To determine the slope of the dilated line segment [tex]\(\overline{X^{\prime} Y^{\prime}}\)[/tex], we need to understand how dilation affects a line.
A dilation with respect to the origin involves scaling the distances from the origin to the points on the line by the scale factor, which in this case is 1.3. This transformation preserves the direction of the line but changes the magnitude of the lengths proportionally.
The slope of a line, represented by [tex]\(m\)[/tex], is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. When we dilate a line segment [tex]\(\overline{X Y}\)[/tex] by a scale factor of 1.3, both the vertical change and the horizontal change of the line segment are multiplied by 1.3, as the points are moved proportionally further from the origin.
However, since both the rise and run are scaled by the same factor, their ratio (which defines the slope) remains unchanged. Therefore, the slope of [tex]\(\overline{X^{\prime} Y^{\prime}}\)[/tex] will be the same as the slope of [tex]\(\overline{X Y}\)[/tex].
Given that the slope of [tex]\(\overline{X Y}\)[/tex] is [tex]\(m\)[/tex], the slope of [tex]\(\overline{X^{\prime} Y^{\prime}}\)[/tex] also remains [tex]\(m\)[/tex].
Thus, the correct answer is:
[tex]\[ D. \quad m \][/tex]
