Objectives
- Use slope relationships to determine whether two lines are parallel or perpendicular, and find the equation of lines parallel or perpendicular to a given line through a given point.
Section 3.4 Parallel and Perpendicular Lines (LF4)
Subsection 3.4.1 Activities
Activity 3.4.1.
Let’s revisit Activity 3.2.1 to investigate special types of lines.
(a)
What is the slope of line A?
- \(\displaystyle 1\)
- \(\displaystyle 2\)
- \(\displaystyle \frac{1}{2} \)
- \(\displaystyle -2\)
(b)
What is the slope of line B?
- \(\displaystyle 1\)
- \(\displaystyle 2\)
- \(\displaystyle \frac{1}{2}\)
- \(\displaystyle -2\)
(c)
What is the \(y\)-intercept of line A?
- \(\displaystyle -2\)
- \(\displaystyle -1.5\)
- \(\displaystyle 1 \)
- \(\displaystyle 3\)
(d)
What is the \(y\)-intercept of line B?
- \(\displaystyle -2\)
- \(\displaystyle -1.5\)
- \(\displaystyle 1 \)
- \(\displaystyle 3\)
(e)
What is the same about the two lines?
(f)
What is different about the two lines?
Remark 3.4.2.
Notice that in Activity 3.4.1 the two lines never touch.Definition 3.4.3.
Parallel lines are lines that always have the same distance apart (equidistant) and will never meet. Parallel lines have the same slope, but different \(y\)-intercepts.
Activity 3.4.4.
Suppose you have the function,
\begin{equation*}
f(x)=-\frac{1}{2}x-1
\end{equation*}
(a)
What is the slope of \(f(x)\text{?}\)
- \(\displaystyle -1\)
- \(\displaystyle 2\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
(b)
Applying Definition 3.4.3, what would the slope of a line parallel to \(f(x)\) be?
- \(\displaystyle -1\)
- \(\displaystyle 2\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
(c)
Find the equation of a line parallel to \(f(x)\) that passes through the point \((-4,2)\text{.}\)
Activity 3.4.5.
Consider the graph of the two lines below.
(a)
What is the slope of line A?
- \(\displaystyle 3\)
- \(\displaystyle 2\)
- \(\displaystyle -\frac{1}{2} \)
- \(\displaystyle -2\)
(b)
What is the slope of line B?
- \(\displaystyle 3\)
- \(\displaystyle 2\)
- \(\displaystyle -\frac{1}{2} \)
- \(\displaystyle -2\)
(c)
What is the \(y\)-intercept of line A?
- \(\displaystyle -2\)
- \(\displaystyle -\frac{1}{2}\)
- \(\displaystyle 2 \)
- \(\displaystyle 3\)
(d)
What is the \(y\)-intercept of line B?
- \(\displaystyle -2\)
- \(\displaystyle -\frac{1}{2}\)
- \(\displaystyle 1 \)
- \(\displaystyle 3\)
(e)
If you were to think of slope as "rise over run," how would you write the slope of each line?
(f)
How would you compare the slopes of the two lines?
Remark 3.4.6.
Notice in Activity 3.4.5, that even though the two lines have different slopes, the slopes are somewhat similar. For example, if you take the slope of Line A \(\left(-\frac{1}{2}\right)\) and flip and negate it, you will get the slope of Line B \(\left(\frac{2}{1}\right)\text{.}\)Definition 3.4.7.
Perpendicular lines are two lines that meet or intersect each other at a right angle. The slopes of two perpendicular lines are negative reciprocals of each other (given that the slope exists!).
Activity 3.4.8.
Suppose you have the function,
\begin{equation*}
f(x)=3x+5
\end{equation*}
(a)
What is the slope of \(f(x)\text{?}\)
- \(\displaystyle -\frac{1}{3}\)
- \(\displaystyle 3\)
- \(\displaystyle 5\)
- \(\displaystyle -\frac{1}{5}\)
(b)
Applying Definition 3.4.7, what would the slope of a line perpendicular to \(f(x)\) be?
- \(\displaystyle -\frac{1}{3}\)
- \(\displaystyle 3\)
- \(\displaystyle 5\)
- \(\displaystyle -\frac{1}{5}\)
(c)
Find an equation of the line perpendicular to \(f(x)\) that passes through the point \((3,6)\text{.}\)
Activity 3.4.9.
For each pair of lines, determine if they are parallel, perpendicular, or neither.
(a)
\begin{equation*}
f(x)=-3x+4
\end{equation*}
\begin{equation*}
g(x)=5-3x
\end{equation*}
(b)
\begin{equation*}
f(x)=2x-5
\end{equation*}
\begin{equation*}
g(x)=6x-5
\end{equation*}
(c)
\begin{equation*}
f(x)=6x-5
\end{equation*}
\begin{equation*}
g(x)=\frac{1}{6}x+8
\end{equation*}
(d)
\begin{equation*}
f(x)=\frac{4}{5}x+3
\end{equation*}
\begin{equation*}
g(x)=-\frac{5}{4}x-1
\end{equation*}
Activity 3.4.10.
Consider the linear equation, \(f(x)=-\frac{2}{3}x-4\) and the point A: \((-6,4)\text{.}\)
(a)
Find an equation of the line that is parallel to \(f(x)\) and passes through the point A.
(b)
Find an equation of the line that is perpendicular to \(f(x)\) and passes through the point A.
Activity 3.4.11.
Consider the line, \(y=2\text{,}\) as shown in the graph below.
(a)
What is the slope of the line \(y=2\text{?}\)
- undefined
- \(\displaystyle 0\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
(b)
What is the slope of a line that is parallel to \(y=2\text{?}\)
- undefined
- \(\displaystyle 0\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
(c)
Find an equation of the line that is parallel to \(y=2\) and passes through the point \((-1,-4)\text{.}\)
(d)
What is the slope of a line that is perpendicular to \(y=2\text{?}\)
- undefined
- \(\displaystyle 0\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
(e)
Find an equation of the line that is perpendicular to \(y=2\) and passes through the point \((-1,2)\text{.}\)
Exercises 3.4.2 Exercises
Exercises available at
https://tbil.org/precalculus/preview/exercises/#/bank/LF4/.
