To determine which statement is part of the proof that the dilation of line segment \( \overline{CD} \) 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 \( \overline{CD} \), we can analyze the provided options.
When a line segment is dilated with a scale factor \( k \) from a center point (in this case, the origin), the resulting length of the line segment becomes \( k \) times the original length. Therefore, if \( k = \frac{1}{3} \), the new segment after dilation will be \( \frac{1}{3} \) the length of the original segment.
Now, let's analyze the statements:
- "The length of \( C'D' \) is approximately 1.41 units."
- "The length of line segment \( \overline{C'D'} \) is approximately 1.41 units."
- "C' is at (9,−18) and D' is at (18,−9)."
- "The length of \( C'D' \) is approximately 4.24."
- "The length of line segment \( \overline{C'D'} \) is approximately 4.24."
- "C' is at (2,−1) and D' is at (1,−2)."
For the dilation to result in a line segment that is \( \frac{1}{3} \) the length of the original line segment, the significant pieces of information would include:
- The original length of \( \overline{CD} \) before dilation.
- The resulting length after dilation, which should be \( \frac{1}{3} \) times the original length.
If we consider the lengths given:
- If the original length of \( \overline{CD} = 4.24 \) units, then after dilation, it should be \( 4.24 \cdot \frac{1}{3} \approx 1.41 \) units.
So, the accurate statements associated with the dilation process that would show this relationship are:
- "The length of \( C'D' \) is approximately 1.41 units."
- "The length of line segment \( \overline{C'D'} \) is approximately 1.41 units."
Therefore, we can conclude that statements indicating a length of approximately \( 1.41 \) units would indeed be part of the proof and confirm that \( \overline{C'D'} \) is \( \frac{1}{3} \) the length of \( \overline{CD} \).