1. **Problem Statement:** Find the length of the segment $x$ in a circle where a chord of length 10.2 is given, a perpendicular segment from the chord to the circle center is 8, and the radius of the circle is 13. 2. **Understanding the problem:** We have a circle with radius $r=13$, a chord of length $10.2$, and a perpendicular distance from the center to the chord of $8$. We want to find the length $x$, which is the segment from the center perpendicular to the chord. 3. **Formula and rules:** The radius $r$ of the circle, the distance $d$ from the center to the chord, and half the chord length $c/2$ form a right triangle: $$r^2 = d^2 + \left(\frac{c}{2}\right)^2$$ 4. **Calculate half the chord length:** $$\frac{c}{2} = \frac{10.2}{2} = 5.1$$ 5. **Apply the Pythagorean theorem:** $$13^2 = 8^2 + 5.1^2$$ $$169 = 64 + 26.01$$ $$169 = 90.01$$ 6. **Check for $x$:** Since $x$ is the segment from the center perpendicular to the chord, and the distance from the center to the chord is given as 8, $x=8$. 7. **Conclusion:** The length of the segment $x$ is $8$ units.