Skip to main content
\(\require{cancel}\newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\point}[2]{\left(#1,#2\right)} \newcommand{\highlight}[1]{{\color{blue}{{#1}}}} \newcommand{\lowlight}[1]{{\color{lightgray}{#1}}} \newcommand{\attention}[1]{\mathord{\overset{\downarrow}{#1}}} \newcommand{\substitute}[1]{{\color{blue}{{#1}}}} \newcommand{\addright}[1]{{\color{blue}{{{}+#1}}}} \newcommand{\addleft}[1]{{\color{blue}{{#1+{}}}}} \newcommand{\subtractright}[1]{{\color{blue}{{{}-#1}}}} \newcommand{\multiplyright}[2][\cdot]{{\color{blue}{{{}#1#2}}}} \newcommand{\multiplyleft}[2][\cdot]{{\color{blue}{{#2#1{}}}}} \newcommand{\divideunder}[2]{\frac{#1}{{\color{blue}{{#2}}}}} \newcommand{\divideright}[1]{{\color{blue}{{{}\div#1}}}} \newcommand{\apple}{\text{🍎}} \newcommand{\banana}{\text{🍌}} \newcommand{\pear}{\text{🍐}} \newcommand{\cat}{\text{🐱}} \newcommand{\dog}{\text{🐶}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section4.8Horizontal, Vertical, Parallel, and Perpendicular Lines

Horizontal and vertical lines have some special features worth our attention. Also if a pair of lines are parallel or perpendicular to each other, we have some interesting things to say about them. This section looks at these geometric features that lines may have.

Figure4.8.1Horizontal Line
Figure4.8.2Vertical Line
Figure4.8.3Parallel Lines
Figure4.8.4Perpendicular Lines

Subsection4.8.1Horizontal Lines and Vertical Lines

Figure4.8.5Alternative Video Lessons

We learned in Section 4.7 that all lines can be written in standard form (4.7.1). When either \(A\) or \(B\) equal \(0\text{,}\) we end up with a horizontal or vertical line, as we will soon see. Let's take the standard form (4.7.1) line equation, let \(A=0\) and \(B=0\) one at a time and simplify each equation.

\begin{align*} Ax+By\amp=C\amp Ax+By\amp=C\\ \substitute{0}x+By\amp=C\amp Ax+\substitute{0}y\amp=C\\ By\amp=C\amp Ax\amp=C\\ y\amp=\divideunder{C}{B}\amp x\amp=\divideunder{C}{A}\\ y\amp=k\amp x\amp=h \end{align*}

At the end we just renamed the constant numbers \(\frac{C}{B}\) and \(\frac{C}{A}\) to \(k\) and \(h\) because of tradition. What is important, is that you view \(h\) and \(k\) (as well as \(A\text{,}\) \(B\text{,}\) and \(C\)) as constants: numbers that have some specific value and don't change in the context of one problem.

Think about just one of these last equations: \(y=k\text{.}\) It says that the \(y\)-value is the same no matter where you are on the line. If you wanted to plot points on this line, you are free to move far to the left or far to the right on the \(x\)-axis, but then you always move up (or down) to make the \(y\)-value equal \(k\text{.}\) What does such a line look like?

Example4.8.6

Let's plot the line with equation \(y=3\text{.}\) (Note that this is the same as \(0x+1y=3\text{.}\))

To plot some points, it doesn't matter what \(x\)-values we use. All that matters is that \(y\) is always \(3\text{.}\)

A line like this is horizontal, parallel to the horizontal axis. All lines with an eqaution in the form

\begin{equation*} y=k \end{equation*}

(or, in standard form, \(0x+By=C\)) are horizontal.

a horizontal line at y=3
Figure4.8.7\(y=3\)
Example4.8.8

Let's plot the line with equation \(x=5\text{.}\) (Note that this is the same as \(1x+0y=5\text{.}\))

The line has \(x=5\text{,}\) so to plot points, we are required to move over to \(x=5\text{.}\) From there, we have complete freedom to move however far we like up or down.

