To find the scale factor used in the dilation of line segment \( AB \) (from points \( A(2, 3) \) and \( B(4, 1) \)) to line segment \( A'B' \) (with \( A'(6, 9) \) and \( B'(12, 3) \)), we can use the formula for scale factor in dilation, which relates the coordinates of the original points and the dilated points.
The scale factor \( k \) can be determined by comparing the coordinates of points A and A' (or B and B').
Steps:
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Find the scale factor for point A to A':
- Original coordinates of A: \( (2, 3) \)
- Dilated coordinates of A': \( (6, 9) \)
For the x-coordinate: \[ k_x = \frac{x'}{x} = \frac{6}{2} = 3 \]
For the y-coordinate: \[ k_y = \frac{y'}{y} = \frac{9}{3} = 3 \]
Thus, the scale factor from A to A' is \( k = 3 \).
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Now, let's check the scale factor for point B to B':
- Original coordinates of B: \( (4, 1) \)
- Dilated coordinates of B': \( (12, 3) \)
For the x-coordinate: \[ k_x = \frac{x'}{x} = \frac{12}{4} = 3 \]
For the y-coordinate: \[ k_y = \frac{y'}{y} = \frac{3}{1} = 3 \]
The scale factor from B to B' is also \( k = 3 \).
Conclusion:
Since both calculations yield the same scale factor, the scale factor used in the dilation of line segment \( AB \), centered at the origin, is
\[ \boxed{3} \]