Open In App

Perpendicular Lines

Improve
Improve
Like Article
Like
Save
Share
Report

Perpendicular Lines in Mathematics are pairs of lines that always intersect each other at right angles, i.e. perpendicular lines are always intersecting lines that intersect at 90°. The perpendicular lines are readily seen by us, the corners of the walls, the corners of the desk, and others represent the parallel line. For perpendicular lines, we say that they intersect each other at right angles. The shortest distance between two lines is given using the perpendicular distance between them, i.e. the perpendicular line between two points gives the shortest distance between them.

In this article, we will learn about Perpendicular Lines, their properties, and others in detail.

What is Perpendicular?

Perpendicular is defined as a line that makes a right angle with another line. In other words, perpendicular line means the lines that make an angle of 90 degrees. The shortest distance between the point and the line is the perpendicular line between them. A perpendicular makes 90 degrees with the other line. The line AB and PQ as shown in the image below are perpendicular to each other because they intersect each other at 90 degrees.

The line AB and CD added in the image below shows two perpendicular lines.

Perpendicular lines

What are Perpendicular Lines?

Perpendicular Lines means the lines that intersect each other at an angle equal to 90 degrees i.e. if two lines meet at a right angle they are called Perpendicular lines. Take the figure added below here, the line l and line m intersect each other at point O and the angle made by them is 90 degrees.

Perpendicular Line l and m

Thus, we can say that l is a line perpendicular to m line or line m is Perpendicular to line l. We represent this condition as, l ⊥ m. Now any line parallel to line l is perpendicular to the line m. The shortest distance between the point and the line is always the perpendicular distance between them.

Note: Not all the intersecting lines are perpendicular lines but all the perpendicular lines are intersecting lines.

Perpendicular Sign

Perpendicular lines are represented using the symbol, ‘⊥‘. If lines l and m are perpendicular to each other, i.e. they intersect each other at 90 degrees then they are called perpendicular lines and they are represented as, l ⊥ m. The point of intersection is called the foot of the perpendicular.

Perpendicular Shapes

Prependicular shapes can be seen around us in our daily life. In perpendicular shapes are the shapes in which the at least one angle is 90°. Various shapes that have perpendicular lines (perpendicular shapes) are,

Properties of Perpendicular Lines

Any two intersecting lines intersecting at an angle of 90 degrees are called perpendicular lines. Perpendicular lines have different properties than the intersecting lines and the general properties of the intersecting lines are,

  • Perpendicular lines are the lines that always intersect each other at the right angle.
  • If two lines are perpendicular to the same line, then these two lines are always parallel to each other.

Slope of Perpendicular Lines

The slope of any line is the tan of the angle formed by the line with the positive x-axis and the slope in the case of the perpendicular lines has a particular relation between them.

Suppose we have two lines PQ and RS that are perpendicular to each other. Now, the slope of line PQ is say m1 and the slope of line RS is say m2, then the product of the slopes is equal to the -1. The statement for the same is,

Statement: Two lines are perpendicular to each other iff the product of their slope is -1.

This can be represented as,

m1.m2 = -1

Perpendicular Lines Formula

The two basic perpendicular line formulas are discussed below,

Statement 1: The product of the Slope of a Perpendicular line with the Slope of the Original line is always -1.

Proof:

Lets the original line makes an angle of θ with the X-axis. 

Then, the line perpendicular to the line will make an angle of θ + 90° or θ – 90° with the X-axis.

Now, the slope of the original line is equal to tan θ

The slope of the perpendicular line is equal to either tan (θ + 90o) or tan (θ – 90o)

tan (θ + 90o) =  tan (θ – 90o) = -cot θ

Thus, the slope of the perpendicular line is -cot θ

Now,

Product of Slopes = tan θ × (-cot θ) = -1

Hence Proved

Statement 2: If the equation of a line is ax + by + c = 0

Then the equation of a line perpendicular to the given line is,

– bx + ay + d = 0

where, c and d are any constant values

Proof:

Equation of line is ax + by + c = 0

Slope of the line is -a/b

Let slope of the perpendicular line is m

We know that product of slope of two perpendicular lines is -1

m × (-a / b) = – 1

m = b / a

Now, if the perpendicular line passes through a point (x1, y1), then the equation of the perpendicular line is,

(y – y1) / (x – x1) = b / a

y – y1 = (b / a) × (x – x1)

ay – ay1 = bx – bx1

– bx + ay + (bx1 – ay1) = 0 {let bx1 – ay1 = d}

Thus, required equation of the line is,

– bx + ay + d = 0

How to Draw Perpendicular Lines?

