To determine the scale factor of the dilation of triangle ABC to triangle A'B'C', we need to compare the coordinates of corresponding vertices before and after dilation.
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Calculate the distances from the origin to the points of triangle ABC:
- For point A(2, 1): \[ \text{Distance from A to origin} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]
- For point B(−2, 2): \[ \text{Distance from B to origin} = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
- For point C(−3, −1): \[ \text{Distance from C to origin} = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]
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Calculate the distances from the origin to the points of triangle A'B'C':
- For point A'(4, 2): \[ \text{Distance from A' to origin} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]
- For point B'(-4, 4): \[ \text{Distance from B' to origin} = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
- For point C'(-6, -2): \[ \text{Distance from C' to origin} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]
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Determine the scale factor \( k \): The scale factor for a dilation centered at the origin can be found by comparing the distance from the origin to a point before and after the dilation: \[ k = \frac{\text{Distance from the origin to the dilated point}}{\text{Distance from the origin to the original point}} \]
We can check the scale factor using any of the corresponding points. Let's use point A and A':
\[ k = \frac{\text{Distance from A' to origin}}{\text{Distance from A to origin}} = \frac{2\sqrt{5}}{\sqrt{5}} = 2 \]
By repeating the process for the other pairs of points (B with B' and C with C'), you will find the same scale factor of \( k = 2 \).
Final answer: The scale factor for the dilation is \( \boxed{2} \).
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