imc

It shouldn't be hard to use trigonometry to bash this and find the height, but there is a much easier way. Extend $\overline{AD}$ and $\overline{BC}$ to meet at point $E$: Since $\overline{AB}||\overline{CD}$ we have $△AEB\sim △DEC$, with the ratio of proportionality being $\frac{39}{52}=\frac{3}{4}$. Thus $$\begin{array}{rl}\frac{CE}{CE+12}=\frac{3}{4}& ⟹CE=36\\ \frac{DE}{DE+5}=\frac{3}{4}& ⟹DE=15\end{array}$$ So the sides of $△CDE$ are $15,36,39$, which we recognize to be a $5-12-13$ right triangle. Therefore (we could simplify some of the calculation using that the ratio of areas is equal to the ratio of the sides squared), $$[ABCD]=[ABE]-[CDE]=\frac{1}{2}\cdot 20\cdot 48-\frac{1}{2}\cdot 15\cdot 36=\boxed{210}$$