When line \( AB \) is dilated with a scale factor of 3 and a center of dilation at the origin, the following effects occur:
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Change in Length: The length of line segment \( AB \) will be multiplied by the scale factor of 3. So, if the original length of line \( AB \) is \( L \), the new length of line \( A'B' \) will be \( 3L \).
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Position of Points: Each point on line \( AB \) will be moved away from the origin by a factor of 3. Specifically, if point \( A \) has coordinates \( (x_A, y_A) \), after dilation, the new coordinates of point \( A' \) will be \( (3x_A, 3y_A) \). The same applies for point \( B \).
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Orientation and Slope: The slope of line \( AB \) will remain the same because dilation is a uniform scaling transformation. The line will still be parallel to its original configuration but simply stretched away from the origin.
If line \( AB \) were dilated with a scale factor of \( \frac{1}{3} \) instead, the following changes would occur:
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Change in Length: The length of line segment \( AB \) would now be multiplied by the scale factor of \( \frac{1}{3} \). So, the new length of line \( A'B' \) would be \( \frac{1}{3}L \).
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Position of Points: Each point on line \( AB \) would move closer to the origin. For example, point \( A \) at \( (x_A, y_A) \) would be transformed to \( \left(\frac{1}{3}x_A, \frac{1}{3}y_A\right) \). The same applies for point \( B \).
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Orientation and Slope: The slope of line \( AB \) would remain unchanged as well, with the line remaining parallel to its original configuration but scaled down towards the origin.
In summary, a dilation of \( AB \) with a scale factor greater than 1 (like 3) results in the line segment being stretched away from the origin, while a scale factor less than 1 (like \( \frac{1}{3} \)) compresses the segment toward the origin. The orientation remains consistent in both scenarios.
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