7.2: Parallel Lines
In a coordinate plane, parallel lines are lines that do not meet or intersect. They are always the same distance apart. Moreover, parallel lines have the same slope .
Find the slope of the line \(l\) that passes through \((2, 0)\) and \((4, −3)\) and the slope of the line \(q\) that passes through \((2, −3)\) and \((4, −6)\). Determine if the lines are parallel.
Solution
Use the slope of the line formula to find the slope of line \(l\), \(m_l\), and the slope of line \(q\), \(m_q\), as follows,
\(\begin{array} &&m_l = \dfrac{y_2 − y_1}{x_2 − x_1}\;\;\;\;\;\;\;\;\;\; &m_q = \dfrac{y_2 − y_1}{x_2 − x_1} \\ &= \dfrac{−3 − 0}{4 − 2}\;\;\;\;\;\;\;\;\;\; &= \dfrac{−6 − (−3)}{4 − 2} \\ &= \dfrac{−3}{2}\;\;\;\;\;\;\;\;\;\; &= \dfrac{−3}{2} \end{array}\)
Since the two slopes are equal, then, lines \(l\) and \(q\) are parallel.
Determine whether the given lines are parallel:
- The line \(l\) that passes through the points \((2, 2)\) and \((3, 3)\) and the line \(q\) that passes through the points \((4, 1)\) and \((0, 5)\).
- The line \(l\) that passes through the points \((1, 3)\) and \((6, −2)\) and the line \(q\) that passes through the points \((−2, −7)\) and \((10, 5)\).
- The line \(l\) that passes through the points \((−6, 5)\) and \((2, −1)\) and the line \(q\) that passes through the points \((−4, 0)\) and \((0, −3)\).