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Open Resources for Community College Algebra

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Section 3.8 Horizontal, Vertical, Parallel, and Perpendicular Lines

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Figure 3.8.1. Alternative Video Lessons
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The equations of horizontal and vertical lines distinguish them from other line equations enough to merit a special investigation. In addition, pairs of lines that are parallel or perpendicular to each other have interesting features and properties. This section examines the geometric features of these types of lines.
the graph of a horizontal line passing through the point (0,3)
Figure 3.8.2. Horizontal Line
the graph of a vertical line passing through the point (3,0)
Figure 3.8.3. Vertical Line
the graph of two parallel slanted lines
Figure 3.8.4. Two Parallel Lines
the graph of a slanted line and the line that crosses it at a 90 degree angle
Figure 3.8.5. Two Perpendicular Lines
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Subsection 3.8.1 Horizontal Lines and Vertical Lines

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We learned in Section 7 that all lines can be written in standard form. When either A or B equal 0, we end up with a horizontal or vertical line, as we will soon see. Let’s take the standard form line equation, Ax+Bx=C, and one at a time let A=0 and B=0 and simplify each equation.
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Ax+By=CAx+By=C0x+By=CAx+0y=CBy=CAx=Cy=CBx=CAy=kx=h
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At the end we just renamed the constant numbers CB and CA to k and h because of tradition. What is important, is that you view h and k (as well as A, B, and C) as constants: numbers that have some specific value and don’t change in the context of one problem.
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Think about one of these equations: y=k. 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. What does such a line look like?
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Example 3.8.6.

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Let’s plot the line with equation y=3. (Note that this is the same as 0x+1y=3.) To plot some points, it doesn’t matter what x-values we use. All that matters is that y is always 3.
a horizontal line at y=3
Figure 3.8.7. y=3
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A line like this is horizontal, parallel to the horizontal axis. All lines with an equation in the form
y=k
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(or, in standard form, 0x+By=C) are horizontal.
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Example 3.8.8.

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Let’s plot the line with equation x=5. Points on the line always have x=5, so if we wanted to make a table for plotting points, we are required to make all of the x-values be 5. From there, we have complete freedom to let y take any value. Here we take some random y-values.
x y
5 −6
5 −2
5 1
5 5
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These points are plotted in Figure 9.
a vertical line at x=5
Figure 3.8.9. x=5
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Note that the equation for this line is the same as x+0y=5. An alternative for making a table is to choose our y-values first and substitute them into the equation.
y x+0y=5⟹x=5 Ordered Pair
−6 x+0(−6)=5⟹x=5 (5,−6)
−2 x+0(−2)=5⟹x=5 (5,−2)
1 x+0(1)=5⟹x=5 (5,1)
5 x+0(5)=5⟹x=5 (5,5)
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In each case no matter what value y is, we find that the equation tells us that x=5.
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A line like this is vertical, parallel to the vertical axis. All lines with an equation in the form
x=h
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(or, in standard form, Ax+0y=C) are vertical.
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Example 3.8.10. Zero Slope.

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In Checkpoint 3.4.17, we learned that a horizontal line’s slope is 0, because the distance doesn’t change as time moves on. So the numerator in the slope formula is 0. Now, if we know a line’s slope and its y-intercept, we can use slope-intercept form to write its equation:
y=mx+by=0x+by=b
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This provides us with an alternative way to think about equations of horizontal lines. They have a certain y-intercept b, and they have slope 0.
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We use horizontal lines to model scenarios where there is no change in y-values, like when Kato stopped for 12 hours (he deserved a rest)!
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Checkpoint 3.8.11. Plotting Points.

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Suppose you need to plot the equation y=−4.25. Since the equation is in “y=” form, you decide to make a table of points. Fill out some points for this table.
y
Explanation.
We can use whatever values for x that we like, as long as they are all different. The equation tells us the y-value has to be −4.25 each time.
x
 y
−2 −4.25
−1 −4.25
0 −4.25
1 −4.25
2 −4.25
Now that we have a table, we could use its values to assist with plotting the line.
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Example 3.8.12. Slope of a Vertical Line.

