[tex]5^2 + 12^2 = 13^2[/tex]
is the second or third Pythagorean triple on the list, which gives us the answer directly.
But if we didn't know that, let x be the short leg:
[tex] x^2 + (x+7)^2 = 13^2[/tex]
[tex]2x^2 + 14 x + 49 - 169 = 0[/tex]
[tex]x^2 + 7x - 60 = 0[/tex]
[tex](x+12)(x-5)=0[/tex]
Rejecting the negative root, x=5, other side 12, the Pythagorean triple we found.
