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Line AB is on the coordinate plane and does not pass through the origin. Line AB is dilated with a scale factor of 3 and a center of dilation at the origin to create line A′B′ . Describe the effects of the dilation on Line AB . If line AB was dilated with a scale factor of 13 , how would your answer change?

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When a line in the coordinate plane is dilated with a scale factor and a center of dilation at the origin, several effects can be observed.

Dilation with Scale Factor of 3

  1. Change in Coordinates: Every point on line AB will be transformed by multiplying its coordinates by the scale factor (in this case, 3). If a point \( (x, y) \) is on line AB, after the dilation, the new point \( (x', y') \) on line A′B′ will be: \[ (x', y') = (3x, 3y) \]

  2. Parallel Lines: The resulting line A′B′ will be parallel to line AB. This is because dilation does not change the slope of the line; it merely stretches or shrinks it away from or towards the center of dilation.

  3. Change in Distance from the Origin: The distance from the origin to any point on line A′B′ will be three times that of the corresponding point on line AB. This means that the entire line A′B′ will be farther away from the origin compared to line AB.

  4. Same Direction: The direction of the line remains unchanged after dilation.

Dilation with Scale Factor of 13

If the line AB is dilated with a scale factor of 13, the effects would be similar to those described for the scale factor of 3, with a few differences in magnitude:

  1. Change in Coordinates: Each point on line AB will now be transformed by multiplying the coordinates by 13. The new point \( (x', y') \) on line A′B′ will be: \[ (x', y') = (13x, 13y) \]

  2. Parallel Lines: Line A′B′ will still be parallel to line AB, just as in the previous case.

  3. Change in Distance from the Origin: The distance from the origin to any point on line A′B′ will be 13 times that of the corresponding point on line AB, making it significantly farther away.

  4. Same Direction: The direction of the line remains unchanged after dilation regardless of the scale factor.

Conclusion

In both cases, regardless of whether the scale factor is 3 or 13, the key effects of dilation are that line A′B′ will be parallel to line AB, the distances to the origin will be scaled according to the factor used, and the direction of the line remains the same. The main difference will be the magnitude of the change—dilating by 13 spreads line A′B′ much farther away from the origin compared to dilating by 3.