Question 1147941
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Let x be the length of the short side, and let y be the length of the long side.<br>
The diagonal is then {{{sqrt(x^2+y^2)}}}.<br>
The problem says the diagonal is less than the semi-perimeter by 1/3 the length of the longer side; that is {{{(x+y)-(1/3)y = x+(2/3)y}}}.  So<br>
{{{sqrt(x^2+y^2) = x+(2/3)y}}}
{{{3*sqrt(x^2+y^2) = 3x+2y}}}
{{{9x^2+9y^2 = 9x^2+12xy+4y^2}}}
{{{5y^2-12xy = 0}}}
{{{y(5y-12x) = 0}}}<br>
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.<br>
ANSWER: The minimum length of the shorter side is 5.<br>