If we call the length of the middle horizontal x, we can use the formula for the radius of the incircles as before to determine the height of each trapezoid in terms of x.

Since we are told the angle on the right is a right angle, we can find two similar triangles above and below it (in fact they are congruent, but we don’t need to assume that for our solution).

So (x-5)(x-12)=10x/(x+5) * 24x/(x=12)

(x+5)(x-5)(x+12)(x-12)=240x^2

(x^2-25)(x^2-144)=240x^2

Let’s make a substitution and call x^2 y

(y-25)(y-144)=240y

y^2-409y+3600=0

y=9 or 400

x therefore is +/-3 or +/-20

In the diagram, it is clearly bigger than 12, so must be equal to 20.

Plugging this value into the height expressions above gives the overall height as 8+15=23.