jmc

A triangle can't have two right angles, so a right triangle with two congruent angles must have congruent acute angles. That is, $△PQR$ must be an isosceles right triangle with acute angles at $Q$ and $R$. Therefore, $PQ=PR=6\sqrt{2}$, and $[QRP]=(QP)(RP)/2=(6\sqrt{2})(6\sqrt{2})/2=(6\cdot 6\cdot \sqrt{2}\cdot \sqrt{2})/2=\boxed{36}$.