Section 3.8 Horizontal, Vertical, Parallel, and Perpendicular Lines
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.
Subsection 3.8.1 Horizontal Lines and Vertical Lines
We learned in Section 7 that all lines can be written in standard form. When either \(A\) or \(B\) equal \(0\text{,}\) we end up with a horizontal or vertical line, as we will soon see. Letâs take the standard form line equation, \(Ax+Bx=C\text{,}\) and one at a time let \(A=0\) and \(B=0\) 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 donât change in the context of one problem.
Think about one of these 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?
Example 3.8.6.
Letâs plot the line with equation \(y=3\text{.}\) (Note that this is the same as \(0x+1y=3\text{.}\)) To plot some points, it doesnât 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 equation in the form
\begin{equation*}
y=k
\end{equation*}
(or, in standard form, \(0x+By=C\)) are horizontal.
Example 3.8.8.
Letâs plot the line with equation \(x=5\text{.}\) Points on the line always have \(x=5\text{,}\) so if we wanted to make a table for plotting points, we are required to make all of the \(x\)-values be \(5\text{.}\) From there, we have complete freedom to let \(y\) take any value. Here we take some random \(y\)-values.
| \(x\) | \(y\) |
| \(5\) | \(-6\) |
| \(5\) | \(-2\) |
| \(5\) | \(1\) |
| \(5\) | \(5\) |
These points are plotted in Figure 9.
Note that the equation for this line is the same as \(x+0y=5\text{.}\) An alternative for making a table is to choose our \(y\)-values first and substitute them into the equation.
| \(y\) | \(x+0y=5\implies x=5\) | Ordered Pair |
| \(-6\) | \(x+0(-6)=5\implies x=5\) | \((5,-6)\) |
| \(-2\) | \(x+0(-2)=5\implies x=5\) | \((5,-2)\) |
| \(1\) | \(x+0(1)=5\implies x=5\) | \((5,1)\) |
| \(5\) | \(x+0(5)=5\implies x=5\) | \((5,5)\) |
In each case no matter what value \(y\) is, we find that the equation tells us that \(x=5\text{.}\)
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.
Example 3.8.10. Zero Slope.
In Checkpoint 3.4.17, we learned that a horizontal lineâs slope is \(0\text{,}\) because the distance doesnât change as time moves on. So the numerator in the slope formula is \(0\text{.}\) Now, if we know a lineâs slope and its \(y\)-intercept, we can use slope-intercept form 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 Kato stopped for \(12\) hours (he deserved a rest)!
Checkpoint 3.8.11. Plotting Points.
Suppose you need to plot the equation \(y=-4.25\text{.}\) Since the equation is in â\(y=\)â form, you decide to make a table of points. Fill out some points for this table.
\(x\) |
\(y\) |
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.
Example 3.8.12. Slope of a Vertical Line.
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 ârunâ of \(1\) unit.
If we continued making the line steeper and steeper until it was vertical, the slope triangle would still have a ârunâ of \(1\text{,}\) but the âriseâ would become larger and larger with no upper limit. The slope would be \(m=\frac{\text{very large}}{1}\text{.}\) Actually if the line is vertical, the âriseâ segment weâve drawn, will never intercept the line. So the slope of a vertical line can be thought of as âinfinitely large.â We usually say that the slope of a vertical line is undefined. Some people say that a vertical line has no slope.
Fact 3.8.14.
The slope of a vertical line is undefined.
Remark 3.8.15.
Be careful not to mix up âno slopeâ (which means âits slope is undefinedâ) with âhas slope \(0\text{.}\)â If a line has slope \(0\text{,}\) it does have a slope.
If you are familiar with NBA basketball, some players wear number \(0\text{.}\) Thatâs not the same thing as ânot having a numberâ. This is similar to the situation with having slope \(0\) versus not having slope.
Checkpoint 3.8.16. Plotting Points.
Suppose you need to plot the equation \(x=3.14\text{.}\) You decide to try making a table of points. Fill out some points for this table.
\(x\) |
\(y\) |
Explanation.