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What is the slope of a vertical line? Figure 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 “run” of 1 unit.
a line with a slope of 1/1=1
a line with a slope of 2/1=2, which is steeper than the first line
a line with a slope of 4/1=4, which is steeper than the previous two lines
Figure 3.8.13.
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If we continued making the line steeper and steeper until it was vertical, the slope triangle would still have a “run” of 1, but the “rise” would become larger and larger with no upper limit. The slope would be m=very large1. Actually if the line is vertical, the “rise” segment we’ve drawn, will never intercept the line. So the slope of a vertical line can be thought of as “infinitely large.” We usually say that the slope of a vertical line is undefined. Some people say that a vertical line has no slope.
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Remark 3.8.15.

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Be careful not to mix up “no slope” (which means “its slope is undefined”) with “has slope 0.” If a line has slope 0, it does have a slope.
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If you are familiar with NBA basketball, some players wear number 0. That’s not the same thing as “not having a number”. This is similar to the situation with having slope 0 versus not having slope.
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Checkpoint 3.8.16. Plotting Points.

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Suppose you need to plot the equation x=3.14. You decide to try making a table of points. Fill out some points for this table.
y
Explanation.
Since the equation says x is always the number 3.14, we have to use this for the x value in all the points. This is different from how we would plot a “y=” equation, where we would use several different x-values. We can use whatever values for y that we like, as long as they are all different.
x
 y
3.14 −2
3.14 −1
3.14 0
3.14 1
3.14 2
The reason we made a table was to help with plotting the line.
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Example 3.8.17.

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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?
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Well, there is no getting around the fact that x=650. As for y, 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 18.
the vertical line x=650, starting at the point (650,0) with an arrow pointing upward
Figure 3.8.18. New TV: hours watched versus purchase price; negative y-values omitted since they make no sense in context
Horizontal Lines Vertical Lines
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A line is horizontal if and only if its equation can be written
y=k
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for some constant k.
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A line is vertical if and only if its equation can be written
x=h
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for some constant h.
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In standard form, any line with equation
0x+By=C
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is horizontal.
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In standard form, any line with equation
Ax+0y=C
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is vertical.
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If the line with equation y=k is horizontal, it has a y-intercept at (0,k) and has slope 0.
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If the line with equation x=h is vertical, it has an x-intercept at (h,0) and its slope is undefined. Some say it has no slope, and some say the slope is infinitely large.
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In the slope-intercept form, any line with equation
y=0x+b
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is horizontal.
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It’s impossible to write the equation of a vertical line in slope-intercept form, because vertical lines do not have a defined slope.
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Figure 3.8.19. Summary of Horizontal and Vertical Line Equations
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Subsection 3.8.2 Parallel Lines

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Example 3.8.20.

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Two trees were planted in the same year, and their growth over time is modeled by the two lines in Figure 21. Use linear equations to model each tree’s growth, and interpret their meanings in this context.
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We can see Tree 1’s equation is y=23x+2, and Tree 2’s equation is y=23x+5. 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, 23 feet per year, or 2 feet every 3 years.
This is a Cartesian grid with two parallel lines. Tree 1 has a slope triangle from (0,2) to (3,4) for a slope of 2/3. Tree 2 has a slope triangle from (0,5) to (3.7) for a slope of 2/3.
Figure 3.8.21. Two Trees’ Growth Chart
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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.
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Checkpoint 3.8.23.

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A line ℓ is parallel to the line with equation y=17.2x−340.9, but ℓ has y-intercept at (0,128.2). What is an equation for ℓ?
Explanation.
Parallel lines have the same slope, and the slope of y=17.2x−340.9 is 17.2. So ℓ has slope 17.2. And we have been given that ℓ’s y-intercept is at (0,128.2). So we can use slope-intercept form to write its equation as
y=17.2x+128.2.
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Checkpoint 3.8.24.

