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Perpendicular Lines

Perpendicular Lines

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, perpendicular lines, StudySmarterPerpendicular 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

Perpendicular lines, non-perpendicular lines, StudySmarterNon-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, perpendicular lines gradient, StudySmarterPerpendicular 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, perpendicular lines graph, StudySmarterPerpendicular 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 getfor 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 atare called perpendicular lines.
  • The slope of the perpendicular lines are negative reciprocal of each other.
  • The slopes of the perpendicular lines using the formula

Frequently Asked Questions about Perpendicular Lines

Two distinct straight lines which intersect at 90° are called perpendicular lines.

Perpendicular lines are found by checking the slopes of both lines.

Equations of perpendicular lines are found by taking the negative reciprocal of both the slopes.

y=3x+2, y=-1/3x+2 is one example of perpendicular lines.

The formula to calculate the perpendicular line is y=mx+b, such that (m1)(m2)=-1.

Final Perpendicular Lines Quiz

Question

What are perpendicular lines?

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Answer

Two distinct straight lines which intersect at 90° are called perpendicular lines.

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Question

What is the intersecting angle measure for perpendicular lines?

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Answer

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Question

Are all the four angles of perpendicular lines measure 90°?

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Answer

Yes

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Question

The slopes of perpendicular lines are the same.

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Answer

True

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Question

What is the relation between the slopes of perpendicular lines?

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Answer

The slopes of perpendicular lines are negative reciprocal of each other.

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Question

The y-intercept in perpendicular lines are always the same.

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Answer

True

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Question

If the slopes of two lines are negative, is it possible that both lines are perpendicular?

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Answer

Yes

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Question

The line passing through (5,2) and perpendicular to line x=0 is?

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Answer

x+2y=4

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