What is the area of $PXQY$ in the rectangle $ABCD$ in the following diagram?

Question

In $ABCD$ rectangle $AB=6$, $AD=8$, $AE=ED$, $BF=FC$, $EP=PQ=QF$. Find the area of $PXQY$.

Source: Bangladesh Math Olympiad 2014 Junior Category

I cant prove that which type of quadrilateral it is. How can I get the diagonal or the sides of $PQXY$? Please help me with a hint.

Answer 1

$PF=FC$ and $PF\perp FC,$ which says $$\measuredangle QPY=\measuredangle FPC=45^{\circ},$$ which gives that $PXQY$ is a square and $$S_{PXQY}=\frac{XY\cdot PQ}{2}=\frac{2^2}{2}=2.$$

Answer 2

with Q being in the middle of PF, and the symmetry of the graph making XY cut the middle of PQ, you can actually visualise triangle PBC as the following diagram:

Can you continue?

Answer 3

As $\triangle XAB\sim \triangle XQP$ and $AB= 3PQ$, the distance from $X$ to $AB$ is equal to 3 times the distance from $X$ to $PQ$.

Therefore, the distance from $X$ to $PQ$ is $\displaystyle 8\times\frac{1}{2}\times\frac{1}{4}=1$.

$PQ=6\div3=2$.

The area of $PXQY$ is $\displaystyle 2\times\frac{1}{2}\times2\times 1=2$.