To determine which statement is part of the proof that the dilation of line segment \( \overline{CD} \) with a scale factor of 13 and a center of dilation at the origin will result in a line segment that is 13 times the length of \( \overline{CD} \), we need to identify the statements relating to the properties of dilation and the resulting lengths.
Based on the provided responses, we focus on the following two options that mention the length of \( C'D' \):
- "The length of \( C'D' \) is approximately 4.24."
- "The length of line segment \( C'D' \) is approximately 4.24."
- "The length of \( C'D' \) is approximately 1.41 units."
A key property of dilation states that when a segment is dilated by a scale factor \( k \), the new length becomes \( k \) times the original length. Therefore, if the original length is \( L \), the new length after dilation would be \( kL \).
Since there is a mention of dilation with a scale factor of 13, we need to check if any of the lengths mentioned could be consistent with this scale factor. Since we do not know the original length, we can analyze each given length:
- If the length of \( C'D' \) is approximately 4.24, the original length before dilation could be \( 4.24 / 13 \).
- If the length of \( C'D' \) is approximately 1.41 units, the original length would be \( 1.41 / 13 \).
To summarize, the correct statement for the proof must indicate the relationship between the lengths before and after dilation according to the scale factor of 13.
The correct statement is:
The length of \( C'D' \) is approximately 4.24.
This statement implies that after dilation, \( C'D' \) is indeed 13 times the original length, confirming the proof requirement.
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