2011 Fermat Problem 16

Six identical rectangles with height h and width w are arranged as shown. Line segment PQ intersects the vertical side of one rectangle at X and the horizontal side of another rectangle at Z. If right-angled \triangle XYZ has YZ = 2XY, then \dfrac{h}{w} equals

\textbf{(A)}\ \frac{2}{3}\quad \textbf{(B)}\ \frac{1}{2}\quad \textbf{(C)}\ \frac{3}{8}\quad \textbf{(D)}\ \frac{1}{3}\quad \textbf{(E)}\ \frac{3}{4}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.