SOLUTION: The length of the diagonal of a rectangle is shorter than the semi-perimeter by 3/4 length of the shorter side. If the dimensions of all sides are integers, find the minimum length

Algebra ->  Length-and-distance -> SOLUTION: The length of the diagonal of a rectangle is shorter than the semi-perimeter by 3/4 length of the shorter side. If the dimensions of all sides are integers, find the minimum length      Log On


   

Question 1100018: The length of the diagonal of a rectangle is shorter than the semi-perimeter by 3/4 length of the shorter side. If the dimensions of all sides are integers, find the minimum length of the longer side.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


let x = shorter side
let y = longer side

The semiperimeter is then x+y; and then the length of the diagonal is
%28x%2By%29-%283%2F4%29%28x%29+=+%281%2F4%29x%2B4y

Then by the Pythagorean Theorem,
%281%2F4%29x%2By+=+sqrt%28%28x%29%5E2%2B%28y%29%5E2%29
%281%2F16%29x%5E2%2B%281%2F2%29xy%2By%5E2+=+x%5E2%2By%5E2
%281%2F2%29xy+=+%2815%2F16%29x%5E2
8xy+=+15x%5E2
8y+=+15x [note dividing by x is okay here, because we know the problem would make no sense if x were equal to 0]

The smallest solution in integers is y=15, x=8, making the diagonal 17.

The answer to the problem is the minimum length of the longer side, which is 15.