Derivation of Distance Formula:
The distance formula is derived from the Pythagorean theorem in a coordinate plane. It is used to calculate the distance between two points, which can be represented as (x1, y1) and (x2, y2).

Pythagorean Theorem:
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, it is represented as:
c^2 = a^2 + b^2

Distance Formula Derivation:
To derive the distance formula, we consider two points in a coordinate plane. Let these points be A (x1, y1) and B (x2, y2).

Step 1:
Calculate the horizontal and vertical distances between the points:
Horizontal distance (Δx) = x2 - x1
Vertical distance (Δy) = y2 - y1

Step 2:
Apply the Pythagorean theorem to find the distance between the two points (AB):
AB^2 = (Δx)^2 + (Δy)^2
AB^2 = (x2 - x1)^2 + (y2 - y1)^2

Step 3:
Take the square root on both sides to get the distance formula:
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
This formula gives the distance between two points in a coordinate plane. It is a fundamental concept in geometry and is widely used in various fields such as mathematics, physics, and engineering.