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?
- \(\displaystyle 1\)
- \(\displaystyle 2\)
- \(\displaystyle \frac{1}{2} \)
- \(\displaystyle -2\)
Answer.
B
(b)
What is the slope of line B?
- \(\displaystyle 1\)
- \(\displaystyle 2\)
- \(\displaystyle \frac{1}{2}\)
- \(\displaystyle -2\)
Answer.
B
(c)
What is the \(y\)-intercept of line A?
- \(\displaystyle -2\)
- \(\displaystyle -1.5\)
- \(\displaystyle 1 \)
- \(\displaystyle 3\)
Answer.
D
(d)
What is the \(y\)-intercept of line B?
- \(\displaystyle -2\)
- \(\displaystyle -1.5\)
- \(\displaystyle 1 \)
- \(\displaystyle 3\)
Answer.
A
(e)
What is the same about the two lines?
Answer.
Both lines have the same slope (\(m=2\)).
(f)
What is different about the two lines?
Answer.
The lines have different \(y\)-intercepts.
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 \(f(x)\text{?}\)
- \(\displaystyle -1\)
- \(\displaystyle 2\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
Answer.
D
(b)
Applying Definition 3.4.3, what would the slope of a line parallel to \(f(x)\) be?
- \(\displaystyle -1\)
- \(\displaystyle 2\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
Answer.
D
(c)
Find the equation of a line parallel to \(f(x)\) that passes through the point \((-4,2)\text{.}\)
Answer.
\(y-2=-\frac{1}{2}(x+4)\) or
\(y=-\frac{1}{2}x\)
Activity 3.4.5.
Consider the graph of the two lines below.
(a)
What is the slope of line A?
- \(\displaystyle 3\)
- \(\displaystyle 2\)
- \(\displaystyle -\frac{1}{2} \)
- \(\displaystyle -2\)
Answer.
C
(b)
What is the slope of line B?
- \(\displaystyle 3\)
- \(\displaystyle 2\)
- \(\displaystyle -\frac{1}{2} \)
- \(\displaystyle -2\)
Answer.
B
(c)
What is the \(y\)-intercept of line A?
- \(\displaystyle -2\)
- \(\displaystyle -\frac{1}{2}\)
- \(\displaystyle 2 \)
- \(\displaystyle 3\)
Answer.
D
(d)
What is the \(y\)-intercept of line B?
- \(\displaystyle -2\)
- \(\displaystyle -\frac{1}{2}\)
- \(\displaystyle 1 \)
- \(\displaystyle 3\)
Answer.
C
(e)
If you were to think of slope as "rise over run," how would you write the slope of each line?
Answer.
Line A could be written as \(-\frac{1}{2}\) and Line B could be written as \(\frac{2}{1}\text{.}\)
(f)
How would you compare the slopes of the two lines?
Answer.
Students might notice that when writing the slopes of Line A and Line B, the slopes are negative reciprocals of each other.
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 \(f(x)\text{?}\)
- \(\displaystyle -\frac{1}{3}\)
- \(\displaystyle 3\)
- \(\displaystyle 5\)
- \(\displaystyle -\frac{1}{5}\)
Answer.
B
(b)
Applying Definition 3.4.7, what would the slope of a line perpendicular to \(f(x)\) be?
- \(\displaystyle -\frac{1}{3}\)
- \(\displaystyle 3\)
- \(\displaystyle 5\)
- \(\displaystyle -\frac{1}{5}\)
Answer.
A
(c)
Find the equation of a line perpendicular to \(f(x)\) that passes through the point \((3,6)\text{.}\)
Answer.
\(y-6=-\frac{1}{3}(x-3)\) or
\(y=-\frac{1}{3}x+7\)
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*}
Answer.
Parallel. The slope of \(f(x)\) is \(-3\) and the slope of \(g(x)\) is \(-3\text{.}\)
(b)
\begin{equation*}
f(x)=2x-5
\end{equation*}
\begin{equation*}
g(x)=6x-5
\end{equation*}
Answer.
Neither. The slope of \(f(x)\) is \(2\) and the slope of \(g(x)\) is \(6\text{.}\) These lines do, however, have the same \(y\)-intercept.
(c)
\begin{equation*}
f(x)=6x-5
\end{equation*}
\begin{equation*}
g(x)=\frac{1}{6}x+8
\end{equation*}
Answer.
Neither. The slope of \(f(x)\) is \(6\) and the slope of \(g(x)\) is \(\frac{1}{6}\text{.}\) Although they are reciprocals of one another, they are not negative reciprocals.
(d)
\begin{equation*}
f(x)=\frac{4}{5}x+3
\end{equation*}
\begin{equation*}
g(x)=-\frac{5}{4}x-1
\end{equation*}
Answer.
Perpendicular. The slope of \(f(x)\) is \(\frac{4}{5}\) and the slope of \(g(x)\) is \(-\frac{5}{4}\) (and are negative reciprocals of one another).
Activity 3.4.10.
Consider the linear equation, \(f(x)=-\frac{2}{3}x-4\) and the point A: \((-6,4)\text{.}\)
(a)
Find the equation of a line that is parallel to \(f(x)\) and passes through the point A.
Answer.
\(y-4=-\frac{2}{3}(x+6)\) or
\(y=-\frac{2}{3}x\)
(b)
Find the equation of a line that is perpendicular to \(f(x)\) and passes through the point A.
Answer.
\(y-4=\frac{3}{2}(x+6)\) or
\(y=\frac{3}{2}x+13\)
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
- \(\displaystyle 0\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
Answer.
B
(b)
What is the slope of a line that is parallel to \(y=2\text{?}\)
- undefined
- \(\displaystyle 0\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
Answer.
B
(c)
Find the equation of a line that is parallel to \(y=2\) and passes through the point \((-1,-4)\text{.}\)
Answer.
\(y=-4\text{.}\) Students might need the graph to help them visualize why the equation is in the form \(y=\) number.
(d)
What is the slope of a line that is perpendicular to \(y=2\text{?}\)
- undefined
- \(\displaystyle 0\)
- \(\displaystyle 1\)
- \(\displaystyle -\frac{1}{2}\)
Answer.
A. You might need to help students see why the slope is undefined by showing that \(-\frac{1}{0}\) is not defined.
(e)
Find the equation of a line that is perpendicular to \(y=2\) and passes through the point \((-1,2)\text{.}\)
Answer.
\(x=-1\text{.}\) Students might need the graph to help them visualize why the equation is in the form \(x=\) number.
Subsection 3.4.2 Videos
It would be great to include videos down here, like in the Calculus book!
