To analyze the effects of dilating line AB with a scale factor of 3 from the origin, we start by understanding how dilation works in the coordinate plane.
Dilation with a Scale Factor of 3:
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Effect on Points: Each point \( (x, y) \) on line AB will be transformed to \( (3x, 3y) \) on line A′B′. This means that every point moves away from the origin by a factor of 3.
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Slope and Position: The slope of line AB will remain the same, as dilation doesn't change the angle of the line, it only changes the distance from the origin. However, since every point is scaled outwards from the origin, line A′B′ will be parallel to line AB but farther away from the origin.
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Distance from the Origin: If line AB originally does not pass through the origin, after dilation, line A′B′ will also not pass through the origin. It will be further away from the origin relative to its original distance.
Thus, the overall result is that line A′B′ is a parallel line to line AB, scaled away from the origin, maintaining the same slope but at a greater distance from the origin.
Dilation with a Scale Factor of 1/3:
Now, let's consider the case with a scale factor of \( \frac{1}{3} \).
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Effect on Points: Each point \( (x, y) \) on line AB will now be transformed to \( \left( \frac{1}{3}x, \frac{1}{3}y \right) \) on line A′B′. This means that every point moves closer to the origin by a factor of \( \frac{1}{3} \).
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Slope and Position: Similar to the previous case, the slope of line AB remains unchanged. The line A′B′ will also be parallel to line AB, but now it will be closer to the origin.
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Distance from the Origin: Since line AB does not pass through the origin, after dilation, line A′B′ will still not pass through the origin, but it will be much closer than line AB was originally.
In summary, for a scale factor of \( \frac{1}{3} \), the resulting line A′B′ is again parallel to line AB but is now closer to the origin compared to line AB, maintaining the same slope.
Summary:
- Dilation with Scale Factor 3: Line A′B′ is parallel to line AB and further from the origin.
- Dilation with Scale Factor \( \frac{1}{3} \): Line A′B′ is parallel to line AB and closer to the origin.