When a line AB on the coordinate plane is dilated with a scale factor of 3 and a center of dilation at the origin, the coordinates of any point \( (x, y) \) on line AB will be transformed to \( (3x, 3y) \). This dilation will have the following effects:
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Positioning: Every point on line AB will move away from the origin. Since the scale factor is greater than 1, the distance from the origin to each point on the line will triple.
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Slope: The slope of the line AB will remain the same. Dilation is uniform—it will stretch the line but will not change how steep or flat the line is.
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Length and Width: The line itself does not have width in a traditional sense, but if you consider the distance between any two points on line AB, that distance will also be multiplied by the scale factor, resulting in greater separation between points.
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Overall Shape: The resulting line A′B′ will be parallel to line AB since the direction and slope of the line do not change.
Now, if line AB is dilated with a scale factor of \( \frac{1}{3} \):
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Positioning: Every point on line AB will move closer to the origin. Since the scale factor is less than 1, the distance from the origin to each point on the line will be reduced to one-third.
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Slope: Just like before, the slope of the line will remain unchanged. Dilation affects size but not orientation or steepness.
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Length and Width: The distance between any two points on the line AB will be reduced to one-third of its original value.
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Overall Shape: The resulting line (let's call it A′B′ in this case as well) will also be parallel to line AB, maintaining the same directional characteristics.
In summary, for both dilations (scale factor of 3 and \( \frac{1}{3} \)), the orientation and slope of line AB remain unchanged; however, the first dilation expands the distance of all points from the origin, while the second dilation contracts it.