To find the coordinates of the points A′ and B′ after dilation of the line segment AB with a scale factor of \( \frac{1}{3} \), centered at the origin, we will apply the dilation transformation to each point.
The formula for dilation centered at the origin is:
\[ (x', y') = (k \cdot x, k \cdot y) \]
where \( k \) is the scale factor, and \( (x, y) \) are the original coordinates.
The endpoints provided are:
- A(−6, 3)
- B(−12, 9)
Applying the dilation with scale factor \( k = \frac{1}{3} \):
- For point A(−6, 3):
\[ A' = \left(\frac{1}{3} \cdot -6, \frac{1}{3} \cdot 3\right) = \left(-2, 1\right) \]
- For point B(−12, 9):
\[ B' = \left(\frac{1}{3} \cdot -12, \frac{1}{3} \cdot 9\right) = \left(-4, 3\right) \]
Thus, the coordinates after dilation are:
\[ A'(-2, 1) \quad \text{and} \quad B'(-4, 3) \]
The correct response is:
A′(−2,1) and B′(−4,3).