Introduction to the Distance Formula
The Distance Formula is an important mathematical tool that helps us compute the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is essential for solving problems in geometry, physics, computer science, and various realworld applications.
Consider two points, and , on the Cartesian plane. The distance between these points can be found using the Distance Formula:
In this article, we will explore how this formula is derived, how to apply it, and where it can be used in everyday life.
We will cover the following topics:
 Derivation of the Distance Formula
 StepbyStep Application
 RealWorld Examples
 Practical Exercises
Derivation of the Distance Formula
To derive the Distance Formula, let us consider a rightangled triangle where the two points, and , form the vertices of the hypotenuse. The legs of the triangle are parallel to the xaxis and yaxis.
Let us plot the points and . The lengths of the legs can be calculated as follows:
 Horizontal leg:
 Vertical leg:
Using the Pythagorean theorem, which states that for a rightangled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides, we have:
Here, is the distance between points and , while and are the lengths of the horizontal and vertical legs:
Taking the square root of both sides gives us the Distance Formula:
StepbyStep Application
Let us look at a stepbystep process to apply the Distance Formula:

 Identify the coordinates of the two points. For example, let be and be .
 Substitute the coordinates into the Distance Formula:

 Simplify the terms inside the square root:

 Calculate the square root:
Thus, the distance between points and is 5 units.
RealWorld Examples
Understanding the Distance Formula can be beneficial in a number of realworld situations. Here are a few examples:
1. GPS Navigation:
When using GPS technology to find the shortest path from one location to another, the Distance Formula helps calculate the direct distance between two geographic coordinates. This helps in determining the most efficient routes.
2. Sports:
In sports like golf, calculating the distance between the ball’s current location and the hole can help players decide on the right club and strike force.
3. Computer Graphics:
In video game design, the Distance Formula helps determine the distance between objects in 2D or 3D space, aiding in collision detection and rendering.
Practical Exercises
Practicing with a variety of problems will strengthen your understanding of the Distance Formula. Here are a few exercises to get you started:
 Find the distance between the points and .
 A triangle is formed by the points , , and . Find the lengths of all three sides.
 A drone flies from point to point . What distance does it cover?
These exercises will help you become proficient in applying the Distance Formula in various contexts.
Happy calculating!