We have learned the concept of lines. When considering two lines, we get a particular form of lines. Like the type of lines, you get to see on the railway track crossing sign, intersecting edges of floor and wall, or the plus sign on the first aid kit. These types of lines are perpendicular lines.
Here we will take a look at perpendicular lines and understand the different concepts related to them.
Perpendicular lines meaning
Perpendicular lines are the lines that intersect each other at a certain angle. As the name says, a perpendicular is formed between the two lines. Perpendicular is a right angle. Hence, both lines intersect at 
Two distinct straight lines which intersect at 
are called perpendicular lines.
Perpendicular lines, StudySmarter Originals
Here straight lines AB and CD intersect at point O and that intersecting angle is 90 degrees. So both the lines AB and CD are perpendicular lines. So, we denote them with a sign 
.
Also, remember that all the four angles in perpendicular lines will be equal to 90 degrees. So, here
Non-perpendicular lines, StudySmarter Originals
Here above both types of lines are not perpendicular lines as the lines in the first figure intersect but not at 
And the lines in the second figure do not intersect at all. Therefore, one should note that not all intersecting lines are perpendicular lines.
Perpendicular lines Gradient
The gradient of perpendicular lines is the slope or the steepness of the lines. As both the perpendicular lines are, in fact, a line in itself, we can represent them in the form of a line equation 
This equation describes the value of y as it varies with x. And m is the slope of that line and b is the y-intercept.
The slope of the perpendicular lines is the negative reciprocal of each other. Suppose the slope of the first line is 
and the slope of the second line is 
The relation between both the perpendicular line slope is 
Hence, we can say that if the product of two slopes is
then both the lines are perpendicular to each other.
Perpendicular lines with gradient relation, StudySmarter Originals
Perpendicular line slope formula
We can find the slope of the perpendicular line with the help of the equation of a line and using the above-mentioned concept of slope. The general form of the equation of a line is represented as 
Then we can simplify this equation as:
We also know that the equation of a line in terms of slope can be written as,
Then comparing equations (1) and (2), we get that 
And from the above theory of slope we know that the product of slopes of perpendicular lines is 
Hence, from the given equation of line 
we can calculate the slopes of the perpendicular lines using the formula 
Suppose a line
is given. Find the slope for the line perpendicular to the given line.
Solution:
It is given that 
Now comparing it with the general equation of line 
we get 
Now we use the above formula to calculate the slope.
Now using the above-mentioned formula in the explanation, the slope of the perpendicular line is,
Hence, the slope for the line perpendicular to
is 
Perpendicular line equation
The perpendicular line equation can be derived from the equation of a line that is written in the form 
We studied, that the slopes of perpendicular lines are the negative reciprocal of each other. So, when writing equations of perpendicular lines, we need to ensure that the slopes of each line when multiplied together get 
If we want to find an equation for a line perpendicular to another line, we must take the negative reciprocal of that line’s slope. This value will be your value for m in the equation. The y-intercept can be anything, as a line can have infinitely many perpendicular lines that intersect with it. So, unless the question states otherwise, you can use any value for b.
Find the equation of a line passing through the point 
such that it is perpendicular to the line 
Solution:
First, we find the slope for the perpendicular line. Here, the equation for one line is given 
Comparing it with the general equation of line 
we get 
Now we take the negative reciprocal of the above slope to find the slope for the other line.
Now it is mentioned in the question that the other line passes through the point 
So the y-intercept for this line will be,
Now finally we substitute all the obtained values in the equation of the line.
Graphically, we can show the obtained perpendicular lines as below.
Perpendicular lines graph, StudySmarter Originals
Perpendicular lines example
Let us take a look at some examples of perpendicular lines.
Check if the given lines are perpendicular or not.
Line 1: 
, Line 2: 
Solution:
To check if the given lines are perpendicular, we will see if the product of the slopes is 
or not. So comparing the given equations of line 
with the general form 
Now we use the formula to calculate the slope for perpendicular lines. Therefore, for the line 1, we get
And for the line 2, the slope is
Here 
are negative reciprocal of each other. So, the product of both of them is
Hence, both the given lines are perpendicular to each other.
Find the equation of the line if it passes through the point 
and is perpendicular to another line 
Solution:
Here, the equation for the first line is given as
. And the second line passes through the point 
Now we simplify the given equation of line such that it looks similar to the form 
So, comparing this obtained equation with the general form of the line from above, we get
for the first line. Now, to find the slope of the second line, we know that it is a negative reciprocal of the slope of the first line.
And as the second line passes through the point
, the y-intercept is,
So putting all the obtained values in the general form of line, we get,
The equation of the line which is perpendicular to 
and passing through 
is 
Perpendicular Lines - Key takeaways
- Two distinct straight lines which intersect at
are called perpendicular lines. - The slope of the perpendicular lines are negative reciprocal of each other.
- The slopes of the perpendicular lines using the formula
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