Question 1134281
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Note that a formal algebraic solution is good exercise; but if the speed of solution is important and formal algebra is not required (as, for example, in a competitive math examination), this problem can be solved very quickly.<br>
The length and width of the rectangle and the diagonal form a right triangle; and lengths of both the diagonal and the perimeter are integers.  That means the three lengths of the sides of the triangle form a Pythagorean Triple.<br>
A Pythagorean Triple with hypotenuse 13 is 5-12-13; if the side lengths are 5 and 12, then the perimeter is 2(5+12) = 34, as required.<br>
So the side lengths are 5 and 12.