Since the perimeter is an integer, we are looking for a pythagorean triple whose sum is 24.
The most used triples (at least in textbooks) are 3-4-5, 5-12-13, 8-15-17, and 7-24-25. None of these sum to 24, but 3-4-5 sums to 12.
A triangle formed by the lengths 6-8-10 will also be a right triangle. This triangle is similar to the 3-4-5 triangle by SSS similarity. Or you could use the converse of the pythagorean theorem -- if the square of the longest side of a triangle equals the sum of the squares of the remaining sides then the triangle is right.
Thus the solution is a triangle with sides 6,8, and 10.