To determine the scale factor used in the dilation of line segment \( AB \) with respect to the origin, we start by finding the lengths of the segments before and after the dilation.
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Find the coordinates of points A and B:
- Point \( A \) is at \( (9, 9) \).
- Point \( B \) is at \( (6, 6) \).
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Calculate the distance between points A and B: The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting \( A(9, 9) \) and \( B(6, 6) \): \[ d_{AB} = \sqrt{(6 - 9)^2 + (6 - 9)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Now find the coordinates of points \( A' \) and \( B' \):
- Point \( A' \) is at \( (3, 3) \).
- Point \( B' \) is at \( (2, 2) \).
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Calculate the distance between points \( A' \) and \( B' \): Again using the distance formula: \[ d_{A'B'} = \sqrt{(2 - 3)^2 + (2 - 3)^2} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Determine the scale factor (k) for the dilation: The scale factor is calculated by the ratio of the distance of \( AB \) to the distance of \( A'B' \): \[ k = \frac{d_{A'B'}}{d_{AB}} = \frac{\sqrt{2}}{3\sqrt{2}} = \frac{1}{3} \]
Thus, the scale factor used in the dilation of line segment \( AB \) from \( A \) and \( B \) to \( A' \) and \( B' \) with respect to the origin is: \[ \frac{1}{3} \]
The correct response is: \[ \frac{1}{3} \]