Problem 175

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

1. Determine the third angle of the triangle:
Given two angles of the triangle are $30^\circ$ and $45^\circ$. The sum of the angles in any triangle is $180^\circ$. Therefore, the third angle is:
$$180^\circ - 30^\circ - 45^\circ = 105^\circ$$

2. **Drop an altitude from the $105^\circ$ angle:**
Dropping an altitude from the $105^\circ$ angle to the side opposite it divides the triangle into two right triangles: a $45^\circ-45^\circ-90^\circ$ triangle and a $30^\circ-60^\circ-90^\circ$ triangle.

3. **Analyze the $30^\circ-60^\circ-90^\circ$ triangle:**
In a $30^\circ-60^\circ-90^\circ$ triangle, the sides are in the ratio $1 : \sqrt{3} : 2$. The side opposite the $30^\circ$ angle is half the hypotenuse. Given that the side opposite the $45^\circ$ angle (which is the hypotenuse of the $30^\circ-60^\circ-90^\circ$ triangle) is $8$, the side opposite the $30^\circ$ angle is:
$$\frac{8}{2} = 4$$

4. **Analyze the $45^\circ-45^\circ-90^\circ$ triangle:**
In a $45^\circ-45^\circ-90^\circ$ triangle, the sides are in the ratio $1 : 1 : \sqrt{2}$. The side opposite the $30^\circ$ angle in the original triangle is the hypotenuse of this $45^\circ-45^\circ-90^\circ$ triangle. Therefore, the length of the side opposite the $30^\circ$ angle is:
$$4 \times \sqrt{2} = 4\sqrt{2}$$

Conclusion:
$$\boxed{4\sqrt{2}}$$