Parallel and Perpendicular Lines

Parallel and Perpendicular Lines are two sides of one coin. Perpendicular lines are intersecting lines, whereas parallel lines never intersect. Parallel lines in geometry are lines that never intersect and are always at the same distance from each other. On the other hand, perpendicular lines are lines that intersect each other at a right angle, forming a 90° angle. In this article, we will discuss these in detail, including examples and differences.

👁 Parallel-and-Perpendicular-Lines

What are Lines?

In geometry, a line is a straight path that extends indefinitely in both directions. It has no thickness or width, and it can be defined by two points on the line.

In other words, a collection of endlessly stretched points in opposing directions makes up a line. There are many types of lines in geometry, some of these types include:

• Horizontal Line
• Vertical Line
• Parallel Lines
• Intersecting Lines
• Perpendicular Lines
• Concurrent Lines
• Ray
• Line Segment

In this article, we will discuss parallel and perpendicular lines, including their differences.

What are Parallel Lines?

Parallel lines in geometry are two lines in the same plane that are at an equal distance from each other and never meet. They can be horizontal, vertical, or diagonal as well.

Parallel lines are denoted by "||" i.e., l || m means line l is parallel to line m. Some examples of parallel lines include railroad tracks, the edges of a bookshelf, the sides of a skyscraper, zebra crossings, or the edges of sidewalks.

Parallel Line Definition

Two straight lines are referred to as parallel lines if they are in the same plane and never cross paths.

Properties of Parallel Lines

Some of the common properties of parallel lines are:

• Parallel lines have the same slope.
• The distance between parallel lines remains constant.
• Parallel Lines never intersect each other at a common point.
• Parallel Lines lie in the same plane.
• Parallel lines are often denoted by a symbol (∥) placed between the lines, indicating that they are parallel.

Equation of Parallel Lines

An equation such as y = mx + c, where c is the y-intercept and "m" is the slope of the line, is utilized to represent a straight line. Since two parallel lines are always have the same steepness i.e., their slopes are always equal. Thus, equation of two parallel lines can be written as

y = mx + c₁ and y = mx + c₂

Where c₁ and c₂ are the y-intercept of lines.

Example: Find the equation of parallel line to y = 4x - 3 which passes through (2, 12).

Solution:

Equation of line passing through (x₁, y₁) with slope m is y − y₁​ = m(x − x₁​).

Given: Equation of line y = 4x - 3

Slope of this line = 4

As we know, for parallel lines slope remains the same thus, m₁ = m₂ = 4.

Therefore, equation of required parallel line is y − 5 = 2(x − 4)

⇒ y − 5 = 2x − 8

⇒ 8 − 5 = 2x - y

⇒ 3 = 2x - y

Thus, the required equation is 2x - y = 3.

What are Perpendicular Lines?

A perpendicular lines in mathematics, are the two lines that intersect each other at a 90° angle. At point of intersection perpendicular lines forms four right angles.

Perpendicular lines are represented with symbol "⊥" i.e., AB ⊥ CD means line segment AD is perpendicular to line segment CD. Some examples of perpendicular lines include the sides of a square, the axes of a coordinate plane (x-axis and y-axis), or the legs of a right triangle.

Perpendicular Line Definition

Two lines are referred to as perpendicular when they cross one another at a 90° angle.

Properties of Perpendicular Lines

Some of the common properties of perpendicular lines include:

• These lines always intersect at a 90° angle.
• Perpendicular lines intersect at a point, forming four right angles where they meet.
• The slopes of perpendicular lines are negative reciprocals of each other i.e., slope of perpendicular lines are m and -1/m.
• Product of the slopes of two perpendicular lines equals -1.
• Perpendicular lines are also referred to as orthogonal lines.
• Perpendicular lines exhibit symmetry about their point of intersection.

Equation of Perpendicular Lines

Equation of a Line with slope m and intercept c is y = mx + c. As we know, that the product of slope of perpendicular lines is -1. Thus, -1/m is the slope of line perpendicular to the given line. Using this we can easily find the equation of perpendicular lines.

Let's consider an example for the same.

Example: Two perpendicular lines intersects at (0, 1). If equation of one line is y = 3x + 2, then find the equation of other line.

Solution:

Given: Equation of line y = 3x + 2.

Slope = 3

Thus, slope of line perpendicular to this line = -1/3

Equation of line passing through (x₁, y₁) with slope m is y − y₁​ = m(x − x₁​).

⇒ y − 1 = (-1/3)(x − 0)

⇒ 3(y − 1 )= -(x − 0)

⇒ 3y − 3 = -x

⇒ x + 3y − 3 = 0

⇒ x + 3y = 3

Thus, equation of the required perpendicular line is x + 3y = 3.

Perpendicular and Parallel Lines in Real Life

There are many instances of parallel and perpendicular lines in our daily lives. Here are a few examples that are given below.

Parallel Lines in Real Life

• Railway Tracks: An amazing example of a parallel line is found in railway tracks. The tracks never cross; instead, they run parallel to one another.
• Shelves: To make the most of available space and effectively arrange space, bookshelves, and libraries typically include examples of shelves that are parallel to one another.

Perpendicular Lines in Real Life

• Buildings Structures: Perpendicular lines are frequently used in architecture to form buildings, with walls meeting floors and ceilings at right angles.
• Room Corners: The corners of rooms in buildings and the walls create a perpendicular angle.

Difference between Perpendicular and Parallel Lines

The key differences between parallel and perpendicular lines are:

Property	Parallel Lines	Perpendicular Lines
Definition	Lines that never intersect, maintaining the same distance apart.	Lines that intersect at a 90° angle, forming right angles.
Slope Relationship	Have the same slope.	Have slopes that are negative reciprocals of each other.
Intersection	Do not intersect, even when extended indefinitely.	Intersect at a 90° angle, forming four right angles.
Symbolic Representation	Denoted by (∥) between the lines.	Denoted by (⊥) between the lines.
Examples	Railroad tracks, sides of a rectangle.	Sides of a square, axes of a coordinate plane.
Slope Relationship Formula	m1​ = m2​, where m1 and m2​​ are slopes.	m1​ ⋅m2​ = −1, where m1 and m2​​ are slopes.

Conclusion

Parallel and Perpendicular Lines, can be essential in various fields like architecture, and engineering whether it can be designing or structuring railway tracks or joining room walls, etc. Parallel lines never meet and are equidistant from each other. Perpendicular lines intersect at a right angle, forming 90°. In this article, we have discussed these lines in detail including examples and differences as well.