Perpendicular Line Formula

Perpendicular Line

Perpendicular lines are the two distinct lines that intersect at each other at 90°. Have you noticed anything common between the joining corners of your walls, or the letter “L”? They are the straight lines known as perpendicular lines that meet each other at a specific angle – the right angle.

We say that a line is perpendicular to another line if the two lines meet at an angle of 90^°. Let us understand the concept of perpendicular lines in detail.

What is Perpendicular?

A perpendicular is a straight line that makes an angle of 90^° with another line. 90^° is also called a right angle and is marked by a little square between two perpendicular lines as shown in the figure. Here, the two lines intersect at a right angle, and hence, are said to be perpendicular to each other.

Now, look at the examples of lines that are not perpendicular. These lines are either not meeting at all, or intersecting at an angle that is not 90^°. Hence, they are not perpendicular.

Perpendicular lines, in math, are two lines that intersect each other and

Some shapes which have perpendicular lines are:

• Square
• Right-angled triangle
• Rectangle

Properties of Perpendicular Line

We have already seen how the perpendicular lines look like. If there is an “L” shape in a figure, the corresponding angle at the vertex is a right angle. Perpendicular lines always intersect each other, however, all intersecting lines are not always perpendicular to each other. The two main properties of perpendicular lines are:

• Perpendicular lines always meet or intersect each other.
• The angle between any two perpendicular lines is always equal to 90^°

How to Draw Perpendicular Lines?

To draw a perpendicular line, all we need is a scale (ruler), a compass or a protractor. We will discuss how to draw the perpendicular lines step by step using a compass and a protractor. So, we can draw perpendicular lines for a given line in two ways.

• Using a protractor
• Using a compass

Drawing a Perpendicular Line Using Protractor

A protractor, in math, is considered an important measuring instrument in the geometry box. This tool not only helps us measure an angle in degrees, but also helps in drawing perpendicular lines. To draw a perpendicular line at point P on the given line, follow the steps given below.

• Step 1: Place the baseline of the protractor along the line such that its center is at P.
• Step 2: Mark a point B at 90^° of the protractor.

Step 3: Remove the protractor and join P and B. So, BP is a line that is perpendicular to the given line.

Drawing a Perpendicular Line Using Compass

We can also draw a perpendicular line using a compass. To draw a perpendicular line at a point P on a line, follow the steps given below.

• Step 1: Adjust the compass to the desired radius.
• Step 2: Placing the pointer of the compass at P, construct a semi-circle that cuts the line at A and B.

Step 3: Without disturbing the radius of the compass, draw two arcs that cut the semi-circle at C and D by placing the pointer of the compass at A and B respectively.

Step 4: Keeping the same radius, draw two intersecting arcs that intersect at Q by placing the pointer of the compass at C and D.

Step 5: Join P and Q, which forms a perpendicular on the given line. Now, PQ is perpendicular to AB.

Perpendicular Lines and Parallel Lines

Two straight lines are said to be parallel if they are equidistant from each other and never meet, no matter how much they may be extended in either directions. Observe the lines shown below to see the difference between perpendicular lines and parallel lines. In the figure, AB is perpendicular to CD and PQ is parallel to RS.

The symbol used to show that the lines are parallel is ||, and we express it as: PQ || RS. You can find parallel lines all around you. For example, zebra crossings on the road or the opposite sides of your ruler, and many others too! The table shown below differentiates between parallel lines and perpendicular lines.

Difference Between Parallel and Perpendicular Lines

Parallel Lines	Perpendicular lines
Parallel lines are those lines that do not intersect anywhere and are always the same distance apart.	Lines that intersect each other forming a right angle are called perpendicular lines.
Example: the steps of a straight ladder; the opposite sides of a rectangle.	Example: the corner of two walls; the letter “L”
The symbol used to denote two parallel lines: ||	The symbol used to denote two perpendicular lines: ⊥

.A perpendicular line is a straight line through a point. It makes an angle of 90 degrees with a particular point through which the line passes. Coordinates and line equation is the prerequisite to finding out the perpendicular line.

Consider the equation of the line is ax + by + c = 0 and coordinates are (x₁, y₁), the slope should be − a/b. If one line is perpendicular to this line, the product of slopes should be -1. Let m₁ and m₂ be the slopes of two lines, and if they are perpendicular to each other, then their product will be -1.

