To analyze the effects of dilation on line AB (which we will describe as being represented by points A and B in the coordinate plane), let's break it down step by step for both scale factors: 3 and 1/3.
Dilation with a Scale Factor of 3
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Location of the Line: Line AB is originally defined by the points A(x₁, y₁) and B(x₂, y₂). Because it does not pass through the origin, both points are located at some coordinates away from (0,0).
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Center of Dilation: The center of dilation is the origin (0, 0).
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Dilation Process: When the line is dilated with a scale factor of 3, both points A and B will be moved further away from the origin. The dilation transforms each point according to the formula: \[ A' = (3x₁, 3y₁), \quad B' = (3x₂, 3y₂) \] This means that for every coordinate of points A and B, the x and y values will be multiplied by 3.
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Effect on the Line:
- The new line A′B′ will be parallel to the original line AB because dilations preserve parallelism.
- The distance from the origin to every point on the new line A′B′ will be three times the distance from the origin to the corresponding point on line AB.
- The slope of line A′B′ will remain the same as that of line AB since the dilation does not affect the angles or the relative positions of the points.
Dilation with a Scale Factor of 1/3
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Dilation Process: If the line AB is dilated with a scale factor of 1/3, the process transforms the points as follows: \[ A' = \left(\frac{1}{3}x₁, \frac{1}{3}y₁\right), \quad B' = \left(\frac{1}{3}x₂, \frac{1}{3}y₂\right) \]
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Effect on the Line:
- The new line A′B′ will still be parallel to the original line AB, as dilations preserve parallelism.
- Each point on A′B′ will be one-third of the distance from the origin compared to the corresponding point on line AB. Therefore, the entire line will be much closer to the origin than line AB.
- The slope of line A′B′ will also remain the same as that of line AB, preserving the angle relative to the x-axis.
Summary of Changes
- In both cases (scale factors of 3 and 1/3), the line remains parallel to the original line.
- The distance from the origin to the line changes. With a scale factor of 3, the line moves further away, whereas with a scale factor of 1/3, the line moves closer to the origin.
- The slope of the line remains invariant under dilation.
Thus, the primary effect of either dilation is that it scales the distance from the center of dilation (the origin) while preserving parallelism and slope.