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.
C=(3, -6)
D=(6, -3)
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¯¯¯¯¯¯¯¯?
Responses

C′ is at (9,−18) and D′ is at (18,−9).
cap c prime is at open paren 9 comma negative 18 close paren and cap d prime is at open paren 18 comma negative 9 close paren. - no response given

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. - not selected, this is the correct answer

The length of C′D′¯¯¯¯¯¯¯¯¯¯¯ is approximately 4.24.
The length of line segment cap c prime cap d prime is approximately 4.24. - incorrect

C′ is at (2,−1) and D′ is at (1,−2).

Answer 1

To find the correct statement that supports the proof that the dilation of segment \( CD \) with a scale factor of 13 and a center of dilation at the origin results in a line segment that is 13 times the length of \( CD \), we first need to calculate the length of the original segment \( CD \) and then its dilated version \( C'D' \).

Given the points:

• \( C = (3, -6) \)
• \( D = (6, -3) \)

First, we calculate the distance of segment \( CD \):

\[ CD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(6 - 3)^2 + (-3 + 6)^2} \] \[ = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \approx 4.24 \]

Now, applying the dilation with a scale factor of 13:

\[ C' = (3 \cdot 13, -6 \cdot 13) = (39, -78) \] \[ D' = (6 \cdot 13, -3 \cdot 13) = (78, -39) \]

Now we calculate the length of the dilated segment \( C'D' \):

\[ C'D' = \sqrt{(78 - 39)^2 + (-39 + 78)^2} \] \[ = \sqrt{(39)^2 + (39)^2} = \sqrt{9 \cdot 39^2} = 39\sqrt{2} \approx 4.24 \cdot 13 \approx 54.92 \]

This gives us the understanding that the new length is indeed 13 times the original length of \( CD \):

Since the original length \( CD \approx 4.24 \), after dilation, the length of segment \( C'D' \) is indeed approximately \( 4.24 \cdot 13 \).

Now looking at the provided responses, the statement "The length of line segment \( C'D' \) is approximately 4.24" is incorrect regarding the nature of the transformation that occurs, although it accurately reflects the original length of the segment \( CD \).

Therefore, no option included correctly reflects the length (which should be calculated post-dilation), but if the intention was to represent the initial calculations before scale transformation, it fits. However, if we take it further as per the scale mention, we would assert the statement regarding the scale factor properties instead—acknowledging the principles behind such transformations in geometry.

Choices verifying the positions secondarily seem to relate to original shorter sections but retrace back to scale measures as we project. Thus, a focus on ensuring replication and maintaining accurate scaling here, hence involving direct lengths, remains critical.

Conclusively, the relevant statement conforms pre-scaled setups, already aiding dimensional proofings, while intertwining ratio-boundary outcomes of vector attributes shows divergence across terms.