To determine the scale factor used in the dilation of line segment \( AB \) (from points \( A(2, 3) \) and \( B(4, 1) \)) to line segment \( A'B' \) (from points \( A'(6, 9) \) and \( B'(12, 3) \)), we need to find the distances from the origin for each of the points \( A \), \( B \), \( A' \), and \( B' \).
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Calculate the distance from the origin to point A: \[ d(A) = \sqrt{(2-0)^2 + (3-0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]
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Calculate the distance from the origin to point B: \[ d(B) = \sqrt{(4-0)^2 + (1-0)^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} \]
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Calculate the distance from the origin to point A': \[ d(A') = \sqrt{(6-0)^2 + (9-0)^2} = \sqrt{6^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117} \]
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Calculate the distance from the origin to point B': \[ d(B') = \sqrt{(12-0)^2 + (3-0)^2} = \sqrt{12^2 + 3^2} = \sqrt{144 + 9} = \sqrt{153} \]
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Find the scale factor for the dilation: The scale factor \( k \) can be determined by comparing the distances of the corresponding points:
\[ k = \frac{d(A')}{d(A)} = \frac{d(B')}{d(B)} \]
Calculating the scale factor using point A: \[ k_A = \frac{\sqrt{117}}{\sqrt{13}} = \sqrt{\frac{117}{13}} = \sqrt{9} = 3 \]
Calculating the scale factor using point B: \[ k_B = \frac{\sqrt{153}}{\sqrt{17}} = \sqrt{\frac{153}{17}} = \sqrt{9} = 3 \]
Since both calculations yield the same scale factor, we can conclude that the scale factor used in the dilation of line segment \( AB \) is \( \boxed{3} \).