Question

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 (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 . 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 (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.

Answer 1

To determine which statement is part of the proof that the dilation of line segment \( \overline{CD} \) with a scale factor of 13 results in a line segment that is 13 times the length of \( \overline{CD} \), we need to focus on the relationship between the original line segment and its image after dilation.

When a point \( (x, y) \) is dilated from the origin by a scale factor \( k \), the new point \( (x', y') \) is given by:

\[ (x', y') = (kx, ky) \]

If we denote the length of the original line segment \( \overline{CD} \) as \( L \), the length of the dilated line segment \( \overline{C'D'} \) will be:

\[ L' = kL \]

When \( k = 13 \), then \( L' = 13L \).

In the options you provided, we need to look for the information that confirms the new length of the line segment after dilation:

1. C′ is at (2,−1) and D′ is at (1,−2). The length of C′D′ is approximately 4.24.
2. C′ is at (9,−18) and D′ is at (18,−9). The length of C′D′ is approximately 1.41 units.

The second statement indicates specific coordinates for points \( C' \) and \( D' \). To check if this aligns with the dilation, we can calculate the length of \( \overline{C'D'} \):

• The length \( L' \) can be calculated using the distance formula:

\[ L' = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

For the coordinates (9, -18) and (18, -9):

\[ L' = \sqrt{(18-9)^2 + (-9+18)^2} = \sqrt{9^2 + 9^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2} \approx 12.73 \]

Thus, if the original segment had a length of \( 12.73 / 13 \approx 0.98 \), then after dilation it is in fact approximately \( 13 \times \) the original length. Therefore, a correct statement about a segment being approximately \( 1.41 \) would not support the proof of dilation at factor of 13 (as it is much smaller).

Overall, we do not have sufficient evidence to confirm the statement definitively without knowing the original length. However, the focus should be on identifying the length after dilation and confirming if it is indeed \( 13 \times \) the original length.

Concisely, the most relevant statement to the proof of dilation by factor of 13 would be the calculations of the length post dilation. However, assessing directly from the options, none show a clear validation of a dilation effect that supports its length increasing by 13 times either.