SOLUTION: The length of the diagonal of a rectangle is shorter than the semi-perimeter by 1/3 the length of the longer side. If the dimensions of all sides are integers, find the minimum len

Algebra ->  Length-and-distance -> SOLUTION: The length of the diagonal of a rectangle is shorter than the semi-perimeter by 1/3 the length of the longer side. If the dimensions of all sides are integers, find the minimum len      Log On


   

Question 1147941: The length of the diagonal of a rectangle is shorter than the semi-perimeter by 1/3 the length of the longer side. If the dimensions of all sides are integers, find the minimum length of the shorter side.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the length of the short side, and let y be the length of the long side.

The diagonal is then sqrt%28x%5E2%2By%5E2%29.

The problem says the diagonal is less than the semi-perimeter by 1/3 the length of the longer side; that is %28x%2By%29-%281%2F3%29y+=+x%2B%282%2F3%29y. So

sqrt%28x%5E2%2By%5E2%29+=+x%2B%282%2F3%29y
3%2Asqrt%28x%5E2%2By%5E2%29+=+3x%2B2y
9x%5E2%2B9y%5E2+=+9x%5E2%2B12xy%2B4y%5E2
5y%5E2-12xy+=+0
y%285y-12x%29+=+0

y=0 would make no sense in the problem, so 5y-12x = 0. Then, if x and y are integers, x=5k and y=12k for some positive integer k. And since we are looking for the minimum length of the shorter side that satisfies the conditions, x=5 and y=12.

ANSWER: The minimum length of the shorter side is 5.