To perform a dilation centered at the origin with a scale factor of \( \frac{1}{2} \), we multiply each coordinate of the original points by the scale factor.
The original coordinates of the vertices are:
- \( P(-10, -10) \)
- \( Q(-10, 6) \)
- \( R(-8, 10) \)
- \( S(-8, -6) \)
Now, we will apply the dilation:
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For \( P' \): \[ P' = \left( -10 \times \frac{1}{2}, -10 \times \frac{1}{2} \right) = \left( -5, -5 \right) \]
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For \( Q' \): \[ Q' = \left( -10 \times \frac{1}{2}, 6 \times \frac{1}{2} \right) = \left( -5, 3 \right) \]
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For \( R' \): \[ R' = \left( -8 \times \frac{1}{2}, 10 \times \frac{1}{2} \right) = \left( -4, 5 \right) \]
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For \( S' \): \[ S' = \left( -8 \times \frac{1}{2}, -6 \times \frac{1}{2} \right) = \left( -4, -3 \right) \]
Thus, the coordinates of the dilated vertices are:
- \( P'(-5, -5) \)
- \( Q'(-5, 3) \)
- \( R'(-4, 5) \)
- \( S'(-4, -3) \)
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