Parallel and perpendicular lines

2020-11-24T11:18:29+02:00

The slope coefficients of lines can be used to examine whether two lines are parallel or perpendicular to one another. Lines that intersect perpendicularly are called perpendiculars.

Parallel and perpendicular lines

Two lines are

parallel to one another if their slope coefficients are the same.
perpendicular to one another if the product of the slope coefficients is [[$ –1 $]] or if one line is parallel to the [[$ x $]] axis and the other is parallel to the [[$ y $]] axis.

Example 1

Examine whether the lines [[$ y = \displaystyle\frac {1} {2}x – 1 $]] and [[$ x – 2y + 6 = 0 $]] are parallel.

Solution:

The slope of the line [[$ y = \displaystyle\frac {1} {2}x – 1 $]] is [[$ \displaystyle\frac {1} {2} $]].

Convert the equation [[$ x - 2 y + 6 = 0 $]] into its slope-intercept form:

[[$ \quad \begin{align} x - 2y + 6 &= 0 \\ - 2y &= –x – 6 \space ||:(–2) \\ y &= \displaystyle\frac {1} {2}x + 3 \end{align} $]]

Thus, the slope of the line [[$ x - 2y + 6 = 0 $]] is [[$ \displaystyle\frac {1} {2} $]].

Answer: Since the two lines have the same slope, the lines are parallel.

Example 2

Determine the equation of a line that passes through point [[$ (-2, 1) $]] and is perpendicular to the line [[$ y = \displaystyle\frac {1} {2}x + 2 $]].

Solution:

The slope of the given line is [[$ \displaystyle\frac {1} {2} $]].

Since the lines are perpendicular to one another, the product of their slope must be [[$ –1 $]]. Based on this, an equation can be used to solve the slope of the second line.

[[$ \quad \begin{align} \displaystyle\frac {1} {2} · k &= –1 \space ||· 2 \\ k &= –2 \end{align} $]]

Thus, the equation of the line is of the form [[$ y = -2x + b $]]. To solve the constant term [[$ b $]], place the point [[$ (-2, 1) $]] on the line expression, and solve the resulting equation for the variable [[$ b $]].

[[$ \quad \begin{align}1 &= -2 · (-2) + b \ \\ 1 &= 4 + b \ \\ - b &= 4 -1 \ \\ - b &= 3 \space ||:(–1) \ \\ b &= -3 \end{align} $]]

Answer: The equation of the line is [[$ y = –2x – 3 $]].

The answer can be checked by drawing the two lines in the same coordinate system:

7. Parallel and perpendicular lines

Parallel and perpendicular lines