To determine the scale factor used in the dilation of line segment AB centered at the origin, we can use the coordinates of points A and A' (or B and B').
The scale factor \( k \) can be found using the following formula:
\[ k = \frac{\text{distance from the center (origin) to the dilated point}}{\text{distance from the center (origin) to the original point}} \]
First, we will calculate the distances from the origin to points A and A':
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Distance to A: \[ d_A = \sqrt{(4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]
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Distance to A': \[ d_{A'} = \sqrt{(12)^2 + (-6)^2} = \sqrt{144 + 36} = \sqrt{180} = 6\sqrt{5} \]
Now, we can calculate the scale factor using the distances:
\[ k = \frac{d_{A'}}{d_A} = \frac{6\sqrt{5}}{2\sqrt{5}} = \frac{6}{2} = 3 \]
Thus, the scale factor used in the dilation of line segment AB, centered at the origin, is \( \boxed{3} \).
To confirm this, we can apply the same to points B and B':
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Distance to B: \[ d_B = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Distance to B': \[ d_{B'} = \sqrt{(9)^2 + (-9)^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2} \]
Checking the scale factor:
\[ k = \frac{d_{B'}}{d_B} = \frac{9\sqrt{2}}{3\sqrt{2}} = \frac{9}{3} = 3 \]
This confirms that the scale factor is indeed \( \boxed{3} \).
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