Diagonal AC of rectangle ABCD creates angle ACD, the cosine of which is 12/13. If the lengths of the sides of rectangle ABCD are all integers, which of the following could be the perimeter of rectangle ABCD?

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The cosine of ACD is equal to the length of CD divided by the length of diagonal AC. Diagonal AC along with sides AD and DC form a right triangle. Use the Pythagorean theorem to find the length of AD: (AD) 2 + 12 2 = 13 2 , (AD) 2 + 144 = 169, (AD) 2 = 25, AD = 5 units. If the length of AD is 5 units, then the length of BC is also 5 units. Since the length of CD is 12 units, the length of AB is also 12 units. The perimeter of ABCD is 5 + 12 + 5 + 12 = 34 units. Notice that this is only one possible perimeter of ABCD. The cosine could have been given in reduced form. The length of CD could be 24, and the length of AC could be 26. However, since the length of every side of ABCD is an integer, the only possible perimeters are multiples of 34. Choice D is the only answer choice that is a multiple of 34.

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