The distance formula is a fundamental mathematical concept that helps us determine the length between two points in a coordinate system. It plays a vital role in several real-world applications, including physics, engineering, and map navigation. In this article, we will discuss how to calculate the distance formula using its derivation from the Pythagorean theorem and provide step-by-step examples.

Derivation of the Distance Formula:

The distance formula is derived from the Pythagorean theorem, which 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.”
Suppose we have two points A(x1, y1) and B(x2, y2) in a two-dimensional coordinate system. By constructing a right-angled triangle with sides parallel to x and y axes, we obtain:

1. The horizontal side’s length = |x2 – x1|

2. The vertical side’s length = |y2 – y1|

Using the Pythagorean theorem, we get:

Distance² = Horizontal side² + Vertical side²

So,

Distance² = (x2 – x1)² + (y2 – y1)²

Taking the square root of both sides yields:

Distance = √((x2 – x1)² + (y2 – y1)²)

Now that we have derived the distance formula let’s demonstrate how it can be applied with an example.

Example:

Given two points A(3, 4) and B(9, 12), find the distance between them using the distance formula.

Step 1: Identify coordinates for (x1, y1) and (x2,y2).

A(3,4): x1 = 3; y1 = 4

B(9,12): x2 = 9; y2 = 12

Step 2: Substitute these values into the distance formula.

Distance = √((9 – 3)² + (12 – 4)²)

Step 3: Solve the equation.

Distance = √((6)² + (8)²)

Distance = √(36 + 64)

Distance = √100

Distance = 10

Thus, the distance between points A and B is 10 units.

Conclusion:

The distance formula, derived from the Pythagorean theorem, is a valuable tool for calculating the length between two points in various applications. By understanding and applying this formula, you will be able to solve complex problems and better comprehend geometric principles.