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, $\overline{QR}$ is the hypotenuse of the triangle, and $QP=RP=\frac{QR}{\sqrt{2}}$, which means $QP=RP=6$ and $[QRP]=(QP)(RP)/2=\boxed{18}$.