To find the dilation of a shape centered at the origin with a scale factor of \( \frac{1}{2} \), you multiply each coordinate of the points by the scale factor.
Given the points:
- \( a = (2, 2) \)
- \( b = (0, 8) \)
- \( c = (4, -6) \)
We apply the dilation transformation \( (x, y) \rightarrow \left( \frac{1}{2}x, \frac{1}{2}y \right) \).
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For point \( a = (2, 2) \): \[ a' = \left( \frac{1}{2} \times 2, \frac{1}{2} \times 2 \right) = (1, 1) \]
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For point \( b = (0, 8) \): \[ b' = \left( \frac{1}{2} \times 0, \frac{1}{2} \times 8 \right) = (0, 4) \]
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For point \( c = (4, -6) \): \[ c' = \left( \frac{1}{2} \times 4, \frac{1}{2} \times -6 \right) = (2, -3) \]
Thus, the dilated points are:
- \( a' = (1, 1) \)
- \( b' = (0, 4) \)
- \( c' = (2, -3) \)
In summary, after performing the dilation by a factor of \( \frac{1}{2} \) centered at the origin, the new coordinates of the points are:
- \( a' = (1, 1) \)
- \( b' = (0, 4) \)
- \( c' = (2, -3) \)