To determine the magnitude of the dilation from parallelogram ABCD to A'B'C'D', we need to follow these steps:
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Identify the coordinates of the vertices of parallelogram ABCD:
- A(-2, 4)
- B(2, 2)
- C(1, -1)
- D(-4, -2)
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Identify the coordinates of the vertices of parallelogram A'B'C'D':
- A'(-1, 2)
- B'(4, 4)
- C'(2, -2)
- D'(-2, -1)
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Calculate the dilation factor (k) between corresponding points from A to A', B to B', C to C', and D to D' by using the formula: \[ k = \frac{distance \ from \ origin \ of \ A'}{distance \ from \ origin \ of \ A} \]
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For point A(-2, 4):
- Distance from origin to A: \(\sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}\)
- Distance from origin to A': \(\sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}\)
- Thus, \(k_A = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2}\)
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For point B(2, 2):
- Distance from origin to B: \(\sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\)
- Distance from origin to B': \(\sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\)
- Thus, \(k_B = \frac{4\sqrt{2}}{2\sqrt{2}} = 2\)
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For point C(1, -1):
- Distance from origin to C: \(\sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}\)
- Distance from origin to C': \(\sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\)
- Thus, \(k_C = \frac{2\sqrt{2}}{\sqrt{2}} = 2\)
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For point D(-4, -2):
- Distance from origin to D: \(\sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\)
- Distance from origin to D': \(\sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}\)
- Thus, \(k_D = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2}\)
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Conclusion: The dilation factor (magnitude of dilation) is different for corresponding vertices, suggesting that the dilation is not uniform across the parallelogram. However, when considering the indicated points and dilations, a common factor of dilation observed among respective distances would indicate aberration.
Furthermore, looking at A (-2, 4) dilated to A'(-1, 2) yields, \[ k = \frac{d(A', O)}{d(A, O)} = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2}. \]
Therefore, the computed dilation factor is \( \frac{1}{2} \).
Thus, the magnitude of the dilation is: \[ \frac{1}{2} \]
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