Solved Example

Question: Check whether 2x + 3y + 5 = 0 and 3x – 2y + 1 = 0 are perpendicular or not.

Solution:

The given equations of lines are:
2x + 3y + 5 = 0 and 3x – 2y + 1 = 0

To check whether they are perpendicular to each other, find out the slopes of both lines. If the product of their slopes is -1, these lines are perpendicular to each other.

Since the product of slope is -1, the given lines are perpendicular to each other.

Equation of a Line Perpendicular to a Line

We will learn how to find the equation of a line perpendicular to a line.

Prove that the equation of a line perpendicular to a given line ax + by + c = 0 is bx – ay + λ = 0, where λ is a constant.

Let m1 be the slope of the given line ax + by + c = 0 and m2

be the slope of a line perpendicular to the given line.

Then,

Algorithm for directly writing the equation of a straight line perpendicular to a given straight line:

To write a straight line perpendicular to a given straight line we proceed as follows:

Step I: Interchange the coefficients of x and y in equation ax + by + c = 0.

Step II: Alter the sign between the terms in x and y of equation i.e., If the coefficient of x and y in the given equation are of the same signs make them of opposite signs and if the coefficient of x and y in the given equation are of the opposite signs make them of the same sign.

Step III: Replace the given constant of equation ax + by + c = 0 by an arbitrary constant.

For example, the equation of a line perpendicular to the line 7x + 2y + 5 = 0 is 2x – 7y + c = 0; again, the equation of a line, perpendicular to the line 9x – 3y = 1 is 3x + 9y + k = 0.

Note:

Assigning different values to k in bx – ay + k = 0 we shall get different straight lines each of which is perpendicular to the line ax + by + c = 0. Thus we can have a family of straight lines perpendicular to a given straight line.

Solved examples to find the equations of straight lines perpendicular to a given straight line

1. Find the equation of a straight line that passes through the point (-2, 3) and perpendicular to the straight line 2x + 4y + 7 = 0.

Solution:

The equation of a line perpendicular to 2x + 4y + 7 = 0 is

4x – 2y + k = 0 …………………… (i) Where k is an arbitrary constant.

According to the problem equation of the perpendicular line 4x – 2y + k = 0 passes through the point (-2, 3)

Then,

4 ∙ (-2) – 2 ∙ (3) + k = 0

⇒ -8 – 6 + k = 0

⇒ – 14 + k = 0

⇒ k = 14

Now putting the value of k = 14in (i) we get, 4x – 2y + 14 = 0

Therefore the required equation is 4x – 2y + 14 = 0.

2. Find the equation of the straight line which passes through the point of intersection of the straight lines x + y + 9 = 0 and 3x – 2y + 2 = 0 and is perpendicular to the line 4x + 5y + 1 = 0.

Solution:  

The given two equations are x + y + 9 = 0 …………………… (i) and 3x – 2y + 2 = 0 …………………… (ii)

Multiplying equation (i) by 2 and equation (ii) by 1 we get

                                                       2x + 2y + 18 = 0

                                                       3x  – 2y +   2 = 0

Adding the above two equations we get, 5x = – 20

⇒ x = – 4

Putting x = -4 in (i) we get, y = -5

Therefore, the co-ordinates of the point of intersection of the lines (i) and (ii) are (- 4, – 5).

Since the required straight line is perpendicular to the line 4x + 5y + 1 = 0, hence we assume the equation of the required line as

5x – 4y + λ = 0 …………………… (iii)

Where λ is an arbitrary constant.

By problem, the line (iii) passes through the point (- 4, – 5); hence we must have,

⇒ 5 ∙ (- 4) – 4 ∙ (- 5) + λ = 0  

⇒ -20 + 20 + λ = 0  

⇒ λ = 0.

Therefore, the equation of the required straight line is 5x – 4y = 0.

Examples on Perpendicular Lines

Solution

Since the two lines AB and CD intersect each other at right angles, there are 4 right angles at the intersecting point. Hence, Angle AOD = Angle DOB = Angle BOC = Angle AOC = 90^°.

Example 2:

In the following figure, AB is perpendicular to CD. If Angle BOC = 90^°, then find the value of x.

