Let's solve the problem step-by-step to find the length of the dilated image [tex]\(\overline{J^{\prime} K^{\prime}}\)[/tex].
1. Understanding the Original Length:
- The given length of [tex]\(\overline{J K}\)[/tex] is denoted as [tex]\(l\)[/tex]. However, in the context of this problem, we see [tex]\(l\)[/tex] is not explicitly mentioned. Instead, the length of [tex]\(\overline{J K}\)[/tex] is usually taken as a unit length (for illustration purposes).
2. Dilation Transformation:
- The dilation transformation scales a figure by a given scale factor. Here, [tex]\(\overline{J K}\)[/tex] is being scaled by a factor of [tex]\(n\)[/tex] with the origin as the center of dilation.
3. Effect of Dilation on Length:
- When a line segment is dilated by a scale factor with respect to the origin, the length of the new line segment is the original length multiplied by the scale factor.
4. Calculating the New Length:
- If the original length of [tex]\(\overline{J K}\)[/tex] is 1 unit, the length of [tex]\(\overline{J^{\prime} K^{\prime}}\)[/tex] will be the original length multiplied by the scale factor [tex]\(n\)[/tex]:
[tex]\[
\text{Length of } \overline{J^{\prime} K^{\prime}} = 1 \times n = n
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{n \times 1} \][/tex]
This corresponds to option D in the given choices. The length of [tex]\(\overline{J^{\prime} K^{\prime}}\)[/tex] is [tex]\(n\)[/tex].
