This is an interesting answer:
Using similar triangles twice, and calling the length of the side of the square x,
in the triangle CYZ, CY/x = 4/3, so CY = 4x/3,
in the triangle WXA, x/XA = 3/4, so XA = 3x/4.
The hypotenuse of the triangle is
CY + YX + XA = 4x/3 + x + 3x/4 = 37x/12 = 5,
so x = 60/37.
(Solved by Guest - Nov 13, 2015)