Solution

Since Angle BOC = 90^°, we can write: x + 63° = 90°. Therefore, x = 27°

Slope of Perpendicular Lines

Slope of perpendicular lines are such that the slope of one line is the negative reciprocal of the slope of another line. If the slopes of the two perpendicular lines are m₁, m₂, then we can represent the relationship between the slope of perpendicular lines with the formula m₁.m₂ = -1. The product of the slope of perpendicular lines is -1.

Let us learn more about the slope of perpendicular lines, their derivation, with the help of examples, FAQs.

Slope of a perpendicular line can be computed from the slope of a given line. The product of the slope of a given line and the slope of the perpendicular line is equal to -1. If the slope of a line is m₁ and the slope of the perpendicular line is m₂, then we have m₁.m₂ = -1.

The equations of two perpendicular lines are such that the coefficients of x and y are interchanged. For the equation of a line ax + by + c₁ = 0, the equation of the perpendicular line is bx – ay + c₂ = 0.

Formula of Slope of Perpendicular Lines: m₁.m₂ = -1

Further, the slope of each of the perpendicular lines can be found from the equations of the lines, or from the points on the line. The slope of a line having slope-intercept form of the equation of a line – y = mx + c is m, and the slope of a line having a general equation of a line ax + by + c = 0 is -a/b. Also, the slope of the line passing through any two points (x₁, y₁), and (x₂, y₂) is m = (y₂ – y₁)/(x₂ – x₁).

The slope of the perpendicular line can be derived from the formula of the angle between two lines. For two lines having slopes m₁ and m₂, the angle between the two lines is obtained using Tanθ.

Tanθ = (m₁ – m₂)/(1 + m₁.m₂)

The angle between two perpendicular lines is 90º, and we have Tan90º= ∞

Tan90º = (m₁ – m₂)/(1 + m₁.m₂)

∞ = (m₁ – m₂)/(1 + m₁.m₂)

n/₂0 = (m₁ – m₂)/(1 + m₁.m₂)

Here the denominator of the right hand side of the expression can be equalized to zero.

1 + m₁.m₂ = 0

m₁.m₂ = -1

m₂ = -1/m₁

_{Thus the slope of the perpendicular line is equal to the negative inverse of the slope of the given line.}

The slope of perpendicular lines can be calculated by knowing the slope of one of the two perpendicular lines. Here we take the equation of one of the perpendicular lines as the general form of the equation of a line. The general form of equation of a line is as follows.

ax + by + c = 0 

Let us convert this above equation into the slope-intercept form of the equation of a line. 

y = -ax/b – c/b

The slope of this line is m₁ = -a/b, and we have the slope of perpendicular lines formula as m₁.m₂ = -1. Thus we can find the slope of the other perpendicular lines as follows.

(-a/b).m₂ = -1

m₂= b/a

Thus the required equation of slope of the perpendicular line is b/a.

Let us understand this with the help of a simple numeric example. The given equation of a line is 5x + 3y + 7 = 0. Now let us try to find the slope of the perpendicular line. 

Comparing this equation 5x + 3y + 7 = 0, with ax + by + c = 0, we have a = 5, b = 3. The slopes of perpendicular lines is m₁ = -a/b = -5/3, and m₂ = b/a = 3/5.

Example 2: Find the equation of a line passing through (4, -3), and has the slope of perpendicular line as 2/3.

Solution:

The given point is $(x_1,y_1)$

= (4, -3), and the slope of the perpendicular line is $m_1=2/3$

.

We know that the product of the slopes of two perpendicular lines is m₁.m₂ = -1.

2/3 . m₂ = -1

m₂ = -1 × 3/2

m₂ = -3/2

The required equation of the line can be found using the formula of point slope form.

$(y-y_1)=m(x-x_1)$

y – (-3) = -3/2(x – 4)

2(y + 3) = -3(x – 4)

2y + 6 = -3x + 12

3x + 2y + 6 – 12 = 0

3x + 2y – 6 = 0

Thus the required equation of the line is 3x + 2y – 6 = 0.

How To Find Equation Of Line From Slope of Perpendicular Line?

, or y = mx + c.

How Do You Derive the Relationship Between the Slopes of Perpendicular Lines?

ACT Math : How to find the equation of a perpendicular line

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