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• Perpendicular Lines – Definition, Symbol, Properties, Examples

Perpendicular Lines – Definition, Symbol, Properties, Examples

Perpendicular is a fundamental concept in geometry that refers to the relationship between two lines that meet at right angles or 90 degrees. In mathematics, architecture, engineering, and other fields, perpendicular lines play an essential role. This article delves into the basics of perpendicular lines with some examples that make the concept more understandable.

What Does Perpendicular Mean?

In geometry, a perpendicular is a 90-degree angle formed by the intersection of two lines, planes, or vectors. Geometry, trigonometry, and calculus, to name a few, rely heavily on the idea of perpendicularity. 

Additionally, a perpendicular line in mathematics intersects another line at a right angle (90 degrees). This perpendicular definition implies that two lines are perpendicular to each other if they intersect at a right angle.

What Is Perpendicular Line?

The two lines are perpendicular when one line meets another at a right angle. These lines intersect the plane at the right angles, totaling 180 degrees since they make a 90 degrees angle each. 

There are a variety of applications for perpendicular lines because of their unique elements. The properties of perpendicular lines are beneficial in various problem-solving contexts because they allow one to quickly and seamlessly establish if two lines are perpendicular.

Perpendicular Symbol

The symbol “⊥” denotes a set of perpendicular lines. Stating that m and n are two lines crossing each other at right angles, one could say they are perpendicular to each other and therefore can be represented with the expression as m⊥n. It suffices to state that the point where perpendicular lines intersect is “the foot of the perpendicular.”

In geometry, the perpendicular sign is used if two lines are perpendicular to one another. The standard phrase for perpendicular lines looks like this: AB ⊥ CD. This attribute graphically represents that line AB is perpendicular to line BC. 

Properties of Perpendicular Lines

In mathematics, lines that meet at an angle are not necessarily perpendicular. However, if the lines at their intersection satisfy the conditions below, we say they are perpendicular. Perpendicular lines have the following two properties:

• First, perpendicular lines only meet at the right angle.
• Secondly, if two lines are perpendicular to the same line, they can never intersect since they are parallel. 

The Slope of Perpendicular Lines

Slopes of perpendicular lines are opposite and reciprocal to each other. Therefore, finding the slope of the line perpendicular to an equation will require you to find the opposite reciprocal.

The perpendicular line’s slope is always the inverse of the original line’s slope. You can calculate the slope of a line by transforming a linear equation into slope-intercept form. Also, you can calculate the slope of a perpendicular line by taking its negative reciprocal. 

A line with a positive slope implies that the perpendicular line has a negative slope. The slope of a perpendicular line is the “negative reciprocal” of the slope of the positive line. The two lines with slopes that are negative reciprocals of each other are perpendicular to each other.

Knowing how to find a perpendicular line is not as complicated as it seems. You can use the technique to find the perpendicular line: if two lines have slopes m1 and m2. They are perpendicular if m1× m2 = -1. The minus sign denotes the right angle formed by the two lines.

If you’re looking to deepen your understanding of the concept of Perpendicular Lines, we recommend checking out the selection of math worksheets offered by Brighterly. These worksheets have been designed to assist in learning and strengthening your comprehension of this important topic in geometry.

Difference between Perpendicular Lines and Parallel Lines

If the distance between any two non-vertical lines remains the same to infinity, those lines are parallel. In geometry, parallel lines are always at the same distance apart. 

On the other hand, a pair of lines are said to be perpendicular to each other if they form four equal angles by their intersection at the right angles. For example, if two lines intersect at 90 degrees, the lines are perpendicular.

When two lines have no chance of intersecting or touching themselves and have the same length, they are parallel. These lines have the same slope, which indicates that the distance between them is constant regardless of the parts. When two lines are parallel, we say that their slopes are equal. 

The slopes of lines that are perpendicular to one another are inverse reciprocals of each other as they meet at right angles.

How to Draw Perpendicular Lines?

