The Distance Formula: Finding the