11. Given a square $A B C D$ with side length 1, point $M$ is the midpoint of side $A D$. A circle $\Gamma$ is drawn with $M$ as the center and $A D$ as the diameter. Point $E$ lies on line segment $A B$, and line $C E$ is tangent to circle $\Gamma$. Find the area of $\triangle C B E$.
Given $E A=E F=x$, then $E B=1-x$, $E C=1+x$, in $\triangle C B E$,
$$
\begin{array}{l}
(1-x)^{2}+1=(1+x)^{2} \Rightarrow-2 x+2=2 x+1 \\
x=\frac{1}{4} \Rightarrow S_{\triangle C B E}=\frac{1}{2} \cdot \frac{3}{4} \cdot 1=\frac{3}{8} .
\end{array}
$$