I honestly can't make heads or tails of your work. It isn't clear to me at all what you are doing. Where did

come from? And you can't use the law of cosines that way--you need 3 sides of the

*same *triangle.

Anyway, here is how I would do it:

First, I suggest finding the height of the "roof" (that is, the length of the perpendicular between

and

). Call it

, and we have:

Now everything follows from the Pythagorean theorem:

is found in much the same way, and

since the two inner right triangles are congruent (two sides and an included angle equal).