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What are Perpendicular Lines?

Perpendicular lines intersect once and always at a right ($90°$) angle.

You can also identify two perpendicular lines by looking at their slopes. If two lines are perpendicular, their slopes are negative reciprocals.

If you write a slope as a fraction (like $\frac{8}{1}$ or $-\frac{4}{5}$, for example), the reciprocal is the flipped over form of the fraction.

Perpendicular Lines

If two lines are not parallel, one other interesting possibility is that the lines are perpendicular, which means the lines form a right angle ($90$ degree angle—a square corner) where they meet. In this case, the slopes when multiplied together will equal $-1$. Solving for one slope leads us to the definition:

Perpendicular Lines

Given two linear equations $f(x)=b+m_1x$ and $g(x)=b+m_2x$.

The lines will be perpendicular if $m_1\cdot m_2=-1$, and so $m_1=\frac{-1}{m_2}$

We often say the slope of a perpendicular line is the "negative reciprocal" of the other line's slope. Example 9

Find the slope of a line perpendicular to a line with: $$\begin{array}{ll} \text{a)}&\text{a slope of }2\\ \text{b)}&\text{a slope of }-4\\ \text{c)}&\text{a slope of }\frac23 \end{array}$$ If the original line had slope $2$, the perpendicular line's slope would be $m_{_2}=\frac{-1}{2}$.

If the original line had slope $-4$, the perpendicular line's slope would be $m_{_2}=\frac{-1}{-4}=\frac14$.

If the original line had slope $\frac23$, the perpendicular line's slope would be $m_{_2}=\frac{-1}{\frac23}=\frac{-3}2$.

Example 10

Find the equation of a line perpendicular to $f(x)=6+3x$ and passing through the point $(3,0)$.

The original line has a slope $m=3$. The perpendicular line will have a slope $m=\frac{-1}{3}$. Using this and the given point, we can find the equation for the line. $$g(x)=b-\frac13x$$ then at $(3,0)$, $$0=b-\frac13(3)\\ b=1$$ This line we're looking for is $g(x)=1-\frac13x$.

Example 12

A line passing through the points $(-2,6)$ and $(4,5)$. Find the equation of a perpendicular line that passes through the point $(4,5)$.

From the two given points on the reference line, we can calculate the slope of that line: $$m_{_1}=\frac{5-6}{4-(-2)}=\frac{-1}{6}$$

The perpendicular line will have a slope $$m_{_2}=\frac{-1}{-\frac16}=6$$

We can then solve for the vertical intercept that makes the line pass through the desired point: $$g(x)=b+6x$$ then at $(4,5)$, $$5=b+6(4)\\ b=-19$$ Giving the line $g(x)=-19+6x$.

(Video) Equations of Parallel and Perpendicular Lines

by ProfRobBob

Here's a good video by ProfRobBob going over some problems you might see involving parallel and perpendicular.

$\large{\text{Summary}}$

Two lines are said to be parallel ($\parallel$) if they have the same slope. Two lines are said to be perpendicular ($\perp$) if their slopes are negative reciprocals of each other.

(Video) Parallel and Perpendicular Lines

by Mahalo Math

Mahalo Math has a good video explaining equations of perpendicular lines and how to find them.

$\large{\text{Summary}}$

Perpendicular lines intersect at a $90$ degree angle. You can tell if two lines are perpendicular based on their slopes. When you multiply slopes together and get $-1$, the lines are perpendicular.

$$2\times-\frac{1}{2}=-1$$

Let's say you have a line $y=3x-\frac{1}{2}$. Take the slope of the existing line and find the negative reciprocal. The negative reciprocal of $3$ is $-\frac{1}{3}$.

(Videos) Perpendicular Line Graphs

by HEGARTYMATHS

Here's some videos by HEGARTYMATHS explaining perpendicular lines and doing some problems.

$\large{\text{Summary}}$

Two lines are said to be parallel if they have the same slope. That is, they never intersect and they're always the same distance from one another. Two lines are said to be perpendicular if their slopes are negative reciprocals of each other. That is they always intersect at right angles of one another.

The type of problems you will see involving perpendicular lines are usually in the form of, "Find the equation of a line perpendicular to $y=mx+b$ that contains the point $(x,y)$." The steps to solve these problems are:

1. Use the equation of the given line to find the slope.
2. Take the negative reciprocal of that slope.
3. Use the equation $y-y_0=m(x-x_0)$.
4. Put the negative reciprocal you found in Step 2 in the $m$ place.
5. Put the point that they give you in the $x_0$ and $y_0$ place.
6. Solve for $y$.

(Video) Parallel and Perpendicular Lines

by GVSUmath

This video by GVSUmath describes parallel and perpendicular lines.

Finding equations for parallel lines

Parallel lines have the same slope.

Perpendicular lines have opposite reciprocal slopes.

You have slope $-3$ and a point $(1,2)$. Let's find $b$. $$2=-3(1)+b$$ So, $b=5$ and we can write the equation as $$y=-3x+5$$

She walks through some more examples in the video. Check them out if you want more!

