2. Let $A B C D$ be an isosceles trapezoid such that $A D=B C, A B=3$, and $C D=8$. Let $E$ be a point in the plane such that $B C=E C$ and $A E \perp E C$. Compute $A E$.
Theorem,
$$
A E^{2}=A C^{2}-C E^{2}=A C^{2}-r^{2}
$$

Let the height of $\triangle A C D$ at $A$ intersect $D C$ at $F$. Once again, by the Pythagorean Theorem,
$$
A C^{2}=F C^{2}+A F^{2}=\left(\frac{8-3}{2}+3\right)^{2}+A D^{2}-D F^{2}=\left(\frac{11}{2}\right)^{2}+r^{2}-\left(\frac{5}{2}\right)^{2}
$$

Plugging into the first equation,
$$
\begin{array}{l}
\qquad A E^{2}=\left(\frac{11}{2}\right)^{2}+r^{2}-\left(\frac{5}{2}\right)^{2}-r^{2} \text {, } \\
\text { so } A E=2 \sqrt{6} \text {. }
\end{array}
$$