I forget how to do this. (Right angle triangles with barely any available measurements)

Question

So I'm designing a roof in Sketchup for school, and I came across this and can't seem to figure it out. The angle between both sides of the roof has to be a 90° angle (see picture). This particular angle has to be situated along line y, which is 48 inches from the right and 240 from the left side. Line x, which runs from point A to B, is what I need to find out. I brute forced it and got something around 107, but it's not precise enough. Please explain how to find the length of x. This is the picture, it won't let me post it because I don't have 10 rep yet... https://i.stack.imgur.com/MjQoL.jpg

Edit: yes, all angles touching point B are right angles. Forgot to add those...

Answer 1

Let call the left most point of your diagram $X$ and the right most point $Y$. Now notice that the angle at $X$ is the same as the angle $BAY$ ; lets call this angle $\phi$ & the perpendicular distance required $\lambda$ \begin{eqnarray*} tan \phi =\frac{\lambda}{240} =\frac{48}{\lambda} \end{eqnarray*} so $\lambda^2=48 \times 240 $. Which gives $\lambda=48 \sqrt{5}$ which is about $107.33$

Answer 2

Use the Geometric mean theorem to compute

$$ \sqrt{240 \cdot 48} = 107.33 \enspace. $$