1. **State the problem:** We need to find the value of $x$ in a circle with a chord and segments labeled 8, 3, and $x$, where two segments labeled 3 are congruent and a perpendicular radius/segment is given. 2. **Identify the relevant theorem:** When a radius or diameter is perpendicular to a chord, it bisects the chord. This means the chord is divided into two equal parts. 3. **Apply the theorem:** The chord length is 8, so each half is $\frac{8}{2} = 4$. 4. **Use the right triangle formed:** The radius (or segment) of length 3 is perpendicular to the chord, forming a right triangle with half the chord (4) and the segment $x$. 5. **Apply the Pythagorean theorem:** $$3^2 + x^2 = 4^2$$ 6. **Calculate:** $$9 + x^2 = 16$$ 7. **Isolate $x^2$:** $$x^2 = 16 - 9$$ $$x^2 = 7$$ 8. **Find $x$:** $$x = \sqrt{7}$$ **Final answer:** $$x = \sqrt{7}$$