To solve for the slope of the image line segment [tex]$\overline{X^{\prime} Y^{\prime}}$[/tex] after dilation, we need to understand the effect of dilation on the slope of a line.
1. Concept of Dilation: Dilation is a transformation that resizes objects by a scale factor while keeping their shape and orientation intact. In this case, the dilation is centered at the origin with a scale factor of 1.3.
2. Slope Transformation During Dilation:
- Dilation changes the size of the geometric figure but does not affect the slope of the line segment. This means that even though the segment length changes, the tilt or steepness of the line remains the same.
3. Given Information:
- The slope of the original line segment [tex]$\overline{X Y}$[/tex] is [tex]$m$[/tex].
- The length of the original line segment is [tex]$I$[/tex].
- The scale factor is 1.3.
4. Effect on the Slope:
- Since dilation does not change the slope of a line segment, the slope of the new line segment [tex]$\overline{X^{\prime} Y^{\prime}}$[/tex] will be the same as the slope of the original line segment [tex]$\overline{X Y}$[/tex].
Thus, the correct answer is [tex]\( m \)[/tex].
Therefore, the slope of [tex]$\overline{X^{\prime} Y^{\prime}}$[/tex] is [tex]\( m \)[/tex].
This matches option D: [tex]\( m \)[/tex].
