My officemate and I discussed this one on-and-off throughout the day, drawing and redrawing many extra lines and triangles. We did finally solve it, but by a kind of questionable meta-method. Here's the basic idea:
Imagine blowing up the circle just a little bit. The line on the left can still be 12 units long. It will still be possible to have another line cut a chord 7 units long. And it will still be possible to have those two lines meet at a point. The angle they form will be different, though, and the value of x may be different. But the angle isn't a given and neither is the diameter of the circle. Therefore that problem is the same problem as this one. Since we know this puzzle has a unique solution, the value of x must actually be a constant and we can make that angle whatever we want. We will make it such that the chord is actually a diameter.
From that point the puzzle is easy and this method did yield the correct answer. The questionable part is assuming there was a unique solution. If we had come across this puzzle "in the wild" this wouldn't have been at all kosher. So how did the first person solve this puzzle? What's the real solution?