A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $105$ and $91$, respectively. What is the volume of the pyramid? $\textbf{(A)}\ 392 \qquad$ $ \textbf{(B)}\ 196\sqrt {6} \qquad$ $ \textbf{(C)}\ 392\sqrt {2} \qquad$ $ \textbf{(D)}\ 392\sqrt {3} \qquad$ $ \textbf{(E)}\ 784$
Solution
Base side $14$; place $A=(0,0,0),B=(14,0,0),C=(14,14,0),D=(0,14,0)$, apex $E=(x,y,z)$.
A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $1296$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $450$ and $522$, respectively. What is the volume of the pyramid?
$\textbf{(A)}\ 392 \qquad$ $ \textbf{(B)}\ 196\sqrt {6} \qquad$ $ \textbf{(C)}\ 392\sqrt {2} \qquad$ $ \textbf{(D)}\ 392\sqrt {3} \qquad$ $ \textbf{(E)}\ 784$ 