Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid?

$\textbf{(A) }600\qquad\textbf{(B) }650\qquad\textbf{(C) }700\qquad\textbf{(D) }750\qquad\textbf{(E) }800$
If you draw altitudes from $A$ and $B$ to $CD,$ the trapezoid will be divided into two right triangles and a rectangle. You can find the values of $a$ and $b$ with the [Pythagorean theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_theorem).
\[a=\sqrt{15^2-12^2}=\sqrt{81}=9\]
\[b=\sqrt{20^2-12^2}=\sqrt{256}=16\]
$ABYX$ is a rectangle so $XY=AB=50.$
\[CD=a+XY+b=9+50+16=75\]
The area of the trapezoid is
\[12\cdot \frac{(50+75)}{2} = 6(125) = \boxed{\textbf{(D)}\ 750}\]