And correct answer, Henry. I follow your and Larry's reasoning and I agree. I haven't checked the trig numbers, but you got the right answer..
Here is a solution with no trig. Using your and Larry's labels:
As Larry pointed out, triangles CXE and CDE are similar. From that, we know that length of CX is the same as CD, or 1.
Similarly, XFE is similar to AFE, so AF = FX. Let's call that length "z".
Consider the triangle FBC. Its hypotenuse is 1+z and its sides are of lengths 1 and 1-z. Pythagoras tells us that
(1+z)^2 = (1-z)^2 + 1^2.
1+2z+z^2 = 1-2z+z^2 + 1
so CF = 1.25