To determine the scale factor used in the dilation of line segment \( AB \) from points \( A(9, 9) \) and \( B(6, 6) \) to points \( A'(3, 3) \) and \( B'(2, 2) \) with a center of dilation at the origin \( (0,0) \), we can use the formula for finding the scale factor \( k \):
\[ k = \frac{\text{new coordinate}}{\text{original coordinate}} \]
We can choose either point \( A \) or point \( B \) to calculate the scale factor. We'll start with point \( A \).
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For point \( A(9, 9) \) and \( A'(3, 3) \):
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The x-coordinates: \[ k = \frac{3}{9} = \frac{1}{3} \]
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The y-coordinates (for verification): \[ k = \frac{3}{9} = \frac{1}{3} \]
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For point \( B(6, 6) \) and \( B'(2, 2) \):
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The x-coordinates: \[ k = \frac{2}{6} = \frac{1}{3} \]
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The y-coordinates (for verification): \[ k = \frac{2}{6} = \frac{1}{3} \]
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Since we have calculated the scale factor to be the same for both points, the scale factor for the dilation of line segment \( AB \) is:
\[ \frac{1}{3} \]
So the answer is:
Start Fraction 1 over 3 End Fraction
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