2016 AMC 10B Problem 20

The dilation scale factor is $\frac32$, since the radius changes from $2$ to $3$. The center of dilation $C$ lies on the line through $A(2,2)$ and $A'(5,6)$. The vector from $A$ to $A'$ is $(3,4)$. If $C$ is the dilation center, then the scale factor gives $$A'-A=\frac12(A-C).$$ Thus $A-C=2(3,4)$, so $C=(2,2)-2(3,4)=(-4,-6).$ Under a scale factor $\frac32$ dilation about $C$, a point moves by half its distance from $C$. Since $CO=\sqrt{(-4)^2+(-6)^2}=\sqrt{52}$, the origin moves $\frac12\sqrt{52}=\sqrt{13}$. Thus, the correct answer is C.