To find the coordinates \( A' \) and \( B' \) after dilation of the line segment \( AB \) with a scale factor of \( 13 \) centered at the origin, we will multiply the coordinates of points \( A \) and \( B \) by the scale factor.
The original coordinates are:
- \( A(-6, 3) \)
- \( B(-12, 9) \)
Now applying the dilation:
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For point \( A \): \[ A' = (x_A \cdot k, y_A \cdot k) = (-6 \cdot 13, 3 \cdot 13) = (-78, 39) \]
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For point \( B \): \[ B' = (x_B \cdot k, y_B \cdot k) = (-12 \cdot 13, 9 \cdot 13) = (-156, 117) \]
Thus, the coordinates after dilation are:
- \( A'(-78, 39) \)
- \( B'(-156, 117) \)
None of the given response options match these calculated coordinates. Please check the responses or provided scale factor and points once again, as it seems they do not correspond correctly.