To find the magnitude of the dilation, we need to compare the coordinates of corresponding points in polygon ABCD and polygon A'B'C'D'. The general formula for dilation with respect to the origin is:
\[ (x', y') = k \cdot (x, y) \]
where \( k \) is the scale factor of the dilation.
We will calculate \( k \) using corresponding points from polygon ABCD and polygon A'B'C'D':
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For point A and A':
- Original A: \((-1, -1)\)
- Dilated A': \((-2, -2)\)
The dilation factor \( k \) can be calculated as:
\[ k = \frac{x'}{x} = \frac{-2}{-1} = 2 \quad \text{and} \quad k = \frac{y'}{y} = \frac{-2}{-1} = 2 \]
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For point B and B':
- Original B: \((-1, 1)\)
- Dilated B': \((-2, 2)\)
The dilation factor \( k \) is:
\[ k = \frac{x'}{x} = \frac{-2}{-1} = 2 \quad \text{and} \quad k = \frac{y'}{y} = \frac{2}{1} = 2 \]
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For point C and C':
- Original C: \((2, 3)\)
- Dilated C': \((2, 3)\)
Here, the dilation factor \( k \) is:
\[ k = \frac{x'}{x} = \frac{2}{2} = 1 \quad \text{and} \quad k = \frac{y'}{y} = \frac{3}{3} = 1 \]
This indicates that point C does not change in position, suggesting that it is at the center of dilation or unaffected by dilation.
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For point D and D':
- Original D: \((2, -1)\)
- Dilated D': \((4, -2)\)
The dilation factor \( k \) is:
\[ k = \frac{x'}{x} = \frac{4}{2} = 2 \quad \text{and} \quad k = \frac{y'}{y} = \frac{-2}{-1} = 2 \]
Conclusion
After comparing all the points, we find that the dilation factor is \( k = 2 \) for points A, B, and D, while point C remains unchanged.
Thus, the magnitude of dilation is:
\[ \boxed{2} \]
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