A line like this is vertical, parallel to the vertical axis. All lines with an equation in the form

\begin{equation*} x=h \end{equation*}

(or, in standard form, \(Ax+0y=C\)) are vertical.

a vertical line at x=5
Figure4.8.9\(x=5\)
Example4.8.10Zero Slope

In Exercise 4.4.22, we learned that a horizontal line's slope is \(0\text{,}\) because the distance doesn't change as time moves on. So the numerator in the slope formula (4.4.3) is \(0\text{.}\) Now, if we know a line's slope and its \(y\)-intercept, we can use slope-intercept form (4.5.1) to write its equation:

\begin{align*} y\amp=mx+b\\ y\amp=0x+b\\ y\amp=b \end{align*}

This provides us with an alternative way to think about equations of horizontal lines. They have a certain \(y\)-intercept \(b\text{,}\) and they have slope \(0\text{.}\)

We use horizontal lines to model scenarios where there is no change in \(y\)-values, like when Tammy stopped for \(12\) hours (she deserved a rest!)

Exercise4.8.11Plotting Points
Example4.8.12Undefined Slope

What is the slope of a vertical line? Figure 4.8.13 shows three lines passing through the origin, each steeper than the last. In each graph, you can see a slope triangle that uses a “rise” of \(4\) each time.

Figure4.8.13

If we continued making the line steeper and steeper until it was vertical, the slope triangle would still have a “rise” of \(4\text{,}\) but the “run” would become smaller and smaller, closer to \(0\text{.}\) And then the slope would be \(m=\frac{4}{\text{very small}}=\text{very large}\text{.}\) So the slope of a vertical line can be thought of as “infinitely large.”

If we actually try to compute the slope using the slope triangle when the run is \(0\text{,}\) we would have \(\frac{4}{0}\text{,}\) which is undefined. So we also say that the slope of a vertical line is undefined. Some people say that a vertical line has no slope.

Remark4.8.14

Be careful not to mix up “no slope” (which means “its slope is undefined”) with “has slope \(0\text{.}\)” If a line has slope \(0\text{,}\) it does have a slope.

Exercise4.8.15Plotting Points
Example4.8.16

Let \(x\) represent the price of a new \(60\)-inch television at Target on Black Friday (which was \(\$650\)), and let \(y\) be the number of hours you will watch something on this TV over its lifetime. What is the relationship between \(x\) and \(y\text{?}\)

Well, there is no getting around the fact that \(x=650\text{.}\) As for \(y\text{,}\) without any extra information about your viewing habits, it could theoretically be as low as \(0\) or it could be anything larger than that. If we graph this scenario, we have to graph the equation \(x=650\) which we now know to give a vertical line, and we get Figure 4.8.17.

Figure4.8.17New TV: hours watched versus purchase price; negative \(y\)-values omitted since they make no sense in context

Subsection4.8.2Parallel Lines

Example4.8.19

Two trees were planted in the same year, and their growth over time is modeled by the two lines in Figure 4.8.20. Use linear equations to model each tree's growth, and interpret their meanings in this context.

This is a Cartesian grid with two lines. A red line passes (0,2), (3,2), (3,4) ... The points (0,2) and (3,4) are plotted. A slope triangle is drawn from (0,2), passes (3,2) and ends at (3,4). Its base is labeled "3-0=3", and its height is labeled "4-2=2". A blue line passes (0,5), (3,5), (3,7) ... The points (0,5) and (3,7) are plotted. A slope triangle is drawn from (0,5), passes (3,5) and ends at (3,7). Its base is labeled "3-0=3", and its height is labeled "7-5=2".
Figure4.8.20Two Trees' Growth Chart

We can see Tree 1's equation is \(y=\frac{2}{3}x+2\text{,}\) and Tree 2's equation is \(y=\frac{2}{3}+5\text{.}\) Tree 1 was \(2\) feet tall when it was planted, and Tree 2 was \(5\) feet tall when it was planted. Both trees have been growing at the same rate, \(\frac{2}{3}\) feet per year, or \(2\) feet every \(3\) years.

An important observation right now is that those two lines are parallel. Why? For lines with positive slopes, the bigger a line's slope, the steeper the line is slanted. As a result, if two lines have the same slope, they are slanted at the same angle, thus they are parallel.

Exercise4.8.22
Exercise4.8.23
Figure4.8.24Alternative Video Lesson

Subsection4.8.3Perpendicular Lines

The slopes of two perpendiular lines have a special relationship too.

