Perpendicular Lines

Perpendicular lines are all around us, forming the foundation for many everyday structures — the corners of books, the intersection of roads at a T-junction, or the edges of windows all showcase perpendicularity. In mathematics, they are crucial for creating right angles in geometric shapes and forming concepts in coordinate geometry. So let’s get deeper into perpendicular lines with this article.  

1.0Definition of Perpendicular Lines 

According to the Definition of Perpendicular Lines, two given lines are perpendicular to each other if they intersect to form a right angle (90°), like this: 

Therefore, this intersection leads to sharp edges and square-like corners forming the basis of various mathematical and real-life figures. Hence, perpendicular lines are fundamental in constructing precise angles in mathematics, as well as geometric structures in real life. 

In mathematics, the perpendicular lines symbol is used when solving geometric problems. It is used as if the line XY is perpendicular to AB. Mathematically, we write it as: 

XY ⊥ AB

2.0Characteristics of Perpendicular Lines

Any pair of perpendicular lines always possesses some characteristic properties, some of which include: 

• Any two lines, if perpendicular, always meet or intersect to form an ‘L’ or ‘T’ shape (inverted and non-inverted) figure. 
• The angle formed by the intersection of perpendicular lines always equals 90°. 
• In the structure of a pair of perpendicular lines, one line will always be horizontal while the other will be vertical in orientation. 

3.0Perpendicular Lines in Geometry: How to Draw These Lines? 

In geometry, the perpendicular lines are constructed using two simple yet core methods. These methods are: 

Using a Protractor

A protractor is a mathematical tool, marked with angles in the shape of a “D” or semicircle. The tool looks something like this: 

This is how this tool is used to draw perpendicular lines: 

• Mark a point on a page, say A. Place the midpoint of the baseline of the protractor on point A. 
• Draw a line through point A, using the protractor.
• Mark a point N at 90°, with the help of the protractor. 
• Once done, remove the protractor and connect the points A and N to form perpendicular lines. 

Using Compass and Ruler

Here is how a compass and ruler are used in the construction of a pair of perpendicular lines: 

• Draw a baseline of any length using a ruler and label it as AB. Also, mark a point N between A and B. 
• Place the needle of the compass point N and draw a semicircle of any radius on the line AB. 
• Now, without changing the radius, place the needle of the compass on any one edge of the semicircle on the line AB. Then, mark two arcs, one from this edge and another from the previous arc. 

• Draw an intersecting arc from these arcs and mark it as M.
• Join points N and M, and your pair of perpendicular lines using a ruler and compass are ready. 

4.0Perpendicular Lines in Coordinate Geometry 

Unlike in geometry, in coordinate geometry, perpendicular lines are analysed and identified by their slopes. A slope is the measure of the steepness of a line symbolised by “m”. That is, two lines are perpendicular if the product of their slopes equals -1. 

Mathematically, if there are given equations for a pair of perpendicular lines, say:

• y = m₁x + c₁
• y = m₂x + c₂ 

Then, the relation between the slope of these lines is given as: 

m₁ × m₂ = –1

Note:

• In case the points, say (x₁,y₁) and (x₂,y₂), from which the lines pass through are given in place of the equation, then the slope is measured as: $m=\frac{y_2-y_1}{x_2-x_1}$
• If one line is horizontal (slope = 0) and the other is vertical (slope is undefined), they are always perpendicular. It is because m is also equal to the tangent of horizontal and vertical lines, and tan90° is an undefined angle. 

5.0Perpendicular Lines vs. Parallel Lines 

Though both involve the basics of straight lines, perpendicular and parallel lines differ significantly in their orientation and interaction. Let’s understand the clear differences between these two prominent concepts of geometry: 

Perpendicular Lines	Parallel Lines
Perpendicular lines always intersect at a right angle.	Parallel lines never intersect with each other and remain equidistant.
The angle between the two lines is exactly 90°.	These lines do not form any angle; that is, the angle will be equal to 0°.
The number of intersections for any two perpendicular lines is exactly 1.	No intersection occurs, meaning the number of intersections for parallel lines is nil.
The slope for a pair of perpendicular lines is related as:   m1 × m2 = –1	The slope for a pair of parallel lines is related as:  m1 = m2
Some real-life perpendicular lines examples include a Wall meeting the floor, a road junction, the edges of paper, etc.	The examples of parallel lines in the practical world include Railway tracks, opposite edges of a ruler or notebook, etc.

6.0Perpendicular Lines Examples: Numericals

Problem 1: Check if the lines L₁: y = 2x + 3 and L₂: y = –½x – 4 are perpendicular.

Solution: Given that, L₁: y = 2x + 3 and L₂: y = –½x – 4

According to the question, 

m₁ = 2 and m₂ = –½ 

For the perpendicularity of two lines, it follows this condition: 

m₁ × m₂ = –1

2 × –½ = –1

–1 = –1

The right-hand side is equal to the left-hand side; hence, the given equations for lines are perpendicular to each other. 

Problem 2: Find the equation of a line perpendicular to y = 3x + 1 and passing through the point (2, 5).

Solution: Given that y = 3x + 1 is perpendicular to the lines with points (2,5). Therefore, it must follow the condition: 

m₁ × m₂ = –1

Here, m₁ = 3 

3 × m₂ = –1

m₂ = –⅓ 

Hence, for another equation: 

y – y₁ = m₂(x–x₂) 

Let (2,5) = (x₁,y₁)

y – 5 = –⅓(x–2)

3y – 15 = –x + 2

x + 3y = 2 + 15

x + 3y = 17

Hence, the other equation in standard form: 

y = –⅓x + 17/3

Problem 3: Check whether the lines formed by the following points are perpendicular or not. 

• Line 1: A(1, 2), B(4, 8)
• Line 2: C(2, 5), D(5, 3)

Solution: According to the question,  

Let the line 1: A(1, 2), B(4, 8) = (x₁,y₁) and (x₂,y₂) for line AB. 

Let the line 2: C(2, 5), D(5, 3) = (x₁,y₁) and (x₂,y₂) for line CD. 

We know, 

$m=\frac{y_2-y_1}{x_2-x_1}$

The slope for line AB (m₁) $=\frac{8-2}{4-1}=\frac{6}{3}=2$

The slope for line CD (m₂) $=\frac{3-5}{5-2}=\frac{-2}{3}$

Now, to check if the lines are perpendicular, they must follow: 

m₁ × m₂ = –1

$2\times \left(-\frac{2}{3}\right)=-\frac{4}{3}$

Here, $-\frac{4}{3}\ne -1$. Therefore, the given points do not belong to perpendicular lines.