When a line segment undergoes dilation, every point on the line segment is moved along a straight line that passes through the center of dilation, in this case, the origin. The key property of dilation is that it alters the distances between points by a certain scale factor (here, 1.3), but it does not change the direction of the line.
The slope of a line represents its steepness and is defined as the ratio of the vertical change to the horizontal change between any two points on the line. During the process of dilation, both the vertical and horizontal changes of the line segment are scaled by the same factor, preserving the ratio between them. Therefore, the slope of the line remains unchanged.
Given:
- The original slope of the line segment [tex]\(\overline{XY}\)[/tex] is [tex]\(m\)[/tex].
- The scale factor of the dilation is 1.3.
Since the dilation does not change the slope of the line, the slope of the image [tex]\(\overline{X'Y'}\)[/tex] is the same as the original slope.
Thus, the slope of [tex]\(\overline{X'Y'}\)[/tex] is:
[tex]\[ \boxed{m} \][/tex]
