In the diagram, what is the perimeter of $\triangle PQR$?
[asy]
draw((0,0)--(28,0)--(8,15)--cycle);
label("$P$",(0,0),SW);
label("$Q$",(8,15),NW);
label("$R$",(28,0),SE);
draw((8,15)--(8,0));
draw((9,0)--(9,1)--(8,1));
label("$8$",(4,0),S); label("$20$",(18,0),S); label("$25$",(18,8),NE);
[/asy]
Each of $\triangle PSQ$ and $\triangle RSQ$ is right-angled at $S$, so we can use the Pythagorean Theorem in both triangles.

In $\triangle RSQ$, we have $QS^2 = QR^2 - SR^2 = 25^2-20^2=625 - 400 = 225$, so $QS=\sqrt{225}=15$ since $QS>0$.

In $\triangle PSQ$, we have $PQ^2 = PS^2 + QS^2 = 8^2 + 225 = 64+225=289$, so $PQ = \sqrt{289}=17$ since $PQ>0$.

Therefore, the perimeter of $\triangle PQR$ is $PQ+QR+RP=17+25+(20+8)=\boxed{70}$.