To solve this problem, we need to understand the concept of dilation in geometry. Dilation is a transformation that stretches or shrinks a figure by a scale factor relative to a fixed point called the center of dilation.
The question states that line segment [tex]\(\overline{JK}\)[/tex] is dilated by a scale factor of [tex]\(n\)[/tex] with the origin as the center of dilation, resulting in the image [tex]\(\overline{J'K'}\)[/tex]. The given slope [tex]\(m\)[/tex] of [tex]\(\overline{JK}\)[/tex] does not affect the length of the segment but rather its steepness. The length of [tex]\(\overline{JK}\)[/tex] is given as [tex]\(l\)[/tex].
When a segment is dilated by a scale factor [tex]\(n\)[/tex], each length of the segment is multiplied by [tex]\(n\)[/tex]. This means that if [tex]\(\overline{JK}\)[/tex] has a length [tex]\(l\)[/tex], its image [tex]\(\overline{J'K'}\)[/tex] after dilation will have a length of:
[tex]\[
\text{Length of } \overline{J'K'} = n \times l
\][/tex]
Given that the result of this calculation is 15, implying that:
[tex]\[
n \times l = 15
\][/tex]
From the options provided:
A. [tex]\(m \times n \times l\)[/tex]
B. [tex]\((m + n) \times 1\)[/tex]
C. [tex]\(m \times 1\)[/tex]
D. [tex]\(n \times 1\)[/tex]
Only option D, [tex]\(n \times 1\)[/tex], aligns with our required format after simplifying the expression as [tex]\(n \times l\)[/tex] or directly [tex]\(15\)[/tex].
Therefore, the correct answer is:
D. [tex]\(n \times 1\)[/tex]
