what is the distance formula

2 min read 12-03-2025

The distance formula is a fundamental concept in coordinate geometry. It allows us to calculate the distance between any two points in a Cartesian coordinate system (a system where points are located using x and y coordinates). Understanding the distance formula is crucial for various mathematical applications, from simple geometry problems to more complex calculations in calculus and physics. This article will break down the formula, explain its derivation, and provide examples to solidify your understanding.

Understanding the Cartesian Coordinate System

Before diving into the distance formula, let's refresh our understanding of the Cartesian coordinate system. This system uses two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), to define the location of any point. Each point is represented by an ordered pair (x, y), where x represents the horizontal distance from the origin (0,0) and y represents the vertical distance.

Deriving the Distance Formula

The distance formula is derived from the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs).

Imagine two points, A(x₁, y₁) and B(x₂, y₂), plotted on a Cartesian plane. If we draw a right-angled triangle with AB as the hypotenuse, the legs will have lengths |x₂ - x₁| (horizontal distance) and |y₂ - y₁| (vertical distance). Applying the Pythagorean theorem, we get:

AB² = (x₂ - x₁)² + (y₂ - y₁)²

Taking the square root of both sides gives us the distance formula:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

How to Use the Distance Formula: Step-by-Step

Here's a step-by-step guide on how to use the distance formula:

Identify the coordinates: Determine the coordinates (x₁, y₁) and (x₂, y₂) of the two points you want to find the distance between.
Substitute the values: Substitute the coordinates into the distance formula: √[(x₂ - x₁)² + (y₂ - y₁)²]
Calculate the differences: Subtract the x-coordinates and the y-coordinates separately.
Square the differences: Square the results obtained in step 3.
Add the squares: Add the squared differences together.
Take the square root: Take the square root of the sum to find the distance.

Examples of Using the Distance Formula

Let's work through a few examples:

Example 1: Find the distance between points A(2, 3) and B(6, 7).

(x₁, y₁) = (2, 3)
(x₂, y₂) = (6, 7)

Distance = √[(6 - 2)² + (7 - 3)²] = √[4² + 4²] = √(16 + 16) = √32 ≈ 5.66

Example 2: Find the distance between points C(-1, 4) and D(3, -2).

(x₁, y₁) = (-1, 4)
(x₂, y₂) = (3, -2)

Distance = √[(3 - (-1))² + (-2 - 4)²] = √[4² + (-6)²] = √(16 + 36) = √52 ≈ 7.21

Applications of the Distance Formula

The distance formula has numerous applications in various fields, including:

Geometry: Calculating the length of line segments, perimeters of polygons, and diagonals.
Calculus: Finding the distance between points on curves.
Physics: Determining the distance between objects in space.
Computer graphics: Calculating distances between pixels on a screen.
Navigation: Determining distances between locations on a map.

Conclusion

The distance formula is a powerful tool for calculating distances between points in a Cartesian coordinate system. Its derivation from the Pythagorean theorem makes it conceptually straightforward. Mastering this formula is essential for success in various mathematical and scientific disciplines. Remember the formula: Distance = √[(x₂ - x₁)² + (y₂ - y₁)²] and practice using it to solve different problems. By understanding its application, you'll unlock a key to solving many geometric and other related problems.