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.


 Feb 7, 2021

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)

 Feb 7, 2021

nah, that was perfect!


Thank You!

notsmart2.0  Feb 7, 2021

Your answer is brilliant!!! Kudos! smiley

jugoslav  Feb 7, 2021

Thanks for the kind words guys smiley

(also, sorry, I made a typo: I meant "Notice that triangles ABO and OCD are similar", not  "Notice that triangles ABO and ODD are similar")

textot  Feb 7, 2021

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