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A line κ is parallel to the line with equation y=−3.5x+17, but κ passes through the point (−12,23). What is an equation for κ?
Explanation.
Parallel lines have the same slope, and the slope of y=−3.5x+17 is −3.5. So κ has slope −3.5. And we know a point that κ passes through, so we can use point-slope form to write its equation as
y=−3.5(x+12)+23.
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Subsection 3.8.3 Perpendicular Lines

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The slopes of two perpendicular lines have a special relationship too.
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Figure 25 walks you through an explanation of this relationship.
the graph of two perpendicular lines, where one has a slope of m
(a) Two generic perpendicular lines, where one has slope m.
the previous graph with a slope triangle added; the rise is m and the run is 1
(b) Since the one slope is m, we can draw a slope triangle with “run” 1 and “rise” m.
the previous graph with the perpendicular line and its slope triangle added; the perpendicular line has a rise of -1 and a run of m which gives a slope of -1/m
(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.
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Figure 3.8.25. The relationship between slopes of perpendicular lines
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The second line in Figure 25 has slope
ΔyΔx=−1m=−1m.
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Not convinced? Here are three pairs of perpendicular lines where we can see if the pattern holds.
the graph of lines y=2x-2 and y=-1/2x+2 showing that they are perpendicular lines.
Figure 3.8.27. Graphing y=2x−2 and y=−12x+2. Note the relationship between their slopes: 2=−1−1/2
the graph of lines y=-3x+4 and y=1/3x-3 showing that they are perpendicular.
Figure 3.8.28. Graphing y=−3x+4 and y=13x−3. Note the relationship between their slopes: −3=−11/3
the graph of lines y=x and y=-x showing that they are perpendicular
Figure 3.8.29. Graphing y=x and y=−x. Note the relationship between their slopes: 1=−1−1
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Example 3.8.30.

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Line A passes through (−2,10) and (3,−10). Line B passes through (−4,−4) and (8,−1). Determine whether these two lines are parallel, perpendicular or neither.
Explanation.
We will use the slope formula to find both lines’ slopes:
Line A's slope=y2−y1x2−x1Line B's slope=y2−y1x2−x1=−10−103−(−2)=−1−(−4)8−(−4)=−205=312=−4=14
Their slopes are not the same, so those two lines are not parallel.
The product of their slopes is (−4)⋅14=−1, which means the two lines are perpendicular.
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Checkpoint 3.8.31.

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Line A and Line B are perpendicular. Line A’s equation is 2x+3y=12. Line B passes through the point (4,−3). Find an equation for Line B.
Explanation.
First, we will find Line A’s slope by rewriting its equation from standard form to slope-intercept form:
2x+3y=123y=12−2x3y=−2x+12y=−2x+123y=−23x+4
So Line A’s slope is −23. Since Line B is perpendicular to Line A, its slope is −1−23=32. It’s also given that Line B passes through (4,−3), so we can write Line B’s point-slope form equation:
y=m(x−x0)+y0y=32(x−4)−3
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Reading Questions 3.8.4 Reading Questions

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1.

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Explain the difference between a line that has no slope and a line that has slope 0.
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2.

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If you make a table of x- and y-values for either a horizontal line or a vertical line, what is going to happen in one of the two columns?
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3.

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If you know two points on one line, and you know two points on a second line, what could you do to determine whether or not the two lines are perpendicular?
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Exercises 3.8.5 Exercises

