5.2.3 The Pythagorean Theorem (continued)

In this application example, the Pythagorean Theorem is used to determine whether a television with a diagonal length of 62 inches can possibly fit within an entertainment center. The diagonal d of the rectangular space of the entertainment center is the hypotenuse of a right triangle with legs 50 in. and 48 in. Substitute these values into a 2 + b 2 = c 2 and solve the equation for d . The length of this diagonal turns out to be d ≈ 69.3 inches. Since the length of the diagonal of the rectangular space is greater than the diagonal of the television, it is possible that the television will fit within the entertainment center. Without knowing the exact length and width of the television, it cannot be assumed that the television will for certain fit within the rectangular space.

Example 3 Identifying Pythagorean Triples Pythagorean triples are sets of three nonzero whole numbers that satisfy the Pythagorean formula.

The Pythagorean Theorem is used to find the length of the hypotenuse of a right triangle in this example. Then, it is determined whether the side lengths form a Pythagorean triple. Leg lengths of 2 and 3 are given. Substitute the values of the lengths of the sides of the triangle into a 2 + b 2 = c 2 . Solve the equation for the hypotenuse. The unknown side has a length of square root of 13, which is not a whole number. Therefore, these sides do not form a Pythagorean triple. The Pythagorean Theorem is used to find the length of a leg of a right triangle in this example. Then it is determined whether the side lengths form a Pythagorean triple. The length of one leg and the hypotenuse are given. Substitute the values of the lengths of the sides of the triangle into a 2 + b 2 = c 2 . Be sure to substitute the correct length (26) for c , the hypotenuse. Solve the equation for the unknown leg. The unknown leg has a length of 10, which is a whole number. Therefore, these sides do form a Pythagorean triple.

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