To determine the length of the dilated segment [tex]\(\overline{J^{\prime}K^{\prime}}\)[/tex] after the dilation with a scale factor [tex]\(n\)[/tex], we must understand how dilation affects lengths.
Given the original length [tex]\(\overline{J K}\)[/tex] is [tex]\(I\)[/tex], and applying a dilation with a scale factor [tex]\(n\)[/tex] from the origin, the length of the image [tex]\(\overline{J^{\prime} K^{\prime}}\)[/tex] can be calculated as follows:
1. Identify the effect of dilation: A dilation with scale factor [tex]\(n\)[/tex] means that every distance from the origin is multiplied by [tex]\(n\)[/tex].
2. Apply the scale factor to the length: The original length [tex]\(\overline{J K}\)[/tex] is multiplied by [tex]\(n\)[/tex].
Thus, the length of the segment [tex]\(\overline{J^{\prime}K^{\prime}}\)[/tex] after dilation is given by:
[tex]\[ n \times I \][/tex]
From the choices provided:
A. [tex]\( m \times n \times 1 \)[/tex] \\
B. [tex]\( (m+n) \times 1 \)[/tex] \\
C. [tex]\( m \times 1 \)[/tex] \\
D. [tex]\( n \times 1 \)[/tex]
Here, [tex]\(I\)[/tex] is given as [tex]\(1\)[/tex].
So, substituting [tex]\(I = 1\)[/tex] into [tex]\( n \times I \)[/tex], we get:
[tex]\[ n \times 1 \][/tex]
The correct answer is:
[tex]\[ \boxed{D \; n \times 1} \][/tex]
