To determine the scale factor used in the dilation of line segment \( AB \), we need to find the ratio of the lengths of the dilated segment to the original segment.
First, we find the lengths of the points \( A(4, -12) \) and \( B(8, -16) \) from the origin \( (0, 0) \).
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Calculate the distance from the origin to point \( A \): \[ OA = \sqrt{(4-0)^2 + (-12-0)^2} = \sqrt{4^2 + (-12)^2} = \sqrt{16 + 144} = \sqrt{160} = 4\sqrt{10} \]
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Calculate the distance from the origin to point \( B \): \[ OB = \sqrt{(8-0)^2 + (-16-0)^2} = \sqrt{8^2 + (-16)^2} = \sqrt{64 + 256} = \sqrt{320} = 8\sqrt{5} \]
Now that we have the distances \( OA \) and \( OB \), we find the scale factor \( k \) by dividing the distances:
\[ k = \frac{OB}{OA} = \frac{8\sqrt{5}}{4\sqrt{10}} = 2 \cdot \frac{\sqrt{5}}{\sqrt{10}} = 2 \cdot \frac{\sqrt{5}}{\sqrt{5} \cdot \sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]
Thus, the scale factor of the dilation is \( \sqrt{2} \).