(Video) Perpendicular Equations of Lines

by mathman1024

Here's a video by mathman1024 showing you how to find an equation for a line given a perpendicular line and a point.

$\large{\text{Summary}}$

Two lines are said to be perpendicular if they meet or cross at a $90$ degree angle. You can tell if two lines are perpendicular if they have negative reciprocal slopes.

The problem he goes over is, "Find the equation of a line perpendicular to $y=5x-4$ and passing through the point $(3,4)$." Since we know our line has to be perpendicular to the line given just take the negative reciprocal of the slope. Since the slope is $5$ the negative reciprocal would be $-\frac15$. Once you have the slope you can use either the slope intercept formula or the point slope formula to find the equation of the line. $$\begin{align} y-y_0&=m(x-x_0)\\ y-4&=-\frac15(x-3)\\ y-4&=-\frac15x+\frac35\\ y-4\color{red}{+4}&=-\frac15x+\frac35\color{red}{+4}\\ y&=-\frac15x+\frac{23}{5} \end{align}$$

Here's a video where he goes over a problem where the line is given in standard form.

The problem he does is, "Find the equation of a line perpendicular to $8x-9y=3$ and passing through the line $(-5,7)$." Solving this is super easy. Take the $Ax+By$ part of the line that was given and re-write it in the form $Bx+Ay$ and change the sign of one of them. Then plug in the point given to find the new $C$ value. $$\begin{align} -9x+8y&=\square\\ \color{red}{+}9x+8y&=\square\\ 9(-5)+8(7)&=\square\\ -45+56&=11 \end{align}$$ So the equation for our line is: $$9x+8y=11$$

(Video) Writing Equations of Perpendicular Lines

by Friendly Math 101

Here's a video by Friendly Math 101 showing you how to find perpendicular lines.

$\large{\text{Summary}}$

Perpendicular lines are lines that intersect at $90^{\circ}$ angles. This means that they will have opposite reciprocal slopes.

Let's look at an example. Find a line perpendicular to $y=\frac23x+1$ through the point $(4,3)$. We know the slope will be the opposite reciprocal of the one given. So if the slope is $m=\frac23$ then the opposite reciprocal will be $m=-\frac32$. With our new slope and the point given, we can use point-slope form to find the equation of the line. $$\begin{align} y-y_1&=m(x-x_1)\\ y-\color{red}{(3)}&=\color{red}{(-\frac32)}(x-\color{red}{(4)})\\ y-3&=-\frac32(x-4)\\ y-3&=-\frac32x+6\\ \color{red}{+3}&=\ \ \ \ \ \ \ \ \ \ \color{red}{+3}\\ \hline y&=-\frac32x+9 \end{align}$$

Two perpendicular lines have opposite, reciprocal slopes. What does this mean? Let's break it up into parts.

The word opposite can be used to mean many different things. In math world, opposite of a number refers to the number having the opposite sign (i.e. positive or negative). For example, the opposite of $2$ is $-2$, and the opposite of $-1.5$ is $1.5$. Just switch the sign on any number, and you've got its opposite.

The reciprocal of a number is the number with the numerator and denominator flipped. Mathematically, it's what you get when you divide $1$ by the number. With fractions, this is pretty straightforward. The fraction $\frac{2}{3}$ has reciprocal $\frac{3}{2}$ and the fraction $\frac{1}{4}$ has reciprocal $\frac{4}{1}$ (a.k.a the number $4$). When we want to find the reciprocal of a whole number, we have to remember to put it over $1$ first. For example, $5$ is equal to $\frac{5}{1}$, so the reciprocal is $\frac{1}{5}$. Similarly, $-3$ is the same as $\frac{-3}{1}$, so the reciprocal is $\frac{-1}{3}$.

The opposite reciprocal is simply a combination of the two!

In order to find the slope of a line perpendicular to another line, lets call it L1, there are two things we need to do.

1) Find the Reciprocal of the Slope of L1

2) Negate the Reciprocal of L1

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Reciprocal means that the top and bottom of the fraction will be flipped. For example, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$

Negate means put a negative sign in front of the fraction.

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Note: If you are having trouble remembering that the reciprocal means to flip the top and bottom of a fraction, try creating a mnemonic for yourself! They are a little silly, but can really help some people (myself included).

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My Mnemonic Device: For me, I read reciprocal and see the word PRO (reci-PRO-cal) which reminds me of how I always wished I could be a pro water bottle flipper (not really but it helps me remember to flip my reciprocals)!

Find the equation of the line perpendicular to L1:

L1: y = 2x + 3

L2 = ?

Step 1: Find the Reciprocal of the Slope of L1 Slope of L1 = 2 (2 is the same as $\frac{2}{1}$) Reciprocal of L1's Slope= $\frac{1}{2}$

Step 2: Negate the Reciprocal of the Slope of L1 $\frac{-1}{2}$

(just put a negative sign in front of the reciprocal)

So that is our slope for L2 is $\frac{-1}{2}$ !

The slope of L2 (the perpendicular line) is the only thing we need in order to write out the rest of its equation because everything else is the same as L1.

So the equation for L2, a line perpendicular to L1 is: y = $\frac{-1}{2}$x + 3