When line AB is dilated with a scale factor of 3 and a center of dilation at the origin, each point (x, y) on line AB will be transformed to a new point (3x, 3y) on line A'B'. This means that:
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Position: The line A'B' will remain parallel to line AB, as dilation affects distances from the center of dilation but does not change the angles between lines. Since both lines share the same direction, A'B' will be a scaled version of AB.
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Distance: The distance between any two points on line A'B' will be three times the distance between the corresponding points on line AB. This factor will also apply to the entire length of the line segment AB.
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Coordinates: The coordinates of any point on the line are multiplied by 3, moving the points further away from the origin while maintaining their relative positions to one another.
If line AB were to be dilated instead with a scale factor of 1/3, the effects would be as follows:
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Position: Similarly, line A'B' would remain parallel to line AB, but now the distances to the center of dilation (the origin) would be reduced.
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Distance: The distance between any two points on line A'B' would be one-third of the distance between the corresponding points on line AB, reducing the size of the line.
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Coordinates: Each point (x, y) on line AB would be transformed to (1/3x, 1/3y) on line A'B'. This brings the points closer to the origin while keeping their relative spacing.
In summary, a dilation with a scale factor of 3 expands the line away from the origin, while a dilation with a scale factor of 1/3 contracts it towards the origin. In both cases, the lines remain parallel and retain their directional properties.
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