+308 As shown in the diagram, $\overline{PQ}$ is the median of trapezoid $ABCD,$ where $AB=25$ inches and $CD=36$ inches. Find $XY$ in inches.
+506 WLOG, let the vertical height of ABCD be 1 (the trapezoid will then obviously be way more squashed than the image).
Notice that triangles ABO and ODD are similar. Therefore, the ratio of the length of CD to the length of AB is \(36:25\) , so the height of triangle ABO is \(\frac{25}{36+25} = \frac{25}{61}\).
The height of OXY is then just \(\frac{1}{2}-\frac{25}{61}=\frac{11}{122}\). Since OXY and ABO are similar, the length of xy is just \(\frac{11}{122}\cdot61 = \boxed{5.5}\)
(i am not completely sure of this answer as I submitted this very late at night and I am not extremely good at these particular problems)
