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The pythagorean Theorem is
. If you substitute in 5 and 12 for a and b respectively you get
. Thus the ratios of the sides must be 5, 12, and 13. Now we know that in order to maintain this ratio we must multiply the sides by a common factor such that the perimeter equals 90. To do this we can create the following equation:
. We can further simplify this equation so it becomes
. Now we must divide both sides of the equation by 30, so we get
. Now that we know that the common factor is 3 we can simply multiply the side length with a ratio of 13 by 3 and we get
. Thus, the side lengths must be 15, 36, and 39 as
. The longest side of the three is 39, so 39 is the answer.