To determine a line that is perpendicular to a given line, we need to understand the relationship between their slopes. When two lines are perpendicular, the product of their slopes is [tex]\(-1\)[/tex]. This means that if we have a line with a slope [tex]\( m \)[/tex], the slope of the line that is perpendicular to it will be the negative reciprocal of [tex]\( m \)[/tex].
Let's apply this to the given problem step-by-step:
1. Identify the slope of the given line. The problem states that the slope of the given line is [tex]\(\frac{1}{2}\)[/tex].
2. Find the negative reciprocal of the given slope to determine the slope of the line that is perpendicular to it. The negative reciprocal of [tex]\(\frac{1}{2}\)[/tex] is [tex]\(-2\)[/tex].
To calculate the negative reciprocal of a fraction [tex]\(\frac{a}{b}\)[/tex]:
- Invert the fraction, changing [tex]\(\frac{a}{b}\)[/tex] to [tex]\(\frac{b}{a}\)[/tex].
- Change the sign to its opposite. Since [tex]\(\frac{a}{b}\)[/tex] is positive, its negative reciprocal will be negative.
Therefore, the negative reciprocal of [tex]\(\frac{1}{2}\)[/tex] is calculated as follows:
[tex]\[ \text{Negative reciprocal of } \frac{1}{2} = -\frac{2}{1} = -2 \][/tex]
Thus, the slope of the line perpendicular to the given line is [tex]\(-2\)[/tex].
Given the above calculation, the line that is perpendicular to the line with slope [tex]\(\frac{1}{2}\)[/tex] will have a slope of [tex]\(-2\)[/tex]. This means any of the lines listed (line [tex]\(AB\)[/tex], line [tex]\(CD\)[/tex], line [tex]\(FG\)[/tex], or line [tex]\(HJ\)[/tex]) that has a slope of [tex]\(-2\)[/tex] will be perpendicular to the given line.
