1. **State the problem:** We are given the endpoints of a diameter of a circle at points $(3,4)$ and $(5,-2)$, and we need to find the radius of the circle. 2. **Recall the formula:** The radius $r$ of a circle is half the length of its diameter. The length of the diameter can be found using the distance formula between two points $A(x_1,y_1)$ and $B(x_2,y_2)$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate the diameter length:** $$d = \sqrt{(5 - 3)^2 + (-2 - 4)^2} = \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40}$$ 4. **Simplify the square root:** $$\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$$ 5. **Find the radius:** $$r = \frac{d}{2} = \frac{2\sqrt{10}}{2} = \cancel{\frac{2}{2}}\sqrt{10} = \sqrt{10}$$ 6. **Approximate the radius:** $$\sqrt{10} \approx 3.162$$ **Final answer:** The radius of the circle is $\sqrt{10}$, approximately 3.162 units.