jmc

We add $\overline{WY}$ to our diagram and note that because $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp \overline{ZY}$, we have $\overline{WY}\perp \overline{WX}$. Therefore, triangles $WYX$ and $WYZ$ are right triangles.

From right triangle $WYZ$, we have $\tan Z=\frac{WY}{ZY}$, so $WY=ZY\cdot \tan Z=12\cdot 1.5=18$. From right triangle $WXY$, we have $\tan X=\frac{WY}{WX}$, so $$WX=\frac{WY}{\tan X}=\frac{18}{2}=9.$$Finally, the Pythagorean Theorem gives $$\begin{array}{rl}XY& =\sqrt{W{Y}^2+W{X}^2}\\ & =\sqrt{18^2+9^2}\\ & =\sqrt{(2\cdot 9{)}^2+9^2}\\ & =\sqrt{5\cdot 9^2}\\ & =\boxed{9\sqrt{5}}.\end{array}$$