jmc

Let the center of the circle be $(x,0)$. Then we know the distance from the center to $(0,4)$ and from the center to $(1,3)$ are the same. Using the distance formula, we have $$\begin{array}{rl}\sqrt{(x-0{)}^2+(0-4{)}^2}& =\sqrt{(x-1{)}^2+(0-3{)}^2}\\ \Rightarrow \sqrt{x^2+16}& =\sqrt{(x-1{)}^2+9}\\ \Rightarrow x^2+16& =(x-1{)}^2+9\\ \Rightarrow x^2+16& =x^2-2x+1+9\\ \Rightarrow 16& =-2x+10\\ \Rightarrow 6& =-2x\\ \Rightarrow x& =-3\end{array}$$ Now we know the center of the circle is $(-3,0)$, and we need to find the radius. Use the distance formula once more: $$\begin{array}{rl}\sqrt{(-3-0{)}^2+(0-4{)}^2}& =\sqrt{(-3{)}^2+(-4{)}^2}\\ & =\sqrt{9+16}\\ & =\sqrt{25}=\boxed{5}.\end{array}$$