To determine the scale factor used in the dilation of line segment \( AB \) with a center point of dilation at the origin, we first need to find the distance of points A and B from the origin (0,0) for both shapes.
Shape One:
- Point A (9, 9)
- Point B (6, 6)
Calculating the distances from the origin: \[ \text{Distance from origin to A} = \sqrt{(9 - 0)^2 + (9 - 0)^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2} \] \[ \text{Distance from origin to B} = \sqrt{(6 - 0)^2 + (6 - 0)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} \]
Shape Two:
- Point A (3, 3)
- Point B (2, 2)
Calculating the distances from the origin: \[ \text{Distance from origin to A} = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \] \[ \text{Distance from origin to B} = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
Finding the Scale Factor:
Now, we need to calculate the scale factor based on the distances from the origin:
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For point A: \[ \text{Scale Factor for A} = \frac{\text{Distance from origin to A in Shape 2}}{\text{Distance from origin to A in Shape 1}} = \frac{3\sqrt{2}}{9\sqrt{2}} = \frac{3}{9} = \frac{1}{3} \]
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For point B: \[ \text{Scale Factor for B} = \frac{\text{Distance from origin to B in Shape 2}}{\text{Distance from origin to B in Shape 1}} = \frac{2\sqrt{2}}{6\sqrt{2}} = \frac{2}{6} = \frac{1}{3} \]
Since the scale factor is consistent for both points, the scale factor used in the dilation of the line segment \( AB \) is:
\[ \boxed{\frac{1}{3}} \]
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