To find the scale factor used in the dilation of line segment \( AB \) centered at the origin, we need to compare the distances of the points before and after dilation.
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Identify the coordinates of points before dilation:
- \( A (-1, -3) \)
- \( B (-3, -5) \)
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Calculate the distance of segment \( AB \): \[ d_{AB} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \] Substituting the coordinates: \[ d_{AB} = \sqrt{(-3 - (-1))^2 + (-5 - (-3))^2} \] \[ = \sqrt{(-3 + 1)^2 + (-5 + 3)^2} \] \[ = \sqrt{(-2)^2 + (-2)^2} \] \[ = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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Identify the coordinates of points after dilation:
- \( A' (-3, -9) \)
- \( B' (-9, -15) \)
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Calculate the distance of segment \( A'B' \): \[ d_{A'B'} = \sqrt{(x_{B'} - x_{A'})^2 + (y_{B'} - y_{A'})^2} \] Substituting the coordinates: \[ d_{A'B'} = \sqrt{(-9 - (-3))^2 + (-15 - (-9))^2} \] \[ = \sqrt{(-9 + 3)^2 + (-15 + 9)^2} \] \[ = \sqrt{(-6)^2 + (-6)^2} \] \[ = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} \]
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Calculate the scale factor \( k \) of the dilation: \[ k = \frac{d_{A'B'}}{d_{AB}} = \frac{6\sqrt{2}}{2\sqrt{2}} = 3 \]
Thus, the scale factor used in the dilation of line segment \( AB \) centered at the origin is \( \boxed{3} \).
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