Question

Use the image to answer the 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.

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.

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 and a center of dilation at the origin will result in a line segment that is 13 times the length of \( \overline{CD} \), we need to understand how the dilation affects the coordinates of points and their distances.

When points are dilated from the origin by a scale factor of \( k \), the new coordinates of a point \( (x, y) \) become \( (kx, ky) \) and the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is multiplied by \( k \).

Let's analyze the available options:

1. **\( C' \) is at (9,−18) and \( D' \) is at (18,−9)**:
- If \( C' = (9, -18) \) and \( D' = (18, -9) \), we can calculate the length of \( C'D' \) using the distance formula \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). This does not directly provide the correct scale factor relevance unless we calculate it, which isn't done here.

2. **The length of \( C'D' \) is approximately 1.41 units**:
- This is suggesting that the length of the dilated segment is a small value compared to the scale factor, which doesn't align with the concept of dilation where the length should be significantly larger.

3. **\( C' \) is at (2,−1) and \( D' \) is at (1,−2)**:
- Using these coordinates, we can again calculate the length of \( C'D' \), which would help ascertain its relation to the original line segment. However, without further information on the original segment, it's unclear.

4. **The length of \( C'D' \) is approximately 4.24**:
- This statement could make sense if we assume \( \overline{CD} \) has a length that when multiplied by 13 gives a length that is not 4.24. So, this does not seem to support the notion of dilation correctly.

To conclude, none of these options provide a clear proof with calculations or context supporting the assertion that the new length should be 13 times longer than the original. The most relevant response would involve correct scaling computations which tie back to the concept that when you dilate by a certain factor, the lengths reflect that multiplicative factor.

The correct part of the proof can be derived or reframed in the context of showing that the length of the new segment indeed reflects the scale of 13 after performing the dilation on the original segment. Often the length calculations relative to the original segment provide that proof rather than standalone coordinate values.

From what's given, the preferred response likely indicates the lengths must correspond adequately to justify that \( \overline{C'D'} \) has been scaled according to the transformation rules of dilation. Therefore, none of the options presented are fully satisfactory as standalone evidence without additional context or original segment information.