Square $ABCD$ has sides of length $4$, and $M$ is the midpoint of $\overline{CD}$. A circle with radius $2$ and center $M$ intersects a circle with radius $4$ and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\overline{AD}$? $\textbf{(A)}\ 3 \qquad$ $ \textbf{(B)}\ \frac {16}{5} \qquad$ $ \textbf{(C)}\ \frac {13}{4} \qquad$ $ \textbf{(D)}\ 2\sqrt {3} \qquad$ $ \textbf{(E)}\ \frac {7}{2}$
Solution
Connect $MP, MD, DP, AM$; $AM$ meets $DP$ at $N$ and $AM\perp DP$.
Square $ABCD$ has sides of length $12$, and $M$ is the point on $\overline{CD}$ with $DM=4$. A circle with radius $4$ and center $M$ intersects a circle with radius $12$ and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\overline{AD}$?
$\textbf{(A)}\ 3 \qquad$ $ \textbf{(B)}\ \frac {16}{5} \qquad$ $ \textbf{(C)}\ \frac {13}{4} \qquad$ $ \textbf{(D)}\ 2\sqrt {3} \qquad$ $ \textbf{(E)}\ \frac {7}{2}$ 
