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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?(4 points)

1 answer

When line \( AB \) is dilated with a scale factor \( k \) and a center of dilation at the origin, the following effects occur:

  1. Scaling of Points: Each point \( (x, y) \) on line \( AB \) will be moved to a new position \( (kx, ky) \) on line \( A'B' \). This means that the coordinates of each point on line \( AB \) are multiplied by the scale factor \( k \).

  2. Line Positioning: The orientation of the line will not change; it will remain parallel to its original position (line \( AB \)). This is because the dilation preserves angles, and hence the angle the line makes with the axes will remain constant.

  3. Distance from the Origin: As the scale factor increases, each point on line \( A'B' \) will be further away from the origin compared to the corresponding points on line \( AB \). The distance from the origin is multiplied by the scale factor.

Dilating with a Scale Factor of 3

If line \( AB \) is dilated with a scale factor of 3, every point \( (x, y) \) on line \( AB \) would move to \( (3x, 3y) \) on line \( A'B' \). The line will be three times farther away from the origin compared to line \( AB \), but it will maintain its original slope and orientation.

Dilating with a Scale Factor of 13

If line \( AB \) is instead dilated with a scale factor of 13, the same principles apply. Each point on line \( AB \) will be moved to a new position at \( (13x, 13y) \) on line \( A'B' \). The line will be 13 times farther away from the origin than line \( AB \), preserving its orientation and slope just as in the previous case with the scale factor of 3.

Summary of Effects

  • Both dilations maintain the orientation of the line.
  • The distance from the origin is scaled by the respective factor (3 or 13).
  • The shape and angles of the figure formed by line \( A'B' \) remain identical to those of line \( AB \) after dilation.

In conclusion, the primary difference between the two scale factors is how much further away line \( A'B' \) is from the origin, with the distances being 3 times and 13 times greater than the original line \( AB \) respectively.