6. $\mathrm{ABCD}$ is a square with $\mathrm{AB}=a$, and $\mathrm{AEFG}$ is a rectangle such that $\mathrm{E}$ lies on side $\mathrm{BC}$ and $\mathrm{D}$ lies on side $\mathrm{FG}$. If $\mathrm{AE}=b$, what is the length of side $\mathrm{EF}$ ?
(A) $\frac{b}{a}$
(B) $\frac{3 a^{2}}{2 b}$
(C) $\frac{4 a^{2}}{3 b}$
(D) $\frac{\sqrt{2} a^{2}}{b}$
(E) $\frac{a^{2}}{b}$
6. Answer: (E)
Note that
$$
\text { area of } \begin{aligned}
\triangle \mathrm{ABD} & =\text { area of } \triangle \mathrm{AED} \\
& =\text { area of } \triangle \mathrm{AEF} .
\end{aligned}
$$

Therefore
$$
\begin{aligned}
a^{2} & =\text { area of square } A B C D \\
& =2 \times \text { area of } \triangle \mathrm{ABD} \\
& =2 \times \text { area of } \triangle \mathrm{AEF} \\
& =\text { area of rectangle } \mathrm{AEFG} .
\end{aligned}
$$

Hence $\mathrm{EF}=\frac{a^{2}}{b}$.