Question 1100018
<br>
let x = shorter side
let y = longer side<br>
The semiperimeter is then x+y; and then the length of the diagonal is
{{{(x+y)-(3/4)(x) = (1/4)x+4y}}}<br>
Then by the Pythagorean Theorem,
{{{(1/4)x+y = sqrt((x)^2+(y)^2)}}}
{{{(1/16)x^2+(1/2)xy+y^2 = x^2+y^2}}}
{{{(1/2)xy = (15/16)x^2}}}
{{{8xy = 15x^2}}}
{{{8y = 15x}}}  [note dividing by x is okay here, because we know the problem would make no sense if x were equal to 0]<br>
The smallest solution in integers is y=15, x=8, making the diagonal 17.<br>
The answer to the problem is the minimum length of the longer side, which is 15.