When line \( AB \) is dilated with a scale factor of 3 and a center of dilation at the origin, the points on line \( AB \) change as follows:
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Distance from the Origin: Each point on line \( AB \) is moved away from the origin by a factor of 3. For a point \( (x, y) \) on line \( AB \), after dilation, its new coordinates will be \( (3x, 3y) \). This means that all points will be three times as far from the origin compared to their original distance.
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Orientation and Slope: The slope of line \( AB \) will remain the same. A dilation centered at the origin maintains the angles and parallelism of lines, so the new line \( A'B' \) will have the same slope as line \( AB \).
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Position: While the entire line moves outward from the origin, its orientation remains unchanged and the line does not intersect the origin (as given).
If line \( AB \) were to be dilated with a scale factor of 13 instead of 3, the following effects would occur, similar in nature but with a larger magnitude:
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Distance from the Origin: Each point on line \( AB \) would now be moved away from the origin by a factor of 13. For a point \( (x, y) \), after dilation, its new coordinates would be \( (13x, 13y) \).
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Orientation and Slope: Similar to the previous dilation, the slope of the line would remain unchanged, and the new line \( A'B' \) will still be parallel to the original line \( AB \).
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Position: The line would be further away from the origin compared to the previous dilation (scale factor of 3), but still not intersect it.
In summary, the main effects of dilating line \( AB \) with either scale factor (3 or 13) are that the line moves outward from the origin while maintaining its slope and orientation. The only difference between the two dilations is the distance from the origin: a scale factor of 3 results in points being three times farther, while a scale factor of 13 results in points being thirteen times farther.
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