Since the equation says \(x\) is always the number \(3.14\text{,}\) we have to use this for the \(x\) value in all the points. This is different from how we would plot a â\(y=\)â 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.
Example 3.8.17.
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 18.
| Horizontal Lines | Vertical Lines |
A line is horizontal if and only if its equation can be written
\begin{equation*}
y=k
\end{equation*}
for some constant \(k\text{.}\)
|
A line is vertical if and only if its equation can be written
\begin{equation*}
x=h
\end{equation*}
for some constant \(h\text{.}\)
|
If the line with equation \(y=k\) is horizontal, it has a \(y\)-intercept at \((0,k)\) and has slope \(0\text{.}\)
|
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. |
In the slope-intercept form, any line with equation
\begin{equation*}
y=0x+b
\end{equation*}
is horizontal.
|
Itâs impossible to write the equation of a vertical line in slope-intercept form, because vertical lines do not have a defined slope. |
Subsection 3.8.2 Parallel Lines
Example 3.8.20.
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 treeâs growth, and interpret their meanings in this context.
We can see Tree 1âs equation is \(y=\frac{2}{3}x+2\text{,}\) and Tree 2âs equation is \(y=\frac{2}{3}x+5\text{.}\) 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 lineâs 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.
Fact 3.8.22.
Any two vertical lines are parallel to each other. For two non-vertical lines, they are parallel if and only if they have the same slope.
Checkpoint 3.8.23.
A line \(\ell\) is parallel to the line with equation \(y=17.2x-340.9\text{,}\) but \(\ell\) has \(y\)-intercept at \((0,128.2)\text{.}\) What is an equation for \(\ell\text{?}\)
Explanation.
Parallel lines have the same slope, and the slope of \(y=17.2x-340.9\) is \(17.2\text{.}\) So \(\ell\) has slope \(17.2\text{.}\) And we have been given that \(\ell\)âs \(y\)-intercept is at \((0,128.2)\text{.}\) So we can use slope-intercept form to write its equation as
\begin{equation*}
y=17.2x+128.2
\text{.}
\end{equation*}
Checkpoint 3.8.24.
A line \(\kappa\) is parallel to the line with equation \(y=-3.5x+17\text{,}\) but \(\kappa\) passes through the point \((-12,23)\text{.}\) What is an equation for \(\kappa\text{?}\)
Explanation.
Parallel lines have the same slope, and the slope of \(y=-3.5x+17\) is \(-3.5\text{.}\) So \(\kappa\) has slope \(-3.5\text{.}\) And we know a point that \(\kappa\) passes through, so we can use point-slope form to write its equation as
\begin{equation*}
y=-3.5(x+12)+23
\text{.}
\end{equation*}
Subsection 3.8.3 Perpendicular Lines
The slopes of two perpendicular lines have a special relationship too.
Figure 25 walks you through an explanation of this relationship.
The second line in Figure 25 has slope
\begin{equation*}
\frac{\Delta y}{\Delta x}=\frac{-1}{m}=-\frac{1}{m}\text{.}
\end{equation*}
Fact 3.8.26.
A vertical line and a horizontal line are perpendicular. For two lines that are neither vertical nor horizontal, they are perpendicular if and only if the slope of one is the negative reciprocal of the slope of the other. That is, if one has slope \(m\text{,}\) the other has slope \(-\frac{1}{m}\text{.}\)
Another way to say this is that the product of the slopes of two perpendicular lines is \(-1\) (assuming both of the lines have a slope in the first place). That is, if there are two perpendicular lines and we let \(m_1\) and \(m_2\) represent their slopes, then \(m_1\cdot m_2=-1\text{.}\)
Not convinced? Here are three pairs of perpendicular lines where we can see if the pattern holds.
Example 3.8.30.
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.
Explanation.
We will use the slope formula to find both linesâ 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.
Checkpoint 3.8.31.
Line \(A\) and Line \(B\) are perpendicular. Line \(A\)âs equation is \(2x+3y=12\text{.}\) Line \(B\) passes through the point \((4,-3)\text{.}\) Find an equation for Line \(B\text{.}\)
Explanation.
