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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)

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Dilation is a transformation that creates a new figure similar to the original figure, where the distances between points change according to a specified scale factor. In this case, we are dilating line AB from the origin with a scale factor of 3, and then considering its effect with a scale factor of 13.

Effects of Dilation with a Scale Factor of 3:

  1. Positioning: The new points A' and B' will be located at three times the distance from the origin compared to points A and B, respectively. If point A has coordinates (x_A, y_A) and point B has coordinates (x_B, y_B) in the coordinate plane, then point A' will have coordinates (3x_A, 3y_A) and point B' will have coordinates (3x_B, 3y_B).

  2. Slope: The slope of the line AB will remain unchanged. The slope is a ratio of the change in y to the change in x, and dilation does not affect this ratio since both the x-coordinates and y-coordinates are multiplied by the same factor (3).

  3. Length: The distance between points A and B will be multiplied by 3. If the original length of line segment AB is d, then the length of line segment A'B' will be 3d.

  4. Orientation: The orientation of line A'B' in the coordinate plane will remain the same as that of line AB because dilation centered at the origin preserves the angles and parallelism.

Effects of Dilation with a Scale Factor of 13:

  1. Positioning: Similar to the previous case, the new points A' and B' will now be located at thirteen times the distance from the origin. Therefore, point A' will have coordinates (13x_A, 13y_A) and point B' will have coordinates (13x_B, 13y_B).

  2. Slope: The slope of line A'B' will continue to be the same as that of line AB. The ratio remains unchanged because dilation scales both the x-coordinates and the y-coordinates by the same factor (13).

  3. Length: The length of the line segment A'B' will be multiplied by 13. If the original length of segment AB is d, then the length of segment A'B' will be 13d.

  4. Orientation: Just as with the earlier dilation, the orientation of line A'B' will remain the same as that of line AB.

Summary:

In both cases, the dilation with a scale factor (whether 3 or 13) maintains the line’s slope and orientation while enlarging its length (3 times or 13 times, respectively) and moving the entire line away from the origin while preserving its direction. The greater the scale factor, the farther away the dilated line moves from the origin, and the longer the segment becomes.