To find the scale factor used in the dilation of line segment \( AB \) to line segment \( A'B' \), we can use the coordinates of points \( A \), \( B \), \( A' \), and \( B' \).
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Identify the coordinates:
- Point \( A (-3, 4) \)
- Point \( B (2, -4) \)
- Point \( A' (-6, 8) \)
- Point \( B' (4, -8) \)
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Calculate the distances from the origin for points \( A \) and \( A' \):
- Distance to \( A \): \[ d_A = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- Distance to \( A' \): \[ d_{A'} = \sqrt{(-6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
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Calculate the distances from the origin for points \( B \) and \( B' \):
- Distance to \( B \): \[ d_B = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]
- Distance to \( B' \): \[ d_{B'} = \sqrt{(4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5} \]
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Determine the scale factor:
- From \( A \) to \( A' \): \[ \text{Scale Factor} = \frac{d_{A'}}{d_A} = \frac{10}{5} = 2 \]
- From \( B \) to \( B' \): \[ \text{Scale Factor} = \frac{d_{B'}}{d_B} = \frac{4\sqrt{5}}{2\sqrt{5}} = 2 \]
Both scale factors are equal, confirming that the same scale factor was used for the dilation.
Therefore, the scale factor used in the dilation of line segment \( AB \) to line segment \( A'B' \) is \( \boxed{2} \).