jmc

The sum of the areas of the squares is $P{R}^2+P{Q}^2+Q{R}^2$. By the Pythagorean theorem, $P{R}^2=P{Q}^2+Q{R}^2$. Substituting the left-hand side of this equation for the right-hand side, we find that the sum of the areas of the squares is $P{R}^2+P{R}^2=2\cdot P{R}^2$. Setting this equal to 338 square centimeters, we find that $P{R}^2=338/2=\boxed{169}$ square centimeters.