We can easily construct the pair of the perpendicular line, by using the Protractor and the Compass. 

Drawing Perpendicular Lines using Protractor

For drawing a pair of perpendicular lines follow the steps discussed below,

Step 1: First draw a horizontal line AB on the paper using a ruler.

Step 2: Mark any point P on line AB from which we have to draw the perpendicular line.

Step 3: Place the protector on the line and match the midpoint of the protector with point P on the line.

Step 4: Mark the 90-degree angle using the protector. 

Step 5: Join the line using any ruler with the 90 degrees angle, to get a pair of perpendicular line.

Drawing Perpendicular Line using Compass

Following are the steps to make perpendicular lines using a compass

Step 1: Draw a line on the paper using a ruler

Step 2: Take a point on the line and place the needle of the compass on it.

Step 3: Draw an arc (a semicircle) on one side of the line.

Step 4: Without changing the radius of the compass now place the needle on one end of the diameter of the semicircle.

Step 5: Trisect the semicircular arc by cutting it two times. The first cut marks 60° and the second cut marks 120°

Step 6: There is a difference of 60° between the first and second cut. Bisect this gap using the compass without changing its radius.

Step 7: Now join the point of bisection of 60 and 120 with the point assumed initially to draw the semicircular arc.

Step 8: The line so drawn is perpendicular to the initial line.

Perpendicular Lines Examples

Perpendicular lines are the lines that always meet each other at 90 degrees. We see various examples of parallel lines in real life, some of them are,

  • The corners of the rooms are perpendicular to each other.
  • The hands of the clock represent perpendicular lines at 3′ o clock.
  • The corners of the table and the desk represents the perpendicular lines.

Perpendicular and Parallel Lines

Perpendicular lines are the lines that make an angle of 90° with each other where as parallel lines are the lines that are parallel to each other that is they are equidistant from each other and never intersect each others.

Note: Parallel Lines meet at Infinity.

Slope of Parallel and Perpendicular Lines

Slope of parallel lines are equal whereas the product of slope of perpendicular lines is -1.

Equations of Parallel and Perpendicular Lines

If two lines are parallel then their equation of lines are,

  • ax + by + c = 0 and ax + by + d = 0

Whereas the equation of two perpendicular are,

  • ax + by + c = 0, and -bx + ax + d = 0

What are Parallel Lines?

Parallel lines in Geometry are defined as the lines that do not meet each other in the 2-D plane, i.e. they never intersect each other in the 2-D plane. The distance between the two parallel lines is always constant. The image added below shows two pairs of parallel lines.

Parallel lines

The lines a, b, and x, and y are parallel to each other.

Difference Between Parallel Lines and Perpendicular Lines

Parallel lines Vs Perpendicular lines are discussed in the table below.

Parallel Lines

Perpendicular Lines

The lines that do not intersect each other in the 2-D planes are called parallel lines. The distance between two parallel lines is always constant. The lines that intersect each other at 90 degrees in the 2-D planes are called perpendicular lines.
The  “||” symbol is used to represent the parallel line. The “⊥” symbol is used to represent perpendicular lines.
The parallel line never intersects each other. The perpendicular line intersects each other at 90 degrees.
Examples of Parallel Lines: Opposite sides of a square. Examples of Perpendicular Lines: Adjacent sides of a square.

Perpendicular Line Equation

The standard equation of a line is ax + by + c = 0 and the line perpendicular to the given line is given using,

-bx + ay + d = 0

where, d is the constant value and its value is found by using the other condition given.

Perpendicular Line Slope

Suppose we are given a line whose equation is of the form y = mx + c and its slope is m, then the slope of the line perpendicular to the given line is,

Slope of Perpendicular Line = -1/m

Now if the slope of two lines are m1 and m2 then the relation between these two slopes are, m1m2 = -1

Read More,

Perpendicular Lines Examples

Example 1: Are the lines 3x + 2y + 5 = 0 and 2x – 3y + 8 = 0 perpendicular?

Solution:

Slope of the line ax + by + c = 0 is -a/b

  • Slope of the line 3x + 2y + 5 = 0 is m1 = – 3 / 2.
  • Slope of the line 2x – 3y + 8 = 0 is m2 = -2 / (-3) = 2 / 3

We know that lines are perpendicular if their slopes have the condition.

m1×m2 = -1

Now from the above condition,

= (- 3 / 2) × (2 / 3) 

= -1

The product of the slopes is -1 and thus the lines are perpendicular.

Example 2: Find the line perpendicular to the line x + 2y + 5 = 0 and pass through the point (2, 5).

