jmc

By the Pythagorean Theorem in $△PQR$, $$P{Q}^2=P{R}^2-Q{R}^2=13^2-5^2=144,$$so $PQ=\sqrt{144}=12$.

By the Pythagorean Theorem in $△PQS$, $$Q{S}^2=P{S}^2-P{Q}^2=37^2-12^2=1225,$$so $QS=\sqrt{1225}=35$.

Therefore, the perimeter of $△PQS$ is $12+35+37=\boxed{84}$.