Verified Answer :

  • The slope \(m\) of the original line is \(-\frac{5}{6}\).
  • Perpendicular lines satisfy the condition \(m \cdot m_{\perp} = -1\).
  • Solving for the required perpendicular slope gives \(m_{\perp} = -\frac{1}{-\frac{5}{6}} = \frac{6}{5}\).
  • Any line with a slope of \(\boxed{\frac{6}{5}}\) is perpendicular to the given line.

Explanation

  1. Determine the required slope for a perpendicular line.

To determine the slope of a line that is perpendicular to a given line, we rely on the fundamental relationship between the slopes of two perpendicular lines in a 2D coordinate plane. If a line has a slope \(m\), any line perpendicular to it must have a slope \(m_{\perp}\) such that the product of their slopes is \(-1\). Mathematically, this is expressed as:
\[m \cdot m_{\perp} = -1\]
In this problem, we are given a line with a slope of \(m = -\frac{5}{6}\). To find the slope \(m_{\perp}\) of the line perpendicular to it, we substitute the known value into the formula:
\[-\frac{5}{6} \cdot m_{\perp} = -1\]
To isolate \(m_{\perp}\), we multiply both sides of the equation by \(-\frac{6}{5}\):
\[m_{\perp} = (-1) \cdot \left(-\frac{6}{5}\right)\]
\[m_{\perp} = \frac{6}{5}\]
Therefore, any line perpendicular to the given line must have a slope of \(\frac{6}{5}\). While the specific choices (JK, LM, NO, PQ) are provided as labels, the correct answer corresponds to any line that possesses this calculated slope of \(\frac{6}{5}\).

Examples

In urban planning, streets are often designed to intersect at right angles to create efficient grids. If a street follows the path of a line with a slope of \(-\frac{5}{6}\) on a city map, any street constructed perpendicular to it must be mapped with a slope of \(\frac{6}{5}\) to ensure they meet at perfect \(90^{\circ}\) angles, which is essential for orderly traffic flow and proper drainage systems.

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