To find the vertices of polygon A′B′C′D′ after dilating polygon ABCD with a scale factor of \(\frac{1}{4}\), we will apply the scale factor to each vertex.
The formula to apply a dilation with scale factor \( k \) from the origin \( (0, 0) \) is:
\[ (x', y') = (k \cdot x, k \cdot y) \]
Given \( k = \frac{1}{4} \), we will apply this to each vertex of polygon ABCD.
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Vertex A(−4, 6): \[ A' = \left(\frac{1}{4} \cdot -4, \frac{1}{4} \cdot 6\right) = (-1, 1.5) \]
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Vertex B(−2, 2): \[ B' = \left(\frac{1}{4} \cdot -2, \frac{1}{4} \cdot 2\right) = (-0.5, 0.5) \]
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Vertex C(4, −2): \[ C' = \left(\frac{1}{4} \cdot 4, \frac{1}{4} \cdot -2\right) = (1, -0.5) \]
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Vertex D(4, 4): \[ D' = \left(\frac{1}{4} \cdot 4, \frac{1}{4} \cdot 4\right) = (1, 1) \]
Now we can summarize the coordinates of the dilated polygon A′B′C′D′:
- A′(−1, 1.5)
- B′(−0.5, 0.5)
- C′(1, −0.5)
- D′(1, 1)
Looking at the provided options, the correct vertices of polygon A′B′C′D′ are:
A′(−1, 1.5), B′(−0.5, 0.5), C′(1, −0.5), D′(1, 1).