You can draw a perpendicular line by using a protractor or a compass. Regardless of your tool, these perpendicular lines are easy to draw; you only need your pencil, paper, compass, or protractor. The tips below will guide you:

Drawing a Perpendicular Line Using a Protractor

Drawing perpendicular sides using a protractor is easy. Follow the steps to learn:

• Simply make a straight line.
• Set the protractor on top of the line so that its baseline is parallel to the line.
• Draw a little line on the line you measure at the 90-degree mark on the protractor.
• Adjust the baseline by moving the protractor to the end of the freshly created line.
• At the 90-degree mark, make another little parallel line.
• Connect the little lines you made with a straight line. This straight line runs at right angles to the former lines.

Drawing a Perpendicular Line Using a Compass

You can draw a perpendicular line with a compass applying the following tips:

• Get a piece of paper and make a point on it — draw a circle with your compass and center it on the spot.
• Without shifting the size of your compass, make another circle that shares the center of the first circle’s center and cut through it. 
• Connect the two spots where the circles meet by drawing lines. 

Perpendicular lines examples

Below are some examples and illustrations that explain the concept of perpendicular:

Example 1

A perpendicular line intersects at 90 degrees.

Example 2

From the above photo, writing the relationship between the line segments the arrow indicates could be complex. However, the answers are:

AB ⊥ AE

PQ || SR

PQ || NM

AB ⊥ AeD

Example 3

From the illustration, line AB ⊥ CD. Also, CD ⊥ EF. Two lines are parallel if they are perpendicular to the same line. Hence, line AB and EF are parallel.

FAQ

Do perpendicular lines touch each other?

Lines that meet at right angles are said to be perpendicular to one another. Their respective slopes are the inverses of one another. Perpendicular lines, in contrast to parallel ones, must intersect.

What are perpendicular angles?

In geometry, the perpendicular angle is a dominant concept. Perpendicular angle pertains to two lines intersecting at 90° (right angle). Two of the four angles formed by this intersection are very next to one another and are of similar size. Many geometric operations, like ‘building right angles and determining line slopes,’ rely on perpendicular angles. Also, problem-solving skills in geometry and trigonometry depend on a firm grasp of perpendicular angles.

What are some perpendicular shapes?

When lines or segments meet at a right angle (90 degrees), they are perpendicular. Squares, rectangles, triangles, and parallelograms are all instances of shapes that are perpendicular to one another. With their four parallel edges and right angles, squares and rectangles are among the most easily recognized perpendicular shapes.

Also, triangles are perpendicular if one of their angles is a right angle. And when the diagonals of a parallelogram meet at right angles, the four corners of the parallelogram are perpendicular. Multiple perpendicular sides and edges are characteristic of cubes and other three-dimensional structures.

Are all intersecting lines always perpendicular?

Lines that meet at an angle are not necessarily perpendicular to each other. The forming of lines on the right angles, or 90 degrees, is the intersection of perpendicular lines. 

An angle is created at the point where two lines that are not parallel to each other meet. In that case, the angle can be of any degree. However, it cannot be perpendicular if the lines do not meet at a right angle or 90 degrees.

A right angle is formed whenever two perpendicular lines meet. For instance, if you draw a horizontal line and a vertical line, they will meet at a right angle, giving you four angles measuring 90 degrees. As with a square, a rectangle’s edges run perpendicular to one another. 

To sum up, lines that cross each other don’t have to be perpendicular; in fact, perpendicular lines are a notable example of crossing lines that do so at right angles to each other.

What is a perpendicular lines formula?

Two lines are perpendicular if they satisfy the following formula:

The product of the slopes of lines 1 and 2 must be negative. When the lines are perpendicular to one another, the product of the slopes of the two lines, denoted by m1 and m2, is negative. The formula is written in mathematical form as m1 × m2 = -1. 

The formula requires us to know the slopes of the two lines to work. The formula for a line’s slope is: (m): (y2 – y1) / (x2 – x1), where two points on the line are identified as (x1, y1) and (x2, y2).

The perpendicular lines formula can tell us whether two lines are perpendicular to each other if we know their slopes. The lines are perpendicular if and only if the product of their slopes is negative on the number line.