Question

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 ⅓, how would your answer change?

Answer 1

When a line in the coordinate plane, such as line AB, is dilated using a scale factor, the following effects occur:

Dilation with Scale Factor 3:

1. Coordinates Transformation: For any point $(x,y)$ on line AB, the coordinates after dilation to create line A'B' will be $(3x,3y)$. This means that every point on line AB is moved away from the origin by a factor of 3.

2. Line Properties: The slope of line AB remains unchanged because dilation is a linear transformation. Thus, the angle and orientation of the line with respect to the axes do not change. However, the distance from the origin increases.

3. Distance from Origin: Since the scale factor is greater than 1, each point on the line is positioned further from the origin, effectively "stretching" the line away from the origin.

4. Parallel Lines: Line A'B' will remain parallel to line AB because the dilation occurs uniformally around the origin.

Dilation with Scale Factor $\frac{1}{3}$:

1. Coordinates Transformation: With a scale factor of $\frac{1}{3}$, every point on line AB represented by $(x,y)$ will be transformed to $(\frac{1}{3}x,\frac{1}{3}y)$. This moves each point closer to the origin.

2. Line Properties: Similar to the dilation with a scale factor of 3, the slope of line AB remains unchanged. The orientation and angle with respect to the axes do not change either.

3. Distance from Origin: Since the scale factor is less than 1, each point on line AB is positioned closer to the origin, effectively "shrinking" the line towards the origin.

4. Parallel Lines: Line A'B' will again be parallel to line AB since the dilation is uniform around the origin.

Summary:

• Dilation with Scale Factor 3: The line A'B' moves further away from the origin, maintaining its slope, and remains parallel to line AB.
• Dilation with Scale Factor $\frac{1}{3}$: The line A'B' moves closer to the origin, maintains the same slope, and remains parallel to line AB.

In both cases, the line's slope, orientation, and parallelism to the original line remain unchanged, but the distance from the origin is affected in opposite directions based on whether the scale factor is greater than or less than 1.