To find the coordinates \( A' \) and \( B' \) after a dilation of the line segment \( AB \) with a scale factor of \( \frac{1}{3} \), centered at the origin, we use the formula for dilation:
\[ (x', y') = (kx, ky) \]
where \( (x, y) \) are the original coordinates, and \( k \) is the scale factor.
Given:
- Point \( A(-6, 3) \)
- Point \( B(-12, 9) \)
With a scale factor \( k = \frac{1}{3} \):
For point \( A' \):
\[ A' = \left(-6 \cdot \frac{1}{3}, 3 \cdot \frac{1}{3}\right) = \left(-2, 1\right) \]
For point \( B' \):
\[ B' = \left(-12 \cdot \frac{1}{3}, 9 \cdot \frac{1}{3}\right) = \left(-4, 3\right) \]
Thus, the coordinates after dilation are:
\[ A'(-2, 1) \quad \text{and} \quad B'(-4, 3) \]
So the correct choice is:
A′(−2,1) and B′(−4,3).