To find the dilation of a shape centered at the origin by a scale factor of \(\frac{1}{3}\), we multiply each coordinate of the shape by the dilation factor.
Given the points:
- \( A(3, 0) \)
- \( B(9, 0) \)
- \( C(6, 9) \)
We perform the dilation for each point as follows:
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For point \( A(3, 0) \): \[ A' = \left( 3 \cdot \frac{1}{3}, 0 \cdot \frac{1}{3} \right) = (1, 0) \]
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For point \( B(9, 0) \): \[ B' = \left( 9 \cdot \frac{1}{3}, 0 \cdot \frac{1}{3} \right) = (3, 0) \]
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For point \( C(6, 9) \): \[ C' = \left( 6 \cdot \frac{1}{3}, 9 \cdot \frac{1}{3} \right) = (2, 3) \]
Thus, the coordinates of the dilated shape are:
- \( A'(1, 0) \)
- \( B'(3, 0) \)
- \( C'(2, 3) \)
These are the points of the shape after applying the dilation of \(\frac{1}{3}\).