A triangle has angles of $30^\circ$ and $45^\circ$. If the side opposite the $45^\circ$ angle has length $8$, then the side opposite the $30^\circ$ angle has length
$\textbf{(A) }4\qquad \textbf{(B) }4\sqrt{2}\qquad \textbf{(C) }4\sqrt{3}\qquad \textbf{(D) }4\sqrt{6}\qquad  \textbf{(E) }6$
This triangle can be split into smaller 30-60-90 and 45-45-90 triangles. The side opposite the $45^\circ$ angle has length $8,$ so the 30-60-90 triangle has sides $4, 4\sqrt3,$ and $8.$
One of the legs of the 45-45-90 triangles is $4,$ so the hypotenuse is $4\sqrt2.$ This is also the side opposite the $30^\circ$ angle, so the answer is $\textbf{(B)}.$
-edited by coolmath34