To determine the scale factor used in the dilation of line segment \( AB \) with center point of dilation at the origin, we can follow these steps:
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Find the length of the original line segment \( AB \): \[ A(9, 9) \quad \text{and} \quad B(6, 6) \] The length \( AB \) is calculated using the distance formula: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(6 - 9)^2 + (6 - 9)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Find the length of the dilated line segment \( A'B' \): \[ A'(3, 3) \quad \text{and} \quad B'(2, 2) \] The length \( A'B' \) is calculated similarly: \[ A'B' = \sqrt{(2 - 3)^2 + (2 - 3)^2} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Calculate the scale factor: The scale factor \( k \) is given by the ratio of the dilated distance to the original distance: \[ k = \frac{A'B'}{AB} = \frac{\sqrt{2}}{3\sqrt{2}} = \frac{1}{3} \]
Thus, the scale factor used in the dilation is: \[ \frac{1}{3} \]
So the correct response is: \(\frac{1}{3}\) (Start Fraction 1 over 3 End Fraction)