To find the length of chord DF, we can use the properties of a circle. Since point X is the center of the circle, we know that the line segment XB is also a radius of the circle. Therefore, XB = AC = DF.
We also know that BC = 12. Since XB is a radius, we can use the Pythagorean theorem to find the length of AB, which is half of DF. We have:
AB^2 + BC^2 = XB^2
AB^2 + 12^2 = XB^2
AB^2 + 144 = XB^2
But we also know that AB = DF/2, so we can substitute that into the equation above:
(DF/2)^2 + 144 = XB^2
DF^2/4 + 144 = XB^2
Finally, we substitute XB = AC = DF to get:
DF^2/4 + 144 = DF^2
144 = 3DF^2/4
DF^2 = 192
DF = sqrt(192) ≈ 13.86
Therefore, the length of chord DF is approximately 13.86.