7.3: Perpendicular Lines
Two distinct lines \(l\) and \(q\) are perpendicular , written \(l ⊥ q\), if their intersection form four right angles or angles with measure \(90^{\circ}\). The slopes of the perpendicular lines \(l\) and \(q\) are negative reciprocals. That is,
\[m_l = −\dfrac{1}{m_q} \nonumber \]
and
\[m_q = − \dfrac{1}{m_l} \nonumber \]
Determine if the given lines are perpendicular. The line \(l\) that passes through the points \((0, 1)\) and \((1, 3)\), and the line \(q\) that passes through the points \((−1, 4)\) and \((5, 1)\).
Solution
To determine if the lines are perpendicular, first find their slopes using the slope of the line formula. The slope of line \(l\), \(m_l\), that passes through the points \((0, 1)\) and \((1, 3)\) is,
\(\begin{array}s m_l &= \dfrac{3 − 1}{1 − 0} \\ &= \dfrac{2}{1} \\ &= 2 \end{array}\)
The slope of line \(q\), \(m_q\), that passes through the points \((−1, 4)\) and \((5, 1)\), is
\(\begin{array}s m_q &= \dfrac{1 − 4}{5 − (-1)} \\ &= \dfrac{-3}{6} \\ &= \dfrac{-1}{2} \end{array}\)
Now, lines \(l\) and \(q\) are perpendicular if and only if:
\(m_l = −\dfrac{1}{m_q} \text{ and } m_q = −\dfrac{1}{m_l}\)
\(m_l = 2\) and \(m_q = −\dfrac{1}{m_l} = −\dfrac{1}{2}\). Hence, the slopes of the lines are negative reciprocals so it can be concluded that lines \(l\) and \(q\) are perpendicular lines.
Find the slope of a line perpendicular to line \(l\) that passes through the points \((−3, 0)\) and \((3, 4)\).
Solution
Start by finding the slope of line \(l\) that passes through the points \((−3, 0)\) and \((3, 4)\), using the slope of the line formula. Thus,
\(\begin{array} s m_l &= \dfrac{y_2 − y_1}{x_2 − x_1} \\ &= \dfrac{4 − 0}{3 − (−3)} \\ &= \dfrac{4}{6} \\ &= \dfrac{2}{3} \end{array}\)
Any line perpendicular to line \(l\) must have a slope that is negative reciprocal of its slope. Since \(m_l = \dfrac{2}{3}\) then the slope of the line perpendicular to line \(l\) must be \(m = −\dfrac{3}{2}\)
Determine if the given lines are perpendicular.
- The line \(l\) that passes through the points \((0, 4)\) and \((5, 3)\) and the line \(q\) that passes through the points \((1, 5)\) and \((−1, −5)\).
- The line \(l\) that passes through the points \((−2, −5)\) and \((1, 7)\) and the line \(q\) that passes through the points \((−4, 1)\) and \((−3, −3)\).
Find the slope of a line perpendicular to:
- Line \(l\) that passes through the points \((4, 2)\) and \((−1, −2)\).
- Line \(q\) that passes through the points \((7, −8)\) and \((9, 1)\).