1. **State the problem:** Rowan wants to enlarge a shape using the origin $O$ as the center. The ratio of the length $OA$ to the length $OA'$ is $2:1$. Given point $A$ at coordinates $(6,6)$, find both possible coordinates of $A'$. 2. **Understand the ratio and enlargement:** The ratio $OA : OA' = 2 : 1$ means the length from the origin to $A'$ is half the length from the origin to $A$. 3. **Formula for enlargement about the origin:** If $A = (x,y)$ and the scale factor is $k$, then $A' = (kx, ky)$. 4. **Calculate the scale factor:** Since $OA : OA' = 2 : 1$, the scale factor $k = \frac{1}{2}$. 5. **Calculate $A'$ coordinates:** $$A' = \left(\frac{1}{2} \times 6, \frac{1}{2} \times 6\right) = (3, 3)$$ 6. **Consider the other possible pair:** Enlargement can also be a reduction with a negative scale factor, reflecting $A$ through the origin. 7. **Calculate $A'$ with negative scale factor:** $$A' = \left(-\frac{1}{2} \times 6, -\frac{1}{2} \times 6\right) = (-3, -3)$$ **Final answer:** The two possible coordinates of $A'$ are $(3, 3)$ and $(-3, -3)$.