To find the statement that is part of the proof that the dilation of line segment \(\overline{CD}\) with a scale factor of \(\frac{1}{3}\) and a center of dilation at the origin results in a line segment that is \(\frac{1}{3}\) the length of \(\overline{CD}\), let's begin by calculating the length of \(\overline{CD}\).
The coordinates of point \(C\) are \(C(3, -6)\) and the coordinates of point \(D\) are \(D(6, -3)\). The length of line segment \(\overline{CD}\) can be calculated using the distance formula:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates for \(C\) and \(D\):
\[ \text{Length of } \overline{CD} = \sqrt{(6 - 3)^2 + (-3 - (-6))^2} \]
Calculating this step-by-step:
- \(6 - 3 = 3\)
- \(-3 - (-6) = -3 + 6 = 3\)
- Now plug into the equation:
\[ \text{Length of } \overline{CD} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \approx 4.24 \]
Next, when we dilate this segment by a scale factor of \(\frac{1}{3}\), the new length of the segment \(\overline{C'D'}\) will be:
\[ \text{Length of } \overline{C'D'} = \frac{1}{3} \times 4.24 \approx 1.41 \]
Based on the calculations, the correct choice is:
"The length of line segment \(C'D'\) is approximately 1.41 units."
Therefore, the statement that correctly articulates the part of the proof is:
The length of line segment \(C'D'\) is approximately 1.41 units.
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