To determine which statement is part of the proof that the dilation of line 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 CD.
The points C and D are given as:
- C(3, -6)
- D(6, -3)
To find the length of CD, we can use the distance formula: \[ \text{Length of } CD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of points C and D: \[ \text{Length of } CD = \sqrt{(6 - 3)^2 + (-3 + 6)^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \approx 4.24 \text{ units} \]
Now, after dilation with a scale factor of 13, the length of the new line segment C'D' should be: \[ 13 \times \text{Length of } CD \approx 13 \times 4.24 \approx 55.12 \text{ units} \]
Next, to determine the new points C' and D' after dilation: For point C(3, -6), the coordinates after dilation are given by: \[ C' = (3 \times 13, -6 \times 13) = (39, -78) \] For point D(6, -3): \[ D' = (6 \times 13, -3 \times 13) = (78, -39) \]
The length of the new segment C'D' can again be calculated using the distance formula: \[ \text{Length of } C'D' = \sqrt{(78 - 39)^2 + (-39 + 78)^2} = \sqrt{(39)^2 + (39)^2} = \sqrt{1521 + 1521} = \sqrt{3042} = 13 \cdot \sqrt{18} \text{ (since scale factor is 13)}. \]
Now from the provided response options, the only relevant calculations identified must reflect the dilation correctly. The statement that C′ is at (39,−78) and D′ is at (78,−39) results in a length of \[ \text{Length of } C'D' \approx 55.12 \text{ units (13 times 4.24).} \]
None of the provided statements exactly confirm this scenario of dilution or give C' and D' correctly after the 13-fold scale dilation from given points.
Therefore, the correct statement indicating the result of dilation (ideally identifying the accurate length relationship post-dilation) should reflect that the length of the dilated segment should be 13 times the initial length. Hence, assuming option clarity, “The length of C′D′ is approximately 4.24” is closest indirectly regarding proof calculations to understand length.
However, specifically no directly provided statements from queries match excellently hence note them on potential length symbol C'D' or exact coordinate dilations.