To determine the length of the segment [tex]$\overline{J^{\prime} K^{\prime}}$[/tex] after dilation by a scale factor of [tex]$n$[/tex] with the origin as the center of dilation, let's follow these steps:
1. Understand the concept of dilation:
Dilation is a transformation that produces an image that is the same shape as the original, but is a different size. The size change is determined by a scale factor, and the image is positioned relative to a fixed center point.
2. Scale factor and segment length:
If [tex]$\overline{J K}$[/tex] is a segment with a length of 1 unit and it is dilated by a scale factor of [tex]$n$[/tex], the length of the resulting segment [tex]$\overline{J^{\prime} K^{\prime}}$[/tex] changes according to the scale factor.
3. Effect of dilation on length:
When a segment is dilated with a scale factor of [tex]$n$[/tex], the length of the new segment [tex]$\overline{J^{\prime} K^{\prime}}$[/tex] is given by multiplying the length of the original segment [tex]$\overline{J K}$[/tex] by the scale factor [tex]$n$[/tex].
4. Calculation:
- The original length of [tex]$\overline{J K}$[/tex] is 1.
- The scale factor is [tex]$n$[/tex].
- Therefore, the length of [tex]$\overline{J^{\prime} K^{\prime}} = 1 * n = n$[/tex].
Based on this understanding, the correct answer is:
D. [tex]\( n \times 1 \)[/tex]
