The length of a rectangle’s diagonal is 5 17 , and the length of the rectangle’s shorter side is 5

Question

The length of a rectangle&rsquo;s diagonal is 5 17 , and the length of the rectangle&rsquo;s shorter side is 5 . What is the length of the rectangle&rsquo;s longer side?

Accepted Answer

Choice B is correct. A rectangle&rsquo;s diagonal divides a rectangle into two congruent right triangles, where the diagonal is the hypotenuse of both triangles. It&rsquo;s given that the length of the diagonal is 5 17 and the length of the rectangle&rsquo;s shorter side is 5 . Therefore, each of the two right triangles formed by the rectangle&rsquo;s diagonal has a hypotenuse with length 5 17 , and a shorter leg with length 5 . To calculate the length of the longer leg of each right triangle, the Pythagorean theorem, a 2 + b 2 = c 2 , can be used, where a and b are the lengths of the legs and c is the length of the hypotenuse of the triangle. Substituting 5 for a and 5 17 for c in the equation a 2 + b 2 = c 2 yields 5 2 + b 2 = 5 17 2 , which is equivalent to 25 + b 2 = 25 17 , or 25 + b 2 = 425 . Subtracting 25 from each side of this equation yields b 2 = 400 . Taking the positive square root of each side of this equation yields b = 20 . Therefore, the length of the longer leg of each right triangle formed by the diagonal of the rectangle is 20 . It follows that the length of the rectangle&rsquo;s longer side is 20 . Choice A is incorrect and may result from dividing the length of the rectangle&rsquo;s diagonal by the length of the rectangle&rsquo;s shorter side, rather than substituting these values into the Pythagorean theorem. Choice C is incorrect and may result from using the length of the rectangle&rsquo;s diagonal as the length of a leg of the right triangle, rather than the length of the hypotenuse. Choice D is incorrect. This is the square of the length of the rectangle&rsquo;s longer side.

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