1. **Problem Statement:** Given a circle centered at the origin $O(0,0)$, with points $A(3,4)$ and $B$ on the circle, find the length of the chord $AB$. 2. **Understanding the problem:** Point $A$ lies on the circle, so the radius $r$ of the circle is the distance from $O$ to $A$. 3. **Calculate the radius $r$:** $$r = OA = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 4. **Locate point $B$:** Point $B$ lies on the circle on the positive x-axis, so its coordinates are $(5,0)$ because the radius is 5. 5. **Calculate length of chord $AB$:** Use the distance formula between points $A(3,4)$ and $B(5,0)$: $$AB = \sqrt{(5-3)^2 + (0-4)^2} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$$ 6. **Answer:** The length of $AB$ is $2\sqrt{5}$. Therefore, the correct choice is C) $2\sqrt{5}$.