DISCLAIMER: If this is wrong, just shoot me. I spent way too much time here.

First, draw a line through the intersection of the diagonals and perpendicular to both parallel bases. This will give us the heights of the two triangles whose areas we know.

Label the base of triangle A be and the base of triangle B be

The height of triangle A will be and the height of triangle B will be

Now, the areas of the two triangles:

The area of the trapezoid is given as:

Substituting, we get,

by AA postulate

That means:

*

Since we know from the * above,

Now,

Substituting back into our last T equation we finally get: