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

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


   

Question 1187047: The length of the diagonal of a rectangle is shorter than the semi-perimeter by 3/4 the length of the shorter side. If the dimensions of all sides are integers, find the minimum length of the longer side.
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the length of the longer side
Let y be the length of the shorter side

The length of the diagonal is sqrt%28x%5E2%2By%5E2%29

The length is also equal to the semiperimeter minus 3/4 the length of the shorter side: %28x%2By%29-%283%2F4%29y=x%2By%2F4

So

sqrt%28x%5E2%2By%5E2%29=x%2By%2F4
4sqrt%28x%5E2%2By%5E2%29=4x%2By
16%28x%5E2%2By%5E2%29=16x%5E2%2B8xy%2By%5E2
16x%5E2%2B16y%5E2=16x%5E2%2B8xy%2By%5E2
16y%5E2=8xy%2By%5E2
15y%5E2-8xy=0
y%2815y-8x%29=0

y=0 or 15y-8x=0

y=0 makes no sense in the problem, so 15y-8x=0.

With the requirement that x and y be integers, the smallest solution is with x=15 and y=8.

ANSWER: The minimum length of the longer side is x=15.

CHECK:
sqrt%28x%5E2%2By%5E2%29=sqrt%2815%5E2%2B8%5E2%29=sqrt%28225%2B64%29=sqrt%28289%29=17

%28x%2By%29-%283%2F4%29y+=+23-6+=+17