The measures of the smallest angles of three different right triangles sum to $90^\circ$. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $5$-$12$-$13$ and $8$-$15$-$17$. What is the perimeter of the third triangle?

1. The tangent of the smallest angle of a right triangle is the ratio of its shorter leg to its longer leg: $\tan\alpha=\frac5{12}$ and $\tan\beta=\frac8{15}$.
2. If $\gamma$ is the smallest angle of the third triangle, then $\alpha+\beta+\gamma=90^\circ$, so
   $$\tan\gamma=\cot(\alpha+\beta)=\frac{1-\tan\alpha\tan\beta}{\tan\alpha+\tan\beta}=\frac{1-\frac{40}{180}}{\frac{5}{12}+\frac8{15}}=\frac{140/180}{171/180}=\frac{140}{171}.$$
3. Since $\gcd(140,171)=1$ and $140^2+171^2=19600+29241=48841=221^2$, the third triangle is the primitive triple $140$-$171$-$221$.
4. The perimeter is $140+171+221=\boxed{532}$.