1. **Problem Statement:** We are given a circle centered at point A with coordinates $(5, -1)$ and a point B on the circle at coordinates $(5, 1)$. We need to find the radius of the circle. 2. **Formula:** The radius of a circle is the distance from the center to any point on the circle. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate the distance between A and B:** $$d = \sqrt{(5 - 5)^2 + (1 - (-1))^2} = \sqrt{0^2 + (1 + 1)^2} = \sqrt{0 + 2^2} = \sqrt{4}$$ 4. **Simplify:** $$d = 2$$ 5. **Interpretation:** The radius of the circle is the distance from the center A to point B, which is $2$ units. **Final answer:** The radius of the circle is **2 units**.