To find the coordinates of the points A' and B after dilation, we can use the formula for dilation centered at the origin, which is given by:
\[ (x', y') = (k \cdot x, k \cdot y) \]
where \( k \) is the scale factor, \( (x, y) \) are the original coordinates, and \( (x', y') \) are the new coordinates after dilation.
Given that the scale factor \( k = \frac{1}{3} \), we can calculate the new coordinates for points A and B.
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For point A(-6, 3): \[ A' = \left(\frac{1}{3} \cdot (-6), \frac{1}{3} \cdot 3\right) = \left(-2, 1\right) \]
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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: \[ A'(-2, 1) \quad \text{and} \quad B'(-4, 3) \]
Therefore, the correct answer is: A'(-2, 1) and B'(-4, 3).