Problem 25

In the pentagon $UVWXY$, $\angle U = \angle V = 90^\circ$, $UY = VW$ and $YX = XW$. Four equally spaced points are marked between $U$ and $V$, and perpendiculars are drawn through each point, as shown in the diagram. The dark shaded region has an area of $13\mathrm{cm}^2$ and the light shaded region has an area of $10\mathrm{cm}^2$. What is the area, in $\mathrm{cm}^2$, of the entire pentagon?