First, we will find Line \(A\)âs slope by rewriting its equation from standard form to slope-intercept form:
\begin{equation*}
\begin{aligned}
2x+3y\amp=12\\
3y\amp=12\subtractright{2x}\\
3y\amp=-2x+12\\
y\amp=\divideunder{-2x+12}{3}\\
y\amp=-\frac{2}{3}x+4
\end{aligned}
\end{equation*}
So Line \(A\)âs slope is \(-\frac{2}{3}\text{.}\) Since Line \(B\) is perpendicular to Line \(A\text{,}\) its slope is \(-\frac{1}{-\frac{2}{3}}=\frac{3}{2}\text{.}\) Itâs also given that Line \(B\) passes through \((4,-3)\text{,}\) so we can write Line \(B\)âs point-slope form equation:
\begin{equation*}
\begin{aligned}
y\amp=m(x-x_0)+y_0\\
y\amp=\frac{3}{2}(x-4)-3
\end{aligned}
\end{equation*}
Reading Questions 3.8.4 Reading Questions
1.
Explain the difference between a line that has no slope and a line that has slope \(0\text{.}\)
2.
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?
3.
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?
Exercises 3.8.5 Exercises
Review and Warmup
Exercise Group.
1.
Evaluate the following expressions. If the answer is undefined, you may answer with
DNE (meaning âdoes not existâ).- \(\displaystyle{ \frac{0}{6} = }\)
- \(\displaystyle{ \frac{6}{0} = }\)
2.
Evaluate the following expressions. If the answer is undefined, you may answer with
DNE (meaning âdoes not existâ).- \(\displaystyle{ \frac{7}{0} = }\)
- \(\displaystyle{ \frac{0}{7} = }\)
3.
A line passes through the points \((3,5)\) and \((-5,5)\text{.}\) Find this lineâs slope.
4.
A line passes through the points \((1,7)\) and \((-3,7)\text{.}\) Find this lineâs slope.
5.
A line passes through the points \((10,-2)\) and \((10,5)\text{.}\) Find this lineâs slope.
6.
A line passes through the points \((-9,-3)\) and \((-9,2)\text{.}\) Find this lineâs slope.
7.
Consider the equation:
\begin{equation*}
y=1
\end{equation*}
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (3,1)\)
- \(\displaystyle (1,2)\)
- \(\displaystyle (0,10)\)
- \(\displaystyle (-3,1)\)
8.
Consider the equation:
\begin{equation*}
y=1
\end{equation*}
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (1,5)\)
- \(\displaystyle (-5,1)\)
- \(\displaystyle (0,7)\)
- \(\displaystyle (4,1)\)
9.
Consider the equation:
\begin{equation*}
x+1=0
\end{equation*}
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (0,-8)\)
- \(\displaystyle (-1,3)\)
- \(\displaystyle (-1,0)\)
- \(\displaystyle (1,-1)\)
10.
Consider the equation:
\begin{equation*}
x+1=0
\end{equation*}
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (1,-1)\)
- \(\displaystyle (0,-10)\)
- \(\displaystyle (-1,3)\)
- \(\displaystyle (-1,0)\)
Skills Practice
Tables for Horizontal and Vertical Lines.
11.
Fill out this table for the equation \(y=7\text{.}\) The first row is an example.
| \(x\) | \(y\) | Points |
| \(-3\) | \(7\) | \(\left(-3,7\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=7\) |
12.
Fill out this table for the equation \(y=8\text{.}\) The first row is an example.
| \(x\) | \(y\) | Points |
| \(-3\) | \(8\) | \(\left(-3,8\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=8\) |
13.
Fill out this table for the equation \(x=-2\text{.}\) The first row is an example.
| \(x\) | \(y\) | Points |
| \(-2\) | \(-3\) | \(\left(-2,-3\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(x=-2\) |
14.
Fill out this table for the equation \(x=-1\text{.}\) The first row is an example.
| \(x\) | \(y\) | Points |
| \(-1\) | \(-3\) | \(\left(-1,-3\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(x=-1\) |
Line Equations.
15.
A lineâs graph is shown. Write an equation for the line.
16.
