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The side-lengths a, b and c of a right triangle are integers

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by Max@Math Revolution » Mon Aug 19, 2019 12:23 am

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[GMAT math practice question]

The side-lengths a, b and c of a right triangle are integers that satisfy b-a=c-b=k for some positive integer. One of a, b, c is 56. What is the length of the longest side of the triangle?

A. 60
B. 65
C. 70
D. 75
E.80
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by Max@Math Revolution » Wed Aug 21, 2019 12:13 am
=>

Since k>0, we have c > b > a.
Case 1: b = 56
Since c = k + 56, a = 56 - k and c^2 = a^2 + 56, we have c = 70.
Case 2: a = 56
Since 2b = c + 56 and c^2 = 56^2 + b^2,
c = 2b - 56.
Plugging this into c^2 = 56^2 + b^2 yields
(2b - 56)^2 = 56^2 + b^2
and we solve this equation to give
4b^2 - 224 b + 56^2 = 56^2 + b^2
3b^2 - 224 b = 0
b(3b - 224) =0
b = 224/3 since b is positive.
So,
c =2b - 56 = 448/3 - 56 = 280/3, which is not an integer.
Thus, the length of the largest side is 70.
Therefore, the answer is C.
Answer: C
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