1. The problem asks to find the coordinates of points A' and B' after a dilation of line segment AB with scale factor $\frac{1}{3}$ centered at the origin. 2. The formula for dilation centered at the origin is: $$ A' = (k \cdot x, k \cdot y) $$ where $k$ is the scale factor, and $(x,y)$ are the original coordinates. 3. Given points: $$ A(-6, 3), B(-12, 9) $$ Scale factor: $$ k = \frac{1}{3} $$ 4. Calculate $A'$: $$ A' = \left( \frac{1}{3} \times (-6), \frac{1}{3} \times 3 \right) = (-2, 1) $$ 5. Calculate $B'$: $$ B' = \left( \frac{1}{3} \times (-12), \frac{1}{3} \times 9 \right) = (-4, 3) $$ 6. Therefore, the coordinates after dilation are: $$ A'(-2, 1), B'(-4, 3) $$ 7. The correct choice is: A'(-2, 1) and B'(-4, 3).