To prove that the dilation of segment CD with a scale factor of \( \frac{1}{3} \) about the origin results in a segment that is \( \frac{1}{3} \) the length of the original segment CD, we can start with the definition of dilation.
Let's denote the endpoints of segment CD as points \( C(x_1, y_1) \) and \( D(x_2, y_2) \). The coordinates of the dilated points \( C' \) and \( D' \) can be calculated using the formula for dilation:
\[ C' = \left( \frac{1}{3} x_1, \frac{1}{3} y_1 \right) \quad \text{and} \quad D' = \left( \frac{1}{3} x_2, \frac{1}{3} y_2 \right) \]
Next, we find the lengths of the segments before and after the dilation:
Length of original segment CD:
Using the distance formula, the length \( L_{CD} \) of segment CD is given by:
\[ L_{CD} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Length of dilated segment C'D':
Similarly, the length \( L_{C'D'} \) of segment C'D' is:
\[ L_{C'D'} = \sqrt{\left( \frac{1}{3} x_2 - \frac{1}{3} x_1 \right)^2 + \left( \frac{1}{3} y_2 - \frac{1}{3} y_1 \right)^2} \]
This simplifies to:
\[ L_{C'D'} = \sqrt{\left( \frac{1}{3} (x_2 - x_1) \right)^2 + \left( \frac{1}{3} (y_2 - y_1) \right)^2} \] \[ = \frac{1}{3} \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] \[ = \frac{1}{3} L_{CD} \]
Thus, the statement that is part of the proof is:
"The length of the dilated segment C'D' is \( \frac{1}{3} \) the length of the original segment CD."
This confirms that the dilation reduces the length of the segment by the given scale factor \( \frac{1}{3} \).
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