To find the scale factor used in the dilation of line segment \(AB\), we need to compare the coordinates of points \(A\) and \(B\) with the coordinates of points \(A'\) and \(B'\).
The original points are:
- \(A(2, 3)\)
- \(B(4, 1)\)
The dilated points are:
- \(A'(6, 9)\)
- \(B'(12, 3)\)
First, we find the distances from the origin to the points \(A\) and \(A'\) and then to points \(B\) and \(B'\).
Step 1: Calculate distances from origin to points \(A\) and \(A'\)
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Calculate the distance \(OA\): \[ OA = \sqrt{(2-0)^2 + (3-0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]
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Calculate the distance \(OA'\): \[ OA' = \sqrt{(6-0)^2 + (9-0)^2} = \sqrt{6^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117} \]
Step 2: Calculate distances from origin to points \(B\) and \(B'\)
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Calculate the distance \(OB\): \[ OB = \sqrt{(4-0)^2 + (1-0)^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} \]
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Calculate the distance \(OB'\): \[ OB' = \sqrt{(12-0)^2 + (3-0)^2} = \sqrt{12^2 + 3^2} = \sqrt{144 + 9} = \sqrt{153} \]
Step 3: Find the scale factor
The scale factor \(k\) can be calculated using the distances from the origin to the points before and after dilation.
We can find \(k\) using point \(A\): \[ k = \frac{OA'}{OA} = \frac{\sqrt{117}}{\sqrt{13}} = \sqrt{\frac{117}{13}} = \sqrt{9} = 3 \]
Now, let's verify it using point \(B\): \[ k = \frac{OB'}{OB} = \frac{\sqrt{153}}{\sqrt{17}} = \sqrt{\frac{153}{17}} = \sqrt{9} = 3 \]
Both methods yield the same scale factor, confirming the result.
Conclusion
The scale factor used in the dilation of line segment \(AB\), centered at the origin, is: \[ \boxed{3} \]