A lineâs graph is shown. Write an equation for the line.
17.
A lineâs graph is shown. Write an equation for the line.
18.
A lineâs graph is shown. Write an equation for the line.
Exercise Group.
19.
A line passes through the points \((5,0)\) and \((3,0)\text{.}\) Find an equation for this line.
An equation for this line is .
20.
A line passes through the points \((1,3)\) and \((-3,3)\text{.}\) Find an equation for this line.
An equation for this line is .
21.
A line passes through the points \((5,-3)\) and \((5,2)\text{.}\) Find an equation for this line.
An equation for this line is .
22.
A line passes through the points \((7,4)\) and \((7,-5)\text{.}\) Find an equation for this line.
An equation for this line is .
Intercepts.
23.
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.
\begin{equation*}
x = 10
\end{equation*}
| \(x\)-value | \(y\)-value | Location | |
| \(y\)-intercept | |||
| \(x\)-intercept |
| \(x\)-intercept and \(y\)-intercept of the line \(x=10\) |
24.
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.
\begin{equation*}
x = -9
\end{equation*}
| \(x\)-value | \(y\)-value | Location | |
| \(y\)-intercept | |||
| \(x\)-intercept |
| \(x\)-intercept and \(y\)-intercept of the line \(x=-9\) |
25.
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.
\begin{equation*}
y = -7
\end{equation*}
| \(x\)-value | \(y\)-value | Location | |
| \(y\)-intercept | |||
| \(x\)-intercept |
| \(x\)-intercept and \(y\)-intercept of the line \(y=-7\) |
26.
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.
\begin{equation*}
y = -5
\end{equation*}
| \(x\)-value | \(y\)-value | Location | |
| \(y\)-intercept | |||
| \(x\)-intercept |
| \(x\)-intercept and \(y\)-intercept of the line \(y=-5\) |
Graphs of Horizontal and Vertical Lines.
27.
Graph the line \(y=1\text{.}\)
28.
Graph the line \(y+5=0\text{.}\)
29.
Graph the line \(x=2\text{.}\)
30.
Graph the line \(x-3=0\text{.}\)
Parallel or Perpendicular?.
31.
Line \(m\) passes points \((-4,-15)\) and \((-2,-11)\text{.}\)
Line \(n\) passes points \((3,1)\) and \((5,5)\text{.}\)
These two lines are
- parallel
- perpendicular
- neither parallel nor perpendicular
.
32.
Line \(m\) passes points \((-14,2)\) and \((-21,8)\text{.}\)
Line \(n\) passes points \((28,-21)\) and \((-28,27)\text{.}\)
These two linea are
- parallel
- perpendicular
- neither parallel nor perpendicular
.
33.
Line \(m\) passes points \((12,-4)\) and \((8,-3)\text{.}\)
Line \(n\) passes points \((3,6)\) and \((-4,-22)\text{.}\)
These two lines are
- parallel
- perpendicular
- neither parallel nor perpendicular
.
34.
Line \(m\) passes points \((-10,12)\) and \((-15,18)\text{.}\)
Line \(n\) passes points \((12,11)\) and \((-18,-14)\text{.}\)
These two lines are
- parallel
- perpendicular
- neither parallel nor perpendicular
.
35.
Line \(m\) passes points \((3,2)\) and \((-3,8)\text{.}\)
Line \(n\) passes points \((-2,2)\) and \((4,14)\text{.}\)
These two lines are
- parallel
- perpendicular
- neither parallel nor perpendicular
.
36.
Line \(m\) passes points \((-2,9)\) and \((-6,9)\text{.}\)
Line \(n\) passes points \((-5,-7)\) and \((-2,-7)\text{.}\)
These two lines are
- parallel
- perpendicular
- neither parallel nor perpendicular
.
37.
Line \(m\) passes points \((-9,-9)\) and \((-9,3)\text{.}\)
Line \(n\) passes points \((3,-7)\) and \((3,5)\text{.}\)
These two lines are
- parallel
- perpendicular
- neither parallel nor perpendicular
.
38.
