To find the coordinates of the vertices after a dilation with a scale factor of \(\frac{1}{3}\) centered at the origin, you multiply each coordinate of the points by the scale factor.
Given the points:
- \(D (-6, -9)\)
- \(E (3, -9)\)
- \(F (3, -3)\)
- \(G (-6, -3)\)
We can apply the dilation:
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For point \(D\): \[ D' = \left(-6 \times \frac{1}{3}, -9 \times \frac{1}{3}\right) = \left(-2, -3\right) \]
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For point \(E\): \[ E' = \left(3 \times \frac{1}{3}, -9 \times \frac{1}{3}\right) = \left(1, -3\right) \]
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For point \(F\): \[ F' = \left(3 \times \frac{1}{3}, -3 \times \frac{1}{3}\right) = \left(1, -1\right) \]
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For point \(G\): \[ G' = \left(-6 \times \frac{1}{3}, -3 \times \frac{1}{3}\right) = \left(-2, -1\right) \]
Therefore, the coordinates of the vertices after dilation are:
- \(D' = (-2, -3)\)
- \(E' = (1, -3)\)
- \(F' = (1, -1)\)
- \(G' = (-2, -1)\)