Figure4.8.25Alternative Video Lessons

Figure 4.8.26 walks you through an explanation of this realationship.

(a)Two generic perpendicular lines, where one has slope \(m\text{.}\)
(b)Since the one slope is \(m\text{,}\) we can draw a slope triangle with “run” \(1\) and “rise” \(m\text{.}\)
(c)A congruent slope triangle can be drawn for the perpendicular line. It's legs have the same lengths, but in different positions, and one is negative.
Figure4.8.26The relationship between slopes of perpendicular lines

The second line in Figure 4.8.26 has slope

\begin{equation*} \frac{\Delta y}{\Delta x}=\frac{-1}{m}=-\frac{1}{m}\text{.} \end{equation*}

Not convinced? Here are three pairs of perpendicular lines where we can see if the pattern holds.

This is a grid with the graph of line y=2x-2 and y=-1/2x+2.
This is a grid with the graph of line y=-3x+4 and y=1/3x-3.
This is a grid with the graph of line y=x and y=-x.
Figure4.8.28Graphing \(y=2x-2\) and \(y=-\frac{1}{2}x+2\text{.}\) Note the relationship between their slopes: \(2=-\frac{1}{-\sfrac{1}{2}}\)
Figure4.8.29Graphing \(y=-3x+4\) and \(y=\frac{1}{3}x-3\text{.}\) Note the relationship between their slopes: \(-3=-\frac{1}{\sfrac{1}{3}}\)
Figure4.8.30Graphing \(y=x\) and \(y=-x\text{.}\) Note the relationship between their slopes: \(1=-\frac{1}{-1}\)
Example4.8.31

Line \(A\) passes through \((-2,10)\) and \((3,-10)\text{.}\) Line \(B\) passes through \((-4,-4)\) and \((8,-1)\text{.}\) Determine whether these two lines are parallel, perpendicular or neither.

Solution

We will use the slope formula to find both lines' slopes:

\begin{align*} \text{Line }A\text{'s slope}\amp=\frac{y_2-y_1}{x_2-x_1}\amp\text{Line }B\text{'s slope}\amp=\frac{y_2-y_1}{x_2-x_1}\\ \amp=\frac{-10-10}{3-(-2)}\amp\amp=\frac{-1-(-4)}{8-(-4)}\\ \amp=\frac{-20}{5}\amp\amp=\frac{3}{12}\\ \amp=-4\amp\amp=\frac{1}{4} \end{align*}

Their slopes are not the same, so those two lines are not parallel.

The product of their slopes is \((-4)\cdot\frac{1}{4}=-1\text{,}\) which means the two lines are perpendicular.

Exercise4.8.32

Subsection4.8.4Exercises

Creating tables for horizontal and vertical lines.

1

Fill out this table for the equation \(y=2\text{.}\) The first row is an example.

\(x\) \(y\) Points
\(-3\) \(2\) \(\left(-3,2\right)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
2

Fill out this table for the equation \(y=4\text{.}\) The first row is an example.

\(x\) \(y\) Points
\(-3\) \(4\) \(\left(-3,4\right)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
3

Fill out this table for the equation \(x=-6\text{.}\) The first row is an example.

\(x\) \(y\) Points
\(-6\) \(-3\) \(\left(-6,-3\right)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
4

Fill out this table for the equation \(x=-5\text{.}\) The first row is an example.

\(x\) \(y\) Points
\(-5\) \(-3\) \(\left(-5,-3\right)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)

Determining whether a point is on a horizontal or vertical line.

5

Consider the equation \(y=1\text{.}\)

Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.

\( (-5,1)\)

\( (7,1)\)

\( (1,5)\)

\( (0,10)\)

6

Consider the equation \(y=1\text{.}\)

Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.

\( (-7,1)\)

\( (1,4)\)

\( (8,1)\)

\( (0,8)\)

7

Consider the equation \(x+1=0\text{.}\)

Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.

\( (-1,0)\)

\( (1,-1)\)

\( (-1,5)\)

\( (0,-10)\)

8

Consider the equation \(x+1=0\text{.}\)

Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.

\( (-1,0)\)

\( (1,-1)\)

\( (0,-6)\)

\( (-1,5)\)

Given two points, find the equations of the line connecting them.

