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[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]$\overline{J^{\prime}K^{\prime}}$[/tex]. The slope of [tex]$\overline{JK}$[/tex] is [tex]$m$[/tex]. If the length of [tex]$\overline{JK}$[/tex] is [tex]$l$[/tex], what is the length of [tex]$\overline{J^{\prime}K^{\prime}}$[/tex]?

A. [tex]$m \times n \times l$[/tex]
B. [tex]$(m + n) \times l$[/tex]
C. [tex]$m \times l$[/tex]
D. [tex]$n \times l$[/tex]

Sagot :

GinnyAnswer
To solve the problem, let's break it down into manageable steps.

1. Understand the dilation process: Dilation is a transformation that changes the size of a figure but not its shape. The scale factor [tex]\( n \)[/tex] tells us how much the figure is enlarged or reduced.

2. Given data:
- Original line segment [tex]\(\overline{JK}\)[/tex]
- Original length of [tex]\(\overline{JK}\)[/tex] is [tex]\(I\)[/tex]
- Scale factor of dilation is [tex]\(n\)[/tex]
- The center of dilation is the origin, but the point of dilation is not relevant to find the length after dilation.
- The slope of [tex]\(\overline{JK}\)[/tex] is [tex]\(m\)[/tex], but the slope doesn't affect the length after a dilation.

3. Length after dilation: The length of a line segment changes according to the scale factor when dilating. Specifically, the new length of the line segment is given by multiplying the original length by the scale factor.

4. Calculate the new length:
- Original length = [tex]\(I\)[/tex]
- Scale factor = [tex]\(n\)[/tex]
- New length = [tex]\(n \times I\)[/tex]

Based on the given choices, let's analyze the options:

A. [tex]\(m \times n \times 1\)[/tex]
- Incorrect because it involves slope [tex]\(m\)[/tex] which is unrelated to length transformation.

B. [tex]\((m+n) \times 1\)[/tex]
- Incorrect because it adds [tex]\(m\)[/tex] and [tex]\(n\)[/tex] which doesn't correspond to the length of the segment after dilation.

C. [tex]\(m \times 1\)[/tex]
- Incorrect because the length after dilation should involve the scale factor [tex]\(n\)[/tex] and original length [tex]\(I\)[/tex].

D. [tex]\(n \times 1\)[/tex]
- Correct if we consider the length to be scaled by [tex]\(n\)[/tex], but it does not explicitly include [tex]\(I\)[/tex].

Clearly, there might be a mistake in the provided options as they do not reflect the exact calculation involving [tex]\(I\)[/tex]. However, if adjusting to assuming [tex]\(I = 1\)[/tex] (default unit length), the correct answer transforming only by [tex]\(n\)[/tex] would be:
[tex]\[ \text{New length} = n \times 1 \][/tex]

Therefore, the closest correct answer under this special case is:
[tex]\[ \boxed{D} \][/tex]
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