Geometry puzzle

[democracy]

ceptimus - Thanks. It a great problem. I was hoping that we would not need an approximation. I'm not looking for that and I won't measure the lines because there is nothing to learn but the correct answer, which is ho-hum to me. The process is what is intriguing about the problem and solution.

[democracy]

There are two vertical lines that cut through the centers of the two circles and meet at line x. In both of these lines, the middle segment = 3. There are total of three lines that met x with a middle segment = 3. One does not run through center of a circle.

[ceptimus]

As usual, find the value of x.

Two right-angled triangles sharing a common side that is the hypotenuse of neither. One has height 5, the other has height 3. x is the height (perpendicular distance from the common side) of the intersection of the two hypotenuses.

[JoeP]

show spoiler

15/8, 1.875

Let b be the length of the base, and y be the distance along the base to the perpendicular x.

Then because they are straight lines or because they are tangents of congruent angles or any way you wish to see it
x/y = 5/b
and
x/(b-y) = 3/b

Solving gives y = 3b/8 and x = 15/8

[ceptimus]

JoeP wins the prize!

I did it exactly the same way, even using one of the same letters - but I had 'a' in place of your 'y'.