Which line is perpendicular to a line that has a slope of $-\frac{1}{3}$?
Two lines with slopes \(m_1\) and \(m_2\) are perpendicular if and only if the product of their slopes is equal to \(-1\). That is:
\[m_1 \cdot m_2 = -1\]
Given the slope of the original line is \(m_1 = -\frac{1}{3}\), we need to find the slope \(m_2\) of a line perpendicular to it. Substituting \(m_1\) into the equation:
\[-\frac{1}{3} \cdot m_2 = -1\]
Multiplying both sides by \(-3\), we find:
\[m_2 = (-1) \cdot (-3) = 3\]
Thus, any line with a slope of \(3\) is perpendicular to the original line. Since the question asks "Which line is perpendicular..." among choices (line MN, line JK, or line AB) without providing their equations or slopes, the defining characteristic for such a line is that its slope must be \(3\). Without further information (like coordinates or specific equations for MN, JK, or AB), the answer is determined by this property. If the question implies selecting from these based on their properties in a broader context or if this is a general conceptual question, we conclude that the perpendicular line must have a slope of \(3\). If the options are meant to be evaluated against this requirement, then the line with slope \(3\) is the correct one. In a generic sense, the property of being perpendicular is solely defined by this slope relationship.
Perpendicularity is a fundamental concept in engineering and design. For example, if you are designing a table where the legs must be perfectly vertical to the tabletop, you ensure the connection between the leg (represented by one line) and the frame (represented by another line) has slopes that satisfy the perpendicularity condition \(m_1 \cdot m_2 = -1\). If the frame is placed at a slope of \(-\frac{1}{3}\), the legs must be attached at a slope of \(3\) relative to the frame to maintain a perfect right angle, ensuring the table is stable and balanced.