To solve this problem, we need to understand how dilation affects a line segment's slope.
### Understanding Dilation:
Dilation is a transformation that scales objects by a certain factor while preserving the shape and orientation of the object. When a line segment is dilated with respect to the origin by a scale factor, only its length changes, but its slope remains the same.
#### Given Information:
- The line segment [tex]\(\overline{XY}\)[/tex] has a slope of [tex]\(m\)[/tex] and a length of [tex]\(I\)[/tex].
- The dilation factor is 1.3.
- The origin is the center of dilation.
### Computational Details:
1. Effect on Slope:
The slope of a line is defined as the ratio of the vertical change ([tex]\(\Delta y\)[/tex]) to the horizontal change ([tex]\(\Delta x\)[/tex]). The formula for the slope ([tex]\(m\)[/tex]) is:
[tex]\[
m = \frac{\Delta y}{\Delta x}
\][/tex]
When we dilate the line segment by a factor of 1.3, both [tex]\(\Delta y\)[/tex] and [tex]\(\Delta x\)[/tex] are scaled by the same factor (1.3). Thus, the new vertical change ([tex]\(\Delta y'\)[/tex]) and the new horizontal change ([tex]\(\Delta x'\)[/tex]) are:
[tex]\[
\Delta y' = 1.3 \times \Delta y
\][/tex]
[tex]\[
\Delta x' = 1.3 \times \Delta x
\][/tex]
2. Calculating the New Slope:
The slope of the dilated line segment [tex]\(\overline{X'Y'}\)[/tex], which we'll call [tex]\(m'\)[/tex], is:
[tex]\[
m' = \frac{\Delta y'}{\Delta x'} = \frac{1.3 \times \Delta y}{1.3 \times \Delta x} = \frac{\Delta y}{\Delta x} = m
\][/tex]
### Conclusion:
The slope of the line segment [tex]\(\overline{X'Y'}\)[/tex] remains the same post-dilation.
Thus, the correct answer is:
[tex]\[
\boxed{m}
\][/tex]
