Ask a New Question
Menu

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 1\3
, how would your answer change?(4 points)

1 answer

When a line \( AB \) on the coordinate plane undergoes a dilation with a center at the origin and a scale factor greater than 1, such as 3, the following effects occur:

  1. Distance from the Origin: Each point on the line \( AB \) is moved farther away from the origin by a factor of 3. If a point \( (x, y) \) on line \( AB \) is dilated, it transforms to the point \( (3x, 3y) \).

  2. Slope: The slope of line \( AB \) remains unchanged. Since dilation is an operation that pushes points away from the origin without changing their relative orientation, the angle of the line with respect to the axes does not change.

  3. Position: Line \( A'B' \) will be parallel to line \( AB \) because the slope is preserved. However, line \( A'B' \) will be located farther away from the origin than line \( AB \).

  4. Intersection with Axes: If line \( AB \) has an intercept, the intercepts will also be multiplied by the dilation factor. For instance, if line \( AB \) intersects the y-axis at \( c \), then line \( A'B' \) will intersect it at \( 3c \).

Now let's consider the case where line \( AB \) is dilated with a scale factor of \( \frac{1}{3} \):

  1. Distance from the Origin: Each point on line \( AB \) will be moved closer to the origin by a factor of \( \frac{1}{3} \). So, a point \( (x, y) \) on line \( AB \) will transform to \( \left(\frac{1}{3}x, \frac{1}{3}y\right) \).

  2. Slope: Similar to the previous case, the slope of line \( AB \) remains unchanged.

  3. Position: Line \( A'B' \) will still be parallel to line \( AB \), but it will be located closer to the origin than line \( AB \).

  4. Intersection with Axes: The intercepts of line \( AB \) will be multiplied by \( \frac{1}{3} \). Therefore, if line \( AB \) intersects the y-axis at \( c \), then line \( A'B' \) will intersect it at \( \frac{1}{3}c \).

In summary, the dilation with a scale factor of 3 moves line \( AB \) away from the origin, while a dilation with a scale factor of \( \frac{1}{3} \) brings the line closer to the origin. In both cases, the slope remains unchanged, and the lines remain parallel to each other.