To determine the scale factor used in the dilation of line segment AB with center point of dilation at the origin, we need to compare the coordinates of the points before and after the dilation.
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Original points (before dilation):
- Point A: \( A(9, 9) \)
- Point B: \( B(6, 6) \)
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New points (after dilation):
- Point A': \( A'(3, 3) \)
- Point B': \( B'(2, 2) \)
The scale factor \( k \) can be found by comparing the distances from the origin (0, 0) to each set of points.
For point A: \[ k = \frac{\text{Distance from origin to } A'}{\text{Distance from origin to } A} = \frac{\sqrt{3^2 + 3^2}}{\sqrt{9^2 + 9^2}} = \frac{\sqrt{9 + 9}}{\sqrt{81 + 81}} = \frac{\sqrt{18}}{\sqrt{162}} = \frac{3\sqrt{2}}{9\sqrt{2}} = \frac{3}{9} = \frac{1}{3} \]
For point B: \[ k = \frac{\text{Distance from origin to } B'}{\text{Distance from origin to } B} = \frac{\sqrt{2^2 + 2^2}}{\sqrt{6^2 + 6^2}} = \frac{\sqrt{4 + 4}}{\sqrt{36 + 36}} = \frac{\sqrt{8}}{\sqrt{72}} = \frac{2\sqrt{2}}{6\sqrt{2}} = \frac{2}{6} = \frac{1}{3} \]
Since both points A and B give the same scale factor, the scale factor for the dilation is \( \frac{1}{3} \).
Thus, the answer is: Start Fraction 1 over 3 End Fraction