To determine the scale factor used in the dilation of the line segments \( AB \), we can use the coordinates of the points before and after dilation.
For the first segment:
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Original Points:
- \( A(6, 9) \)
- \( B(12, 3) \)
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Center of Dilation: The origin \( O(0, 0) \).
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Calculating the distance from the origin to each point:
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The distance to point \( A \): \[ OA = \sqrt{6^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117} \]
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The distance to point \( B \): \[ OB = \sqrt{12^2 + 3^2} = \sqrt{144 + 9} = \sqrt{153} \]
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Finding the Scale Factor: The scale factor \( k \) can be found by taking the ratio of the distances: \[ k = \frac{OB}{OA} = \frac{\sqrt{153}}{\sqrt{117}} = \frac{\sqrt{153/117}} = \sqrt{\frac{153}{117}} \approx \sqrt{1.30769} \approx 1.14 \] (However, it is better to simplify the ratio.)
To simplify: \[ 153 = 3 \times 51 = 3 \times 3 \times 17 = 9 \cdot 17, \quad 117 = 3 \times 39 = 3 \times 3 \times 13 = 9 \cdot 13 \] So: \[ \frac{153}{117} = \frac{9 \cdot 17}{9 \cdot 13} = \frac{17}{13} \] Thus, the scale factor will be: \[ k = \sqrt{\frac{17}{13}} \]
For the second segment:
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Original Points:
- \( A(2, 3) \)
- \( B(4, 1) \)
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Calculating the distance from the origin to each point:
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The distance to point \( A \): \[ OA = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]
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The distance to point \( B \): \[ OB = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} \]
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Finding the Scale Factor: Using the same approach as before: \[ k = \frac{OB}{OA} = \frac{\sqrt{17}}{\sqrt{13}} = \sqrt{\frac{17}{13}} \]
Conclusion
For both segments, the scale factor for the dilation of line segments \( AB \) centered at the origin is the same:
\[ k = \sqrt{\frac{17}{13}} \text{ or } \frac{\sqrt{17}}{\sqrt{13}} \]
This represents the scale factor used in the dilation.
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