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Which statement is part of the proof that the dilation of CD¯¯¯¯¯¯¯¯ with a scale factor of 1/3 and a center of dilation at the origin will result in a line segment that is 1/3 the length of CD¯¯¯¯¯¯¯¯ ? (1 point) Responses The length of C′D′¯¯¯¯¯¯¯¯¯¯¯ is approximately 1.41 units. The length of line segment cap c prime cap d prime is approximately 1.41 units. C′ is at (9,−18) and D′ is at (18,−9) . upper C prime is at open paren 9 comma negative 18 close paren and upper D prime is at open paren 18 comma negative 9 close paren . The length of C′D′¯¯¯¯¯¯¯¯¯¯¯ is approximately 4.24. The length of line segment cap c prime cap d prime is approximately 4.24. C′ is at (2,−1) and D′ is at (1,−2) .

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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:

  1. "The length of \( C'D' \) is approximately 1.41 units."
  2. "The length of line segment \( \overline{C'D'} \) is approximately 1.41 units."
  3. "C' is at (9,−18) and D' is at (18,−9)."
  4. "The length of \( C'D' \) is approximately 4.24."
  5. "The length of line segment \( \overline{C'D'} \) is approximately 4.24."
  6. "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 \( \overline{CD} \).