To determine which statement is part of the proof that the dilation of segment \( CD \) with a scale factor of 13 results in a line segment that is 13 times the length of \( CD \), we need to focus on the relationship between the original segment and its dilated counterpart.
A dilation with a scale factor \( k \) will increase the length of any segment by a factor of \( k \). Therefore, if the original segment \( CD \) has a length \( L \), after dilation with a scale factor of 13, the new length \( L' \) will be:
\[ L' = k \cdot L = 13 \cdot L \]
Thus, one of the statements in the list must indicate a clear relationship between the length of \( C'D' \) and the original length of \( CD \).
Among the options provided, we interpret the statements as follows:
- The length of \( C'D' \) is approximately 4.24.
- The length of line segment \( c'd' \) is approximately 4.24.
- \( C' \) is at (9,−18) and \( D' \) is at (18,−9).
- The length of \( C'D' \) is approximately 1.41 units.
- The length of line segment \( c'd' \) is approximately 1.41 units.
To find the correct response, we need to ascertain the lengths that result from the dilation. If \( C'D' \) has lengths of 4.24 or 1.41 units, we need to determine which one aligns with the concept of being 13 times the original length.
Let’s assume that the original length (before dilation) was \( L \). If the dilation produced \( C'D' \approx 4.24 \), then before dilation:
\[ L \approx \frac{4.24}{13} \approx 0.327 \]
If \( C'D' \) is \( \approx 1.41 \):
\[ L \approx \frac{1.41}{13} \approx 0.1085 \]
Since the main assertion we need is regarding the length relationship after dilation, the statement that references the length of \( C'D' = 4.24 \) would likely be more relevant as it represents a tangible result of the dilation scaling.
Therefore, a likely correct answer regarding the proof of the dilation would be:
The length of \( C'D' \) is approximately 4.24.