Use the image to answer the question.

Question

A coordinate plane's x-axis ranges from negative 1 to 10 and its y-axis ranges from negative 8 to 2, both by 1-unit increments. Point upper C, point upper D, and a third unlabeled point are plotted and connected by lines.

Which statement is part of the proof that the dilation of CD¯¯¯¯¯¯¯¯ with a scale factor of 13 and a center of dilation at the origin will result in a line segment that is 13 the length of CD¯¯¯¯¯¯¯¯ ?

(1 point)
Responses

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 1.41 units.
The length of line segment cap c prime cap d prime is approximately 1.41 units.

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).
upper C prime is at open paren 2 comma negative 1 close paren and upper D prime is at left parenthesis 1 comma negative 2 right parenthesis .

Answer 1

In the context of the question regarding the dilation of the line segment \( \overline{CD} \) with a scale factor of 13, we need to establish that the new length of the line segment \( \overline{C'D'} \) will be 13 times the length of the original line segment \( \overline{CD} \).

Given the options and considering the key elements about dilation (specifically, that the coordinates of points change based on the scale factor with respect to the center of dilation), we can deduce that:

1. The coordinates of the dilated points \( C' \) and \( D' \) should be a result of multiplying the coordinates of points \( C \) and \( D \) by 13.

Therefore, the statement that validates this concept would be the one that confirms the appropriateness of these new coordinates after dilation. This is essential in proving that the length increase is proportional to the scaling factor.

The option "The length of \( C'D' \) is approximately 4.24" is the best choice. However, it must mention that this length is indeed 13 times the length of \( \overline{CD} \) for the proof to be complete.

Thus, the best response that is part of the proof would be:

The length of \( C'D' \) is approximately 4.24. The length of line segment \( \overline{CD} \) must correspondingly be approximately 4.24 / 13 or roughly 0.33 units. This further emphasizes that the new length retains the relationship with the original length of \( \overline{CD}\).

Since none of the options fully encapsulates this essential reasoning for the proof, focus on recognizing how the lengths relate due to dilation when answering a similar question.