To find the scale factor used in the dilation of line \( AB \) to line \( A'B' \) with the center of dilation at the origin, we can follow these steps:
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Identify the coordinates of points A and B:
Point \( A \) is at \( (-3, 4) \) and point \( B \) is at \( (2, -4) \). -
Identify the coordinates of points A' and B':
Point \( A' \) is at \( (-6, 8) \) and point \( B' \) is at \( (4, -8) \). -
Calculate the distances from the origin to each point:
- For point \( A \): \[ \text{Distance to } A = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- For point \( B \): \[ \text{Distance to } B = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]
- For point \( A' \): \[ \text{Distance to } A' = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
- For point \( B' \): \[ \text{Distance to } B' = \sqrt{4^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5} \]
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Calculate the scale factor:
The scale factor \( k \) can be found by comparing the distances from the origin to the points before and after dilation. The distances scale as follows:- For point \( A \) to \( A' \): \[ k = \frac{\text{Distance to } A'}{\text{Distance to } A} = \frac{10}{5} = 2 \]
- For point \( B \) to \( B' \): \[ k = \frac{\text{Distance to } B'}{\text{Distance to } B} = \frac{4\sqrt{5}}{2\sqrt{5}} = 2 \]
Since both points give the same scale factor, we confirm that the scale factor used in the dilation from line \( AB \) to line \( A'B' \) is:
\[ \boxed{2} \]