Just as you have illustrated on your diagram, using the information provided PQ is equal in length to PS and PR is in common to both of the two triangles PQR and PSR. Given they are both right angled triangles, Pythagoras’ theorem implies QR equals SR and hence we see that the two triangles are congruent: one is simply the mirror image of the other through the line through PR. Points Q and S are mirror images of each other in the line and therefore equally distant from the line. O is the point where the line through QS meets PR (perpendicularly) and the line PR bisects the line QS at O. That is equivalent to saying that O is the midpoint of QS.