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Section 3.4 Parallel and Perpendicular Lines (LF4)
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.
Subsection 3.4.1 Activities
Activity 3.4.1 .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
(a)
What is the slope of line A?
\(\displaystyle \dfrac{1}{2} \)
(b)
What is the slope of line B?
\(\displaystyle \dfrac{1}{2}\)
(c)
What is the \(y\) -intercept of line A?
(d)
What is the \(y\) -intercept of line B?
(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.
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)=-\dfrac{1}{2}x-1
\end{equation*}
(a)
What is the slope of \(f(x)\text{?}\)
\(\displaystyle -1\)
\(\displaystyle 2\)
\(\displaystyle 1\)
\(\displaystyle -\dfrac{1}{2}\)
(b)
Applying
Definition 3.4.3 , what would the slope of a line parallel to
\(f(x)\) be?
\(\displaystyle -1\)
\(\displaystyle 2\)
\(\displaystyle 1\)
\(\displaystyle -\dfrac{1}{2}\)
(c) Find the equation of a line parallel to
\(f(x)\) that passes through the point
\((-4,2)\text{.}\)
Answer .
\(y-2=-\dfrac{1}{2}(x+4)\) or
Activity 3.4.5 .
Consider the graph of the two lines below.
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
(a)
What is the slope of line A?
\(\displaystyle -\dfrac{1}{2} \)
(b)
What is the slope of line B?
\(\displaystyle -\dfrac{1}{2} \)
(c)
What is the \(y\) -intercept of line A?
(d)
What is the \(y\) -intercept of line B?
\(\displaystyle -\dfrac{1}{2}\)
(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
\(-\dfrac{1}{2}\) and Line B could be written as
\(\dfrac{2}{1}\text{.}\)
(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.
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 \(f(x)\text{?}\)
\(\displaystyle -\dfrac{1}{3}\)
\(\displaystyle 3\)
\(\displaystyle 5\)
\(\displaystyle -\dfrac{1}{5}\)
(b)
Applying
Definition 3.4.7 , what would the slope of a line perpendicular to
\(f(x)\) be?
\(\displaystyle -\dfrac{1}{3}\)
\(\displaystyle 3\)
\(\displaystyle 5\)
\(\displaystyle -\dfrac{1}{5}\)
(c) Find an equation of the line perpendicular to
\(f(x)\) that passes through the point
\((3,6)\text{.}\)
Answer .
\(y-6=-\dfrac{1}{3}(x-3)\) or
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)=\dfrac{1}{6}x+8
\end{equation*}
Answer .
Neither. The slope of
\(f(x)\) is
\(6\) and the slope of
\(g(x)\) is
\(\dfrac{1}{6}\text{.}\) Although they are reciprocals of one another, they are not negative reciprocals.
(d)
\begin{equation*}
f(x)=\dfrac{4}{5}x+3
\end{equation*}
\begin{equation*}
g(x)=-\dfrac{5}{4}x-1
\end{equation*}
Answer .
Perpendicular. The slope of
\(f(x)\) is
\(\dfrac{4}{5}\) and the slope of
\(g(x)\) is
\(-\dfrac{5}{4}\) (and are negative reciprocals of one another).
Activity 3.4.10 .
Consider the linear equation,
\(f(x)=-\dfrac{2}{3}x-4\) 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.
Answer .
\(y-4=-\dfrac{2}{3}(x+6)\) or
(b) Find an equation of the line that is perpendicular to
\(f(x)\) and passes through the point A.
Answer .
\(y-4=\dfrac{3}{2}(x+6)\) or
Activity 3.4.11 .
Consider the line,
\(y=2\text{,}\) as shown in the graph below.
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
(a)
What is the slope of the line \(y=2\text{?}\)
undefined
\(\displaystyle 0\)
\(\displaystyle 1\)
\(\displaystyle -\dfrac{1}{2}\)
(b)
What is the slope of a line that is parallel to \(y=2\text{?}\)
undefined
\(\displaystyle 0\)
\(\displaystyle 1\)
\(\displaystyle -\dfrac{1}{2}\)
(c) Find an equation of the 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
\(\displaystyle 0\)
\(\displaystyle 1\)
\(\displaystyle -\dfrac{1}{2}\)
Answer .
A. You might need to help students see why the slope is undefined by showing that
\(-\dfrac{1}{0}\) is not defined.
(e) Find an equation of the 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 Exercises
