1. **State the problem:** Rowan wants to enlarge a shape using the origin $O$ as the center. The ratio of the length $OA$ to $OA'$ is $2:1$. Given $A(5,6)$, find both possible coordinates of $A'$. 2. **Understand the ratio and enlargement:** The ratio $OA : OA' = 2 : 1$ means $OA' = \frac{1}{2} OA$. 3. **Formula for enlargement from origin:** If $A = (x,y)$ and the scale factor is $k$, then $A' = (kx, ky)$. 4. **Calculate $A'$ for scale factor $k = \frac{1}{2}$:** $$A' = \left(\frac{1}{2} \times 5, \frac{1}{2} \times 6\right) = \left(\frac{5}{2}, 3\right) = (2.5, 3)$$ 5. **Consider the other possible pair:** Since the ratio is $2:1$, $OA'$ could also be twice $OA$ if the ratio is reversed (i.e., $OA : OA' = 1 : 2$), meaning $k=2$. 6. **Calculate $A'$ for scale factor $k=2$:** $$A' = (2 \times 5, 2 \times 6) = (10, 12)$$ 7. **Final answer:** The two possible coordinates for $A'$ are: $$\boxed{(2.5, 3) \text{ and } (10, 12)}$$