Solution:

The height of an equilateral triangle is $\frac{\sqrt{3}}{2}$ times the length of each of its sides. Thus, the length of one side of the equilateral triangle is $\frac{2}{\sqrt{3}} (8 \sqrt{3}) = 16$, and its area is
$$\frac{1}{2} (8 \sqrt{3})(16) = 64 \sqrt{3}$$
Since the area of the trapezoid is three times that of the equilateral triangle, we have
$$$$
$$\begin{gathered} \text{(height)} \times (\text{median of the trapezoid}) = 3 \cdot 64 \sqrt{3} \\ 8 \sqrt{3} \times (\text{median of the trapezoid}) = 3 \cdot 64 \sqrt{3} \\ \text{median of the trapezoid} = 24 \end{gathered}$$