To determine which line is perpendicular to a given line with a slope of [tex]\(-\frac{1}{3}\)[/tex], we need to understand the relationship between the slopes of perpendicular lines. The key property is that the slopes of two perpendicular lines are negative reciprocals of each other.
Given slope:
[tex]\[ m_1 = -\frac{1}{3} \][/tex]
1. Find the negative reciprocal of [tex]\( m_1 \)[/tex]:
The reciprocal of [tex]\(-\frac{1}{3}\)[/tex] is [tex]\(-3\)[/tex] (since [tex]\(\frac{1}{-\frac{1}{3}} = -3\)[/tex]).
2. Take the negative of this reciprocal:
Negative reciprocal of [tex]\( -\frac{1}{3} \)[/tex] is:
[tex]\[
m_2 = -(-3) = 3
\][/tex]
Thus, the slope of a line that is perpendicular to a line with a slope of [tex]\(-\frac{1}{3}\)[/tex] is [tex]\(3\)[/tex].
Therefore, the line that is perpendicular to the line with a slope of [tex]\(-\frac{1}{3}\)[/tex] has a slope of [tex]\(3\)[/tex]. To choose the correct line (MN, AB, EF, or JK) for any real-world or problem-specific scenario, it is essential to know the individual slopes of these lines. Given only the problem's requirements and knowing the perpendicular slope must be [tex]\(3\)[/tex], you would confirm that the correct line is the one whose slope is [tex]\(3\)[/tex].
