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7.2: Parallel Lines

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    45196

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    Definition: 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.

    Example 7.2.1

    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.

    Exercise 7.2.1

    Determine whether the given lines are parallel:

    1. 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 \((0, 5)\).
    2. 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 \((10, 5)\).
    3. 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 \((0, −3)\).

    This page titled 7.2: Parallel Lines is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Victoria Dominguez, Cristian Martinez, & Sanaa Saykali (ASCCC Open Educational Resources Initiative) .