1. **State the problem:** We have a line segment $\overline{AB}$ with endpoints $A(1, 3)$ and $B(5, 3)$. A dilation centered at the origin with scale factor 4 is applied to this segment. We need to find the coordinates of the dilated points $A'$ and $B'$. 2. **Formula for dilation centered at the origin:** If a point $(x, y)$ is dilated by a scale factor $k$ centered at the origin, the new coordinates $(x', y')$ are given by: $$ (x', y') = (k \cdot x, k \cdot y) $$ 3. **Apply the formula to point $A(1, 3)$:** $$ A' = (4 \times 1, 4 \times 3) = (4, 12) $$ 4. **Apply the formula to point $B(5, 3)$:** $$ B' = (4 \times 5, 4 \times 3) = (20, 12) $$ 5. **Check possible coordinates:** The possible coordinates given are $\left(\frac{1}{4}, \frac{3}{4}\right), \left(\frac{5}{4}, \frac{3}{4}\right), (4, 12), (20, 12), (4, 3)$. The dilated points $A'$ and $B'$ match $(4, 12)$ and $(20, 12)$ respectively. **Final answer:** $$ A' = (4, 12), \quad B' = (20, 12) $$