jmc

Suppose the side lengths of the triangle are $a$, $b$, and $c$, with $c$ the hypotenuse. Then $c^2=a^2+b^2$ by the Pythagorean Theorem. We are told that $$a^2+b^2+c^2=1800.$$ Since $a^2+b^2=c^2$, then $c^2+c^2=1800$ or $2c^2=1800$ or $c^2=900$ or $c=30$ (since the side lengths are positive). So the hypotenuse has length $\boxed{30}$.