To solve this problem, let's carefully analyze how dilation affects both the slope and the length of a line segment.
1. Understanding Dilation:
- Dilation is a transformation that scales a figure by a certain factor relative to a point (in this case, the origin).
- The scale factor of the dilation is given as 1.3. This factor multiplies the length of the line segment but does not affect its slope.
2. Effect on the Length:
- If the original length of line segment [tex]\(\overline{XY}\)[/tex] is denoted as [tex]\(l\)[/tex], the length of the dilated line segment [tex]\(\overline{X^{\prime}Y^{\prime}}\)[/tex] will be [tex]\(1.3 \times l\)[/tex].
- However, length transformation does not alter the slope.
3. Effect on the Slope:
- The slope [tex]\(m\)[/tex] of a line is fundamentally a ratio of the vertical change to the horizontal change between any two points on the line.
- Dilation uniformly scales both the horizontal and vertical distances by the given factor. Since the ratio of the vertical change to the horizontal change remains constant, the slope remains unchanged.
Thus, the slope of the dilated line segment [tex]\(\overline{X^{\prime}Y^{\prime}}\)[/tex] would be the same as the slope of the original line segment [tex]\(\overline{XY}\)[/tex].
4. Identifying the Correct Answer:
- We know the slope remains unchanged by the dilation process.
- Therefore, the slope of [tex]\(\overline{X^{\prime}Y^{\prime}}\)[/tex] is [tex]\(m\)[/tex], which corresponds to option [tex]\(D\)[/tex].
Given these observations, the slope of [tex]\(\overline{X^{\prime}Y^{\prime}}\)[/tex] is:
[tex]\[
\boxed{m}
\][/tex]