9

A line passes through the points \((-2,-9)\) and \((1,-9)\text{.}\) Find an equation for this line.

An equation for this line is .

10

A line passes through the points \((-5,-6)\) and \((-3,-6)\text{.}\) Find an equation for this line.

An equation for this line is .

11

A line passes through the points \((-4,2)\) and \((-4,-1)\text{.}\) Find an equation for this line.

An equation for this line is .

12

A line passes through the points \((-2,-2)\) and \((-2,4)\text{.}\) Find an equation for this line.

An equation for this line is .

Given linear graphs, find the equations of horizontal and vertical lines.

13

A line’s graph is given.

This line’s equation is

14

A line’s graph is given.

This line’s equation is

15

A line’s graph is given.

This line’s equation is

16

A line’s graph is given.

This line’s equation is

Finding the intercepts of horizontal and vertical lines.

17

Find the \(y\)-intercept and \(x\)-intercept of the line given by the equation

\(\displaystyle{x = 10}\)

If a particular intercept does not exist, enter none into all the answer blanks for that row.

\(x\)-value \(y\)-value Location
\(y\)-intercept
\(x\)-intercept
18

Find the \(y\)-intercept and \(x\)-intercept of the line given by the equation

\(\displaystyle{x = -9}\)

If a particular intercept does not exist, enter none into all the answer blanks for that row.

\(x\)-value \(y\)-value Location
\(y\)-intercept
\(x\)-intercept
19

Find the \(y\)-intercept and \(x\)-intercept of the line given by the equation

\(\displaystyle{y = -7}\)

If a particular intercept does not exist, enter none into all the answer blanks for that row.

\(x\)-value \(y\)-value Location
\(y\)-intercept
\(x\)-intercept
20

Find the \(y\)-intercept and \(x\)-intercept of the line given by the equation

\(\displaystyle{y = -4}\)

If a particular intercept does not exist, enter none into all the answer blanks for that row.

\(x\)-value \(y\)-value Location
\(y\)-intercept
\(x\)-intercept

Given their equation, graph horizontal and vertical lines.

21

Graph the line \(y=1\text{.}\)

22

Graph the line \(y+5=0\text{.}\)

23

Graph the line \(x=2\text{.}\)

24

Graph the line \(x-3=0\text{.}\)

Finding equations of parallel lines.

25

A line passes through the point \((2,-7)\text{,}\) and it’s parallel to the line \(y=-2\text{.}\) Find an equation for this line.

An equation for this line is .

26

A line passes through the point \((-10,7)\text{,}\) and it’s parallel to the line \(y=0\text{.}\) Find an equation for this line.

An equation for this line is .

27

A line passes through the point \((0,-1)\text{,}\) and it’s parallel to the line \(x=3\text{.}\) Find an equation for this line.

An equation for this line is .

28

A line passes through the point \((-7,8)\text{,}\) and it’s parallel to the line \(x=5\text{.}\) Find an equation for this line.

An equation for this line is .

29

Line \(k\) has the equation \(y={4x-8}\text{.}\)

Line \(\ell\) is parallel to line \(k\text{,}\) but passes through the point \((-2,-5)\text{.}\)

Find an equation for line \(\ell\) in both slope-intercept form and point-slope form.

An equation for \(\ell\) in slope-intercept form is: .

An equation for \(\ell\) in point-slope form is: .

30

Line \(k\) has the equation \(y={5x+7}\text{.}\)

Line \(\ell\) is parallel to line \(k\text{,}\) but passes through the point \((2,9)\text{.}\)

Find an equation for line \(\ell\) in both slope-intercept form and point-slope form.

An equation for \(\ell\) in slope-intercept form is: .

An equation for \(\ell\) in point-slope form is: .

31

Line \(k\) has the equation \(y={-{\frac{1}{2}}x+1}\text{.}\)

Line \(\ell\) is parallel to line \(k\text{,}\) but passes through the point \((6,{-7})\text{.}\)

Find an equation for line \(\ell\) in both slope-intercept form and point-slope form.

An equation for \(\ell\) in slope-intercept form is: .

An equation for \(\ell\) in point-slope form is: .

