To determine the scale factor used in the dilation of line segment \( AB \) with the center of dilation at the origin, we can use the coordinates of the points before and after dilation.
The original points are:
- Point \( A(9, 9) \)
- Point \( B(6, 6) \)
The points after dilation (the dilated points) are:
- Point \( A' (3, 3) \)
- Point \( B' (2, 2) \)
To find the scale factor \( k \), we can use the formula:
\[ k = \frac{x'}{x} \]
where \( (x, y) \) are the coordinates before dilation, and \( (x', y') \) are the coordinates after dilation.
Using point \( A \): \[ k = \frac{3}{9} = \frac{1}{3} \]
Using point \( B \): \[ k = \frac{2}{6} = \frac{1}{3} \]
Since both points result in the same scale factor, the scale factor used in the dilation is:
\[ \frac{1}{3} \]
Thus, the answer is \( \frac{1}{3} \).