To determine which statement is part of the proof that the dilation of the line segment \( CD \) with a scale factor of \( \frac{1}{3} \) and a center of dilation at the origin results in a line segment that is \( \frac{1}{3} \) the length of \( CD \), we first need to analyze the given statements.
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C′ is at (9,−18) and D′ is at (18,−9) - These coordinates do not represent points of a dilation about the origin with a scale factor of \( \frac{1}{3} \) for the likely points \( C \) and \( D \).
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The length of C′D′ is approximately 1.41 units. - This is a plausible length after dilation, but without knowing the original length of \( CD \), it's hard to validate this statement in terms of relation to \( \frac{1}{3} \).
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The length of line segment \( C′D′ \) is approximately 4.24. - This suggests that length \( CD \) was approximately \( 12.72 \) before dilation, making the new length appropriate given the scale factor of \( \frac{1}{3} \) (as \( 12.72 \div 3 \approx 4.24 \)).
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C′ is at (2,−1) and D′ is at (1,−2). - These points could potentially come from a dilation of a point relative to the origin, but they do not directly support the proof concerning the resulting length from dilation.
Given these observations, the most relevant statement supporting the proof that the dilation results in a line segment \( \frac{1}{3} \) the length of \( CD \) would depend on establishing similarity through the calculation of lengths before and after dilation.
Thus, the most appropriate statement to select would be:
The length of line segment \( C′D′ \) is approximately 4.24.
This indicates that the length after dilation accurately reflects the scale factor of \( \frac{1}{3} \) in relation to the original length \( CD \).
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