32

Line \(k\) has the equation \(y={-{\frac{2}{9}}x-2}\text{.}\)

Line \(\ell\) is parallel to line \(k\text{,}\) but passes through the point \((9,{-1})\text{.}\)

Find an equation for line \(\ell\) in both slope-intercept form and point-slope form.

An equation for \(\ell\) in slope-intercept form is: .

An equation for \(\ell\) in point-slope form is: .

Determining whether Two Lines Are Parallel or Perpendicular

33

Line \(m\) passes points \((5,8)\) and \((-5,-12)\text{.}\)

Line \(n\) passes points \((-4,-1)\) and \((-2,3)\text{.}\)

Determine how the two lines are related.

These two lines are

parallel

perpendicular

neither parallel nor perpendicular

34

Line \(m\) passes points \((6,-11)\) and \((-9,9)\text{.}\)

Line \(n\) passes points \((6,-13)\) and \((-12,11)\text{.}\)

Determine how the two lines are related.

These two lines are

parallel

perpendicular

neither parallel nor perpendicular

35

Line \(m\) passes points \((8,2)\) and \((-4,5)\text{.}\)

Line \(n\) passes points \((-1,-12)\) and \((4,8)\text{.}\)

Determine how the two lines are related.

These two lines are

parallel

perpendicular

neither parallel nor perpendicular

36

Line \(m\) passes points \((-10,6)\) and \((10,-18)\text{.}\)

Line \(n\) passes points \((6,-5)\) and \((-6,-15)\text{.}\)

Determine how the two lines are related.

These two lines are

parallel

perpendicular

neither parallel nor perpendicular

37

Line \(m\) passes points \((1,-12)\) and \((4,-18)\text{.}\)

Line \(n\) passes points \((5,21)\) and \((-5,-19)\text{.}\)

Determine how the two lines are related.

These two lines are

parallel

perpendicular

neither parallel nor perpendicular

38

Line \(m\) passes points \((4,7)\) and \((-10,7)\text{.}\)

Line \(n\) passes points \((6,2)\) and \((-10,2)\text{.}\)

Determine how the two lines are related.

These two lines are

parallel

perpendicular

neither parallel nor perpendicular

39

Line \(m\) passes points \((10,-4)\) and \((10,-1)\text{.}\)

Line \(n\) passes points \((-3,6)\) and \((-3,0)\text{.}\)

Determine how the two lines are related.

These two lines are

parallel

perpendicular

neither parallel nor perpendicular

40

Line \(m\) passes points \((-9,10)\) and \((-9,8)\text{.}\)

Line \(n\) passes points \((5,-10)\) and \((5,6)\text{.}\)

Determine how the two lines are related.

These two lines are

parallel

perpendicular

neither parallel nor perpendicular

Find a Line's Equation Perpendicular to a Given Equation

41

Line \(k\) has the equation \(y={-x+3}\text{.}\)

Line \(\ell\) is perpendicular to line \(k\text{,}\) and passes through the point \((3,1)\text{.}\)

Find an equation for line \(\ell\) in both slope-intercept form and point-slope form.

An equation for \(\ell\) in slope-intercept form is: .

An equation for \(\ell\) in point-slope form is: .

42

Line \(k\) has the equation \(y={-3x+2}\text{.}\)

Line \(\ell\) is perpendicular to line \(k\) and passes through the point \((-9,{0})\text{.}\)

Find an equation for \(\ell\) in both slope-intercept form and point-slope forms.

An equation for \(\ell\) in slope-intercept form is: .

An equation for \(\ell\) in point-slope form is: .

43

Line \(k\)’s equation is \(y={-{\frac{5}{9}}x-3}\text{.}\)

Line \(\ell\) is perpendicular to line \(k\) and passes through the point \((-15,{-25})\text{.}\)

Find an equation for line \(\ell\) in both slope-intercept form and point-slope forms.

An equation for \(\ell\) in slope-intercept form is: .

An equation for \(\ell\) in point-slope form is: .

44

Line \(k\) has the equation \({x-6y}=-30\text{.}\)

Line \(\ell\) is perpendicular to line \(k\) and passes through the point \((-5,{32})\text{.}\)

Find line \(\ell\)’s equation in both slope-intercept form and point-slope form.

An equation for \(\ell\) in slope-intercept form is: .

An equation for \(\ell\) in point-slope form is: .