Line AB is on the coordinate plane and does not pass through the origin. Line AB

Question

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)

Answer 1

To describe the effects of the dilation on Line \( AB \) with a scale factor of \( 3 \) and a center of dilation at the origin, we need to consider how dilation works in a coordinate plane.

1. Dilation with Scale Factor of 3:

   • When a line is dilated by a scale factor of \( 3 \) with the origin as the center of dilation, every point on line \( AB \) is moved away from the origin by a factor of \( 3 \). Mathematically, if the coordinates of point \( A \) on line \( AB \) are initially \( (x, y) \), after dilation, the new coordinates \( A' \) would be \( (3x, 3y) \). Similarly, for point \( B \) on line \( AB \) with coordinates \( (x', y') \), the new coordinates \( B' \) would be \( (3x', 3y') \).
   • Effect on the Line: The direction of line \( AB \) remains unchanged, but the distance from the origin increases. The line \( A'B' \) will be parallel to line \( AB \) and positioned further away from the origin. The slope of the line remains the same since the dilation maintains the angle with respect to the axes.

2. Dilation with Scale Factor of \( \frac{1}{3} \):

   • If we now dilate line \( AB \) with a scale factor of \( \frac{1}{3} \) from the origin, a similar process occurs. Every point on line \( AB \) is moved towards the origin by a factor of \( \frac{1}{3} \). So, for a point \( A \) with coordinates \( (x, y) \), the new coordinates \( A' \) after dilation will be \( \left(\frac{x}{3}, \frac{y}{3}\right) \). For point \( B \) with coordinates \( (x', y') \), \( B' \) will be \( \left(\frac{x'}{3}, \frac{y'}{3}\right) \).
   • Effect on the Line: Just as with the previous dilation, the direction and slope of line \( AB \) are unchanged. Line \( A'B' \) will also be parallel to line \( AB \), but it will be positioned closer to the origin as each point is scaled down towards the origin.

In summary, both dilations maintain the relationship of parallelism with the original line, keeping the direction and slope the same, but change the distance of the line from the origin based on the dilation factor. A scale factor of \( 3 \) moves the line further away from the origin, while a scale factor of \( \frac{1}{3} \) brings the line closer to the origin.