1. The problem asks to find the coordinates of points A' and B' after a dilation of the line segment AB with a scale factor of 2, centered at the origin. 2. The original points are A(3,7) and B(4,9). 3. The formula for dilation centered at the origin with scale factor $k$ is: $$ A' = (k \cdot x, k \cdot y) $$ where $(x,y)$ are the coordinates of the original point. 4. Applying the formula to point A: $$ A' = (2 \cdot 3, 2 \cdot 7) = (6, 14) $$ 5. Applying the formula to point B: $$ B' = (2 \cdot 4, 2 \cdot 9) = (8, 18) $$ 6. Therefore, the coordinates after dilation are A'(6,14) and B'(8,18). 7. The correct choice is: A' (6, 14) and B' (8, 18).