1. **State the problem:** We have a circle with center $O$ and diameter $FG$. The arc $FXG$ has length $14\pi$. We need to find the length of the line segment $XO$. 2. **Understand the problem:** Since $FG$ is a diameter, the circle's radius is half the length of $FG$. The arc $FXG$ is part of the circle's circumference. 3. **Formula for circumference:** The circumference $C$ of a circle is given by $$C = 2\pi r$$ where $r$ is the radius. 4. **Arc length formula:** The length of an arc is $$\text{arc length} = r \times \theta$$ where $\theta$ is the central angle in radians subtending the arc. 5. **Given arc length:** The arc $FXG$ length is $14\pi$, so $$14\pi = r \times \theta$$ 6. **Diameter and radius:** Since $FG$ is a diameter, the full circumference is $$C = 2\pi r$$ 7. **Arc $FXG$ is more than half the circle:** Because $14\pi$ is more than half the circumference if $r$ is small, let's find $r$ first. 8. **Find $r$ using the diameter:** The diameter $FG$ is twice the radius, so $$FG = 2r$$ 9. **Arc $FXG$ is the major arc between $F$ and $G$ passing through $X$:** The minor arc $FG$ would be $\pi r$ (half the circumference), so the major arc $FXG$ is $$C - \pi r = 2\pi r - \pi r = \pi r$$ 10. **Given arc length $14\pi$ equals major arc length $\pi r$:** $$14\pi = \pi r$$ 11. **Solve for $r$:** $$\cancel{\pi}14 = \cancel{\pi} r \Rightarrow r = 14$$ 12. **Find $XO$:** Since $X$ lies on the circle, $XO$ is a radius, so $$XO = r = 14$$ **Final answer:** $$\boxed{14}$$