To determine the scale factor used in the dilation of line segment \( AB \) centered at the origin, we first need to analyze the original points \( A(4, 1) \) and \( B(2, 3) \) and the dilated points \( A'(12, 3) \) and \( B'(6, 9) \).
Step 1: Calculate distances from origin for original points
- For point \( A(4, 1) \): \[ d_A = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} \]
- For point \( B(2, 3) \): \[ d_B = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]
Step 2: Calculate distances from origin for dilated points
- For point \( A'(12, 3) \): \[ d_{A'} = \sqrt{12^2 + 3^2} = \sqrt{144 + 9} = \sqrt{153} \]
- For point \( B'(6, 9) \): \[ d_{B'} = \sqrt{6^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117} \]
Step 3: Determine the scale factor
To find the scale factor \( k \), we can compare the distances of the dilated points to the original points.
For point \( A \): \[ k = \frac{d_{A'}}{d_A} = \frac{\sqrt{153}}{\sqrt{17}} = \sqrt{\frac{153}{17}} = \sqrt{9} = 3 \]
For point \( B \): \[ k = \frac{d_{B'}}{d_B} = \frac{\sqrt{117}}{\sqrt{13}} = \sqrt{\frac{117}{13}} = \sqrt{9} = 3 \]
Conclusion
The scale factor used in the dilation of line segment \( AB \), centered at the origin, is \( \boxed{3} \).