High School Pythagorean triples that sum to 60

[Mr Davis 97]

I'm trying to find pythagorean triples that sum to 60. Just from memory, I kow that 3-4-5 and 5-12-13, scaled to some factor, will give triples that sum to 60. These seem to be the only ones that sum to 60, but how can I be sure that there aren't more triples that sum to 60?

 

[fresh_42]

What do you mean by "sum up"? I don't see the $60$.
You can find all by $x = u^2 - v^2 \; , \; y = 2uv \; , \; z = u^2 + v^2$.

 

[Infrared]

Let x,y,z be the side lengths of a triangle you describe, so that x^2+y^2=z^2 and x+y+z=60. You can solve for z in the last equation and plug it into the first, getting x^2+y^2=(60-(x+y))^2=3600-120(x+y)+x^2+2xy+y^2. This gives the equation 60x+60y-xy=1800, which you can rearrange into (60-x)(60-y)=1800. Now you just need to look for positive factor pairs of 1800 with both factors smaller than 60 (since 0<x,y<60).