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Review and Warmup

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Exercise Group.
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1.
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Evaluate the following expressions. If the answer is undefined, you may answer with DNE (meaning “does not exist”).
  1. 06=
  2. 60=
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2.
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Evaluate the following expressions. If the answer is undefined, you may answer with DNE (meaning “does not exist”).
  1. 70=
  2. 07=
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3.
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A line passes through the points (3,5) and (−5,5). Find this line’s slope.
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4.
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A line passes through the points (1,7) and (−3,7). Find this line’s slope.
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5.
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A line passes through the points (10,−2) and (10,5). Find this line’s slope.
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6.
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A line passes through the points (−9,−3) and (−9,2). Find this line’s slope.
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7.
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Consider the equation:
y=1
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Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
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  • (3,1)
  • (1,2)
  • (0,10)
  • (−3,1)
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8.
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Consider the equation:
y=1
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Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
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  • (1,5)
  • (−5,1)
  • (0,7)
  • (4,1)
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9.
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Consider the equation:
x+1=0
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Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
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  • (0,−8)
  • (−1,3)
  • (−1,0)
  • (1,−1)
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10.
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Consider the equation:
x+1=0
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Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
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  • (1,−1)
  • (0,−10)
  • (−1,3)
  • (−1,0)
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Skills Practice

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Tables for Horizontal and Vertical Lines.
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11.
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Fill out this table for the equation y=7. The first row is an example.
x y Points
−3 7 (−3,7)
−2
−1
0
1
2
Values of x and y satisfying the equation y=7
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12.
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Fill out this table for the equation y=8. The first row is an example.
x y Points
−3 8 (−3,8)
−2
−1
0
1
2
Values of x and y satisfying the equation y=8
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13.
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Fill out this table for the equation x=−2. The first row is an example.
x y Points
−2 −3 (−2,−3)
−2
−1
0
1
2
Values of x and y satisfying the equation x=−2
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14.
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Fill out this table for the equation x=−1. The first row is an example.
x y Points
−1 −3 (−1,−3)
−2
−1
0
1
2
Values of x and y satisfying the equation x=−1
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Line Equations.
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15.
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A line’s graph is shown. Write an equation for the line.
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16.
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A line’s graph is shown. Write an equation for the line.
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17.
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A line’s graph is shown. Write an equation for the line.
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18.
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A line’s graph is shown. Write an equation for the line.
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Exercise Group.
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19.
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A line passes through the points (5,0) and (3,0). Find an equation for this line.
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An equation for this line is .
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20.
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A line passes through the points (1,3) and (−3,3). Find an equation for this line.
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An equation for this line is .
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21.
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A line passes through the points (5,−3) and (5,2). Find an equation for this line.
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An equation for this line is .
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22.
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A line passes through the points (7,4) and (7,−5). Find an equation for this line.
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An equation for this line is .
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Intercepts.
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23.
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Find the y-intercept and x-intercept of the line given by the equation. If a particular intercept does not exist, enter none into all the answer blanks for that row.
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x=10
x-value y-value Location
y-intercept
x-intercept
x-intercept and y-intercept of the line x=10
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24.
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Find the y-intercept and x-intercept of the line given by the equation. If a particular intercept does not exist, enter none into all the answer blanks for that row.
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x=−9
x-value y-value Location
y-intercept
x-intercept
x-intercept and y-intercept of the line x=−9
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25.
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Find the y-intercept and x-intercept of the line given by the equation. If a particular intercept does not exist, enter none into all the answer blanks for that row.
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y=−7
x-value y-value Location
y-intercept
x-intercept
x-intercept and y-intercept of the line y=−7
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26.
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Find the y-intercept and x-intercept of the line given by the equation. If a particular intercept does not exist, enter none into all the answer blanks for that row.
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y=−5
x-value y-value Location
y-intercept
x-intercept
x-intercept and y-intercept of the line y=−5
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Graphs of Horizontal and Vertical Lines.
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Parallel or Perpendicular?.
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31.
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Line m passes points (−4,−15) and (−2,−11).
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Line n passes points (3,1) and (5,5).
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These two lines are
  • parallel
  • perpendicular
  • neither parallel nor perpendicular
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.
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32.
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Line m passes points (−14,2) and (−21,8).
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Line n passes points (28,−21) and (−28,27).
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These two linea are
  • parallel
  • perpendicular
