To find the coordinates of points \( A' \) and \( B' \) after dilating the line segment \( AB \) with a scale factor of 13, centered at the origin, we use the formula for dilation:
\[ (x', y') = (kx, ky) \]
where \( k \) is the scale factor and \( (x, y) \) are the original coordinates.
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For point \( A(-6, 3) \): \[ A' = (13 \cdot -6, 13 \cdot 3) = (-78, 39) \]
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For point \( B(-12, 9) \): \[ B' = (13 \cdot -12, 13 \cdot 9) = (-156, 117) \]
Thus, the dilated coordinates are \( A'(-78, 39) \) and \( B'(-156, 117) \).
It appears that none of the provided response options match the calculated coordinates. Please double-check the scale factor or the original coordinates provided, as the options listed do not correspond to a scale factor of 13.
If the scale factor should actually be \( \frac{1}{13} \) instead, then the coordinates would be:
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For point \( A(-6, 3) \): \[ A' = \left(-\frac{6}{13}, \frac{3}{13}\right) \approx (-0.46, 0.23) \]
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For point \( B(-12, 9) \): \[ B' = \left(-\frac{12}{13}, \frac{9}{13}\right) \approx (-0.92, 0.69) \]
Again, none of the provided options reflect this as well. Please ensure the numbers are correct or check if another scale factor should be applied.