substitute the values of the lengths of the sides of the triangle into a 2 + b 2 = c 2 . Solve the equation for x. The solution yields x = 15. This value for x could also be used to calculate the lengths of the two unknown sides.

Example 2 Safety Application

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