  • neither parallel nor perpendicular
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.
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33.
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Line m passes points (12,−4) and (8,−3).
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Line n passes points (3,6) and (−4,−22).
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These two lines are
  • parallel
  • perpendicular
  • neither parallel nor perpendicular
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.
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34.
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Line m passes points (−10,12) and (−15,18).
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Line n passes points (12,11) and (−18,−14).
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These two lines are
  • parallel
  • perpendicular
  • neither parallel nor perpendicular
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.
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35.
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Line m passes points (3,2) and (−3,8).
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Line n passes points (−2,2) and (4,14).
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These two lines are
  • parallel
  • perpendicular
  • neither parallel nor perpendicular
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.
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36.
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Line m passes points (−2,9) and (−6,9).
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Line n passes points (−5,−7) and (−2,−7).
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These two lines are
  • parallel
  • perpendicular
  • neither parallel nor perpendicular
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.
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37.
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Line m passes points (−9,−9) and (−9,3).
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Line n passes points (3,−7) and (3,5).
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These two lines are
  • parallel
  • perpendicular
  • neither parallel nor perpendicular
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.
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38.
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Line m passes points (−7,5) and (−7,−9).
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Line n passes points (−2,−7) and (−2,−5).
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These two lines are
  • parallel
  • perpendicular
  • neither parallel nor perpendicular
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.
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Parallel and Perpendicular Line Equations.
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39.
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A line passes through the point (−8,−3), and it’s parallel to the line y=−5. Find an equation for this line.
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40.
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A line passes through the point (8,−10), and it’s parallel to the line y=−2. Find an equation for this line.
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41.
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A line passes through the point (4,−4), and it’s parallel to the line x=0. Find an equation for this line.
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42.
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A line passes through the point (−4,5), and it’s parallel to the line x=2. Find an equation for this line.
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43.
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Line k has the equation y=2x+5.
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Line ℓ is parallel to line k, but passes through the point (5,3).
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Find an equation for line ℓ in both point-slope form and slope-intercept form.
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An equation for ℓ in point-slope form is: .
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An equation for ℓ in slope-intercept form is: .
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44.
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Line k has the equation y=4x+1.
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Line ℓ is parallel to line k, but passes through the point (−5,−15).
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Find an equation for line ℓ in both point-slope form and slope-intercept form.
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An equation for ℓ in point-slope form is: .
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An equation for ℓ in slope-intercept form is: .
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45.
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Line k has the equation y=−94x−1.
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Line ℓ is parallel to line k, but passes through the point (−3,474).
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Find an equation for line ℓ in both point-slope form and slope-intercept form.
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An equation for ℓ in point-slope form is: .
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An equation for ℓ in slope-intercept form is: .
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46.
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Line k has the equation y=−19x−7.
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Line ℓ is parallel to line k, but passes through the point (−3,−23).
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Find an equation for line ℓ in both point-slope form and slope-intercept form.
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An equation for ℓ in point-slope form is: .
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An equation for ℓ in slope-intercept form is: .
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47.
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Line k has the equation y=−x+2.
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Line ℓ is perpendicular to line k, and passes through the point (4,7).
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Find an equation for line ℓ.
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48.
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Line k has the equation y=−3x−3.
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Line ℓ is perpendicular to line k and passes through the point (−3,0).
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Find an equation for line ℓ in both point-slope form and slope-intercept form.
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An equation for ℓ in point-slope form is: .
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An equation for ℓ in slope-intercept form is: .
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49.
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Line k’s equation is y=59x−2.
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Line ℓ is perpendicular to line k and passes through the point (2,−435).
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Find an equation for line ℓ in both point-slope form and slope-intercept form.
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An equation for ℓ in point-slope form is: .
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An equation for ℓ in slope-intercept form is: .
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50.
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Line k has the equation x+6y=−30.
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Line ℓ is perpendicular to line k and passes through the point (−2,−15).
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Find an equation for line ℓ in both point-slope form and slope-intercept form.
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An equation for ℓ in point-slope form is: .
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An equation for ℓ in slope-intercept form is: .
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Challenge

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51.
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Prove that a triangle with vertices at the points (1,1), (−4,4), and (−3,0) is a right triangle.
You have attempted of activities on this page.
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