Solution:

We know that equation of a line perpendicular to the line ax + by + c = 0 is – bx + ay + d = 0.

Given equation of line is x + 2y + 5 = 0

Comparing the line x + 2y + 5 = 0 with ax + by + c = 0 we get,

  • a = 1
  • b = 2
  • c = 5

Thus, the equation of any line perpendicular to this line is – 2x + y + d = 0…(i)

Given, this line passes through (2, 5), 

Thus putting (2, 5) in this equation of the perpendicular line 

-2 × 2 + 5 + d = 0

d = -1

Substituting the value of d in eq(i), we get

-2x + y + (-1) = 0

Thus, the equation of the perpendicular line is -2x + y – 1 = 0

Example 3: Find the slope of the line perpendicular to the line 3x + 9y + 7 = 0.

Solution:

Given, 

Equation of the line is  3x + 9y + 7 = 0

Slope of this line = -a/b = – 3 / 9 = – 1 / 3

Let slope of ine perpendicular to above line is m

Now using the perpendicular line formulla

m × (- 1 / 3) = – 1

m = 3

Thus, the slope of the line perpendicular to the given line is 3.

Example 4: Find the angle of a line perpendicular to the line x + y + 3 = 0.

Solution:

Given line,

 x + y + 3 = 0

Slope of given line = -a/b = – 1 / 1 = – 1

Lets, slope of line perpendicular to the above line is m

From perpendicular line formula,

m × -1 = – 1

m = 1

Angle of line perpendicular to the given line is θ, then

m = tan θ

tan θ = 1

θ = tan-1(1) = 45°

Hence, the angle made by perpendicular line with X-axis is 45°.

Perpendicular Practice Problems

Q1. Find angle of a line perpendicular to the line 3x + 9y – 11 = 0.

Q2. If a line passes through the points (11, –4) and (–1, 8) and another line passes through the points (8, 3) and (–1, -3). Check wether these lines parallel or perpendicular.

Q3. Find the equation for the line that is perpendicular to 5x − 7y = 5 and passing through point (-1, 8).

Q4. Find the equation of line passing through (2, 3) and perpendicular to x-axis.

Perpendicular Lines – FAQs

1. What are the Perpendicular Lines?

If two intersecting lines intersect each other at right angles, i.e. at 90 degrees then these two lines are called perpendicular lines.

2. What are Parallel & Perpendicular Lines?

Parallel lines are the lines that do not meet each other in the 2-D plane. The distance between two parallel lines is always constant. Whereas if two lines meet each other at 90 degrees then these lines are called perpendicular lines.

3. Are Intersecting Lines Always Perpendicular?

No, not all intersecting lines are always perpendicular, they may or may not be perpendicular. The intersecting lines can meet at different angles.

4. What is condition for Slope of Perpendicular Lines?

Suppose the slope of two lines are m1 and m2 then the condition of the slopes of two perpendicular lines is, m1.m2 = -1

5. How many Perpendicular Lines can be Drawn to a line?

We can draw any number of perpendicular lines to a line, i.e. we can have infinite perpendicular lines to any line.

6. When are Two Lines Perpendicular?

Two lines are perpendicular if they intersect at 90°, i.e. perpendicular lines always intersect at the right angle.

7. What is a Perpendicular Triangle?

A triangle that has an angle equal to 90° is called the perpendicular triangle. It is also called the Right-Angled Triangle.

8. What are some Perpendicular Shapes?

Some Shapes that are called the Pependicular Shapes are the shapes that have at least one perpendicular in them. Various examples of the perpendicular shapes are, Square, Rectangle, Right-Angled Triangle

9. What are Perpendicular Angles?

The angles that are equal to 90° are called perpendicular angles. The other name of the perpendicular angles is Right Angles.

10. What is the Perpendicular Symbol?

The symbol or sign that represents perpendicular is, ⟂. We use this symbol to show if two lines are perpendicular. Such as, if it is written A⟂B, where A and B are two lines, then line A is perpendicular to line B and vice-versa.

11. How do you Identify which lines are Perpendicular?

If the angle between two lines is 90°. Then we can say that these two lines are perpendicular. If the slope of the two lines are given as, m1, m2 then we use the perpendicular line formula to find wether they are perpendicular or not. The perpendicular line formula is, m1.m2 = -1

12. How to Find the Slope of the Perpendicular Lines?

The slope of the perpendicular lines can be easily calculated using the slope formula. Suppose we are given a line then we first convert it in the standard form and then use the slope formula to find the slope. The slope formula is, m = -b/a, where a is the coefficient of x and b is the coefficient of y.



Last Updated : 23 Jan, 2024
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads