Three squares have dimensions as indicated in the diagram. What is the area of the shaded quadrilateral?
(A) $\frac{21}{4}$
(B) $\frac{9}{2}$
(C) 5
(D) $\frac{15}{4}$
(E) $\frac{25}{4}$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-143.jpg?height=228&width=399&top_left_y=1851&top_left_x=1313)
Three squares have dimensions as indicated in the diagram. What is the area of the shaded quadrilateral?
(A) $\frac{21}{4}$
(B) $\frac{9}{2}$
(C) 5
(D) $\frac{15}{4}$
(E) $\frac{25}{4}$

## Solution 1

In the first solution, we use similar triangles. We start by labelling the diagram as shown. The objective in this question is to calculate the lengths $E B$ and $F C$ which will allow us to calculate the area of $\triangle A E B$ and $\triangle A F C$. We first note that $\triangle A F C$ and $\triangle A G D$ are similar and that, $\frac{A C}{A D}=\frac{F C}{G D}=\frac{5}{10}=\frac{1}{2}$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-290.jpg?height=290&width=499&top_left_y=2034&top_left_x=1453)

Therefore, $F C=\frac{1}{2} G D=\frac{1}{2}(5)=\frac{5}{2}$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-290.jpg?height=255&width=464&top_left_y=1591&top_left_x=1470)

Using the same reasoning, $\triangle A E B$ and $\triangle A F C$ are also similar triangles meaning that, $\frac{E B}{F C}=\frac{2}{5}$.

Thus, $E B=\frac{2}{5}\left(\frac{5}{2}\right)=1$.

We find the required area to be

$$
\text { area } \begin{aligned}
\triangle A F C-\text { area } \triangle A E B & =\frac{1}{2}(5)\left(\frac{5}{2}\right)-\frac{1}{2}(2)(1) \\
& =\frac{21}{4}
\end{aligned}
$$

## Solution 2

We start by putting the information on a coordinate axes and labelling as shown. The line containing $O D$ has equation $y=\frac{1}{2} x$ while $x=2$ and $x=5$ contains $A E$ and $B F$. Solving the systems $y=\frac{1}{2} x, x=2$ and $y=\frac{1}{2} x$, $x=5$ gives the coordinates of $E$ to be $(2,1)$ and $F$ to be $\left(5, \frac{5}{2}\right)$. This makes $A E=1$ and $B F=\frac{5}{2}$ which now leads to exactly the same answer as in solution 1.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-291.jpg?height=510&width=632&top_left_y=745&top_left_x=1386)