7.3: Perpendicular Lines

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Page ID: 45197

Victoria Dominguez, Cristian Martinez, & Sanaa Saykali
Citrus College via ASCCC Open Educational Resources Initiative

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Definition: 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

\[m_q = − \dfrac{1}{m_l} \nonumber \]

Example 7.3.1

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.

Example 7.3.2

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 \(m = −\dfrac{3}{2}\)

Exercise 7.3.1

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 \((−3, −3)\).

Exercise 7.3.2

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 \((9, 1)\).