When Line AB is dilated with a scale factor of 3 from a center of dilation at the origin, the following effects occur:
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Positioning of Points: Each point on Line AB is moved away from the origin. If a point on Line AB has coordinates (x, y), after the dilation, its new coordinates would be (3x, 3y). This means that all points on Line AB are effectively scaled away from the origin by a factor of three.
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Length Increase: The distances between points on Line AB will be multiplied by 3, making the line three times longer. For example, if the original distance between two points on Line AB was d, after dilation, this distance becomes 3d.
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Slope Preservation: The slope of the line will remain the same since dilation does not alter the angle between the line and the axes.
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Parallel Lines: After dilation, Line A'B' will remain parallel to Line AB since dilation from a point preserves parallelism.
When Line AB is dilated with a scale factor of \( \frac{1}{3} \):
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Positioning of Points: Each point on Line AB is now moved closer to the origin. For a point (x, y) on Line AB, it will be transformed to \( \left(\frac{1}{3}x, \frac{1}{3}y\right) \).
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Length Decrease: The distances between points on Line AB will decrease to one-third of their original lengths. So, if the original distance between two points was d, after dilation, it becomes \( \frac{1}{3}d \).
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Slope Preservation: Similar to the previous case, the slope of the line remains unchanged.
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Parallel Lines: Again, Line A'B' will remain parallel to Line AB, as dilation maintains the parallel property.
In summary, a dilation with a scale factor greater than 1 expands the line away from the origin, increasing its length, while a scale factor less than 1 contracts the line towards the origin, reducing its length. In both cases, the slope and the parallel nature of the lines are preserved.
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