In the figure, $ABC$ and $CDE$ are right-angled and isosceles triangles. Segments $AD$ and $BC$ intersect at $P$, and segments $CD$ and $BE$ intersect at $Q$. (a) Show that segment$ PQ$ is parallel to segment $AE$. (b) If $BP = 4$ and $DQ = 9$, find the measure of segment $BD$.
