Which line is perpendicular to a line that has a slope of $-\frac{5}{6}$?
The slope of the given line is \(m_1 = -\frac{5}{6}\). 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, \(m_1 \cdot m_2 = -1\). This is a fundamental property of perpendicular lines in a Cartesian coordinate system. To find the slope of the perpendicular line, we need to solve for \(m_2\) in the equation \(m_1 \cdot m_2 = -1\).
Substitute the value of \(m_1 = -\frac{5}{6}\) into the perpendicularity condition \(m_1 \cdot m_2 = -1\):
\[\left(-\frac{5}{6}\right) \cdot m_2 = -1\]
To isolate \(m_2\), multiply both sides of the equation by \(-\frac{6}{5}\):
\[m_2 = (-1) \cdot \left(-\frac{6}{5}\right)\]
\[m_2 = \frac{6}{5}\]
Thus, any line with a slope of \(\frac{6}{5}\) is perpendicular to a line with a slope of \(-\frac{5}{6}\).
The line that is perpendicular to the given line must have a slope of \(\frac{6}{5}\). Based on the relationship between slopes and perpendicularity, the target line is the one whose slope satisfies the condition derived.
The final answer is \(\boxed{\frac{6}{5}}\).
In architecture and engineering, ensuring that structural elements or layout lines are perpendicular is essential for stability and aesthetics. For instance, when framing a house, if one wall is set at a specific slope on a blueprint, the perpendicular wall must have the negative reciprocal slope to ensure the corners form precise 90-degree angles. Using the rule \(m_2 = -\frac{1}{m_1}\) allows builders to calculate the correct orientation for intersecting walls precisely.