Line \(m\) passes points \((-7,5)\) and \((-7,-9)\text{.}\)
Line \(n\) passes points \((-2,-7)\) and \((-2,-5)\text{.}\)
These two lines are
- parallel
- perpendicular
- neither parallel nor perpendicular
.
Parallel and Perpendicular Line Equations.
39.
A line passes through the point \((-8,-3)\text{,}\) and itâs parallel to the line \(y=-5\text{.}\) Find an equation for this line.
40.
A line passes through the point \((8,-10)\text{,}\) and itâs parallel to the line \(y=-2\text{.}\) Find an equation for this line.
41.
A line passes through the point \((4,-4)\text{,}\) and itâs parallel to the line \(x=0\text{.}\) Find an equation for this line.
42.
A line passes through the point \((-4,5)\text{,}\) and itâs parallel to the line \(x=2\text{.}\) Find an equation for this line.
43.
Line \(k\) has the equation \(y={2x+5}\text{.}\)
Line \(\ell\) is parallel to line \(k\text{,}\) but passes through the point \((5,{3})\text{.}\)
Find an equation for line \(\ell\) in both point-slope form and slope-intercept form.
An equation for \(\ell\) in point-slope form is: .
An equation for \(\ell\) in slope-intercept form is: .
44.
Line \(k\) has the equation \(y={4x+1}\text{.}\)
Line \(\ell\) is parallel to line \(k\text{,}\) but passes through the point \((-5,{-15})\text{.}\)
Find an equation for line \(\ell\) in both point-slope form and slope-intercept form.
An equation for \(\ell\) in point-slope form is: .
An equation for \(\ell\) in slope-intercept form is: .
45.
Line \(k\) has the equation \(y={-{\frac{9}{4}}x-1}\text{.}\)
Line \(\ell\) is parallel to line \(k\text{,}\) but passes through the point \((-3,{{\frac{47}{4}}})\text{.}\)
Find an equation for line \(\ell\) in both point-slope form and slope-intercept form.
An equation for \(\ell\) in point-slope form is: .
An equation for \(\ell\) in slope-intercept form is: .
46.
Line \(k\) has the equation \(y={-{\frac{1}{9}}x-7}\text{.}\)
Line \(\ell\) is parallel to line \(k\text{,}\) but passes through the point \((-3,{-{\frac{2}{3}}})\text{.}\)
Find an equation for line \(\ell\) in both point-slope form and slope-intercept form.
An equation for \(\ell\) in point-slope form is: .
An equation for \(\ell\) in slope-intercept form is: .
47.
Line \(k\) has the equation \(y={-x+2}\text{.}\)
Line \(\ell\) is perpendicular to line \(k\text{,}\) and passes through the point \((4,{7})\text{.}\)
Find an equation for line \(\ell\text{.}\)
48.
Line \(k\) has the equation \(y={-3x-3}\text{.}\)
Line \(\ell\) is perpendicular to line \(k\) and passes through the point \((-3,{0})\text{.}\)
Find an equation for line \(\ell\) in both point-slope form and slope-intercept form.
An equation for \(\ell\) in point-slope form is: .
An equation for \(\ell\) in slope-intercept form is: .
49.
Line \(k\)âs equation is \(y={{\frac{5}{9}}x-2}\text{.}\)
Line \(\ell\) is perpendicular to line \(k\) and passes through the point \((2,{-{\frac{43}{5}}})\text{.}\)
Find an equation for line \(\ell\) in both point-slope form and slope-intercept form.
An equation for \(\ell\) in point-slope form is: .
An equation for \(\ell\) in slope-intercept form is: .
50.
Line \(k\) has the equation \({x+6y}=-30\text{.}\)
Line \(\ell\) is perpendicular to line \(k\) and passes through the point \((-2,{-15})\text{.}\)
Find an equation for line \(\ell\) in both point-slope form and slope-intercept form.
An equation for \(\ell\) in point-slope form is: .
An equation for \(\ell\) in slope-intercept form is: .
Challenge
51.
Prove that a triangle with vertices at the points \((1, 1)\text{,}\) \((-4, 4)\text{,}\) and \((-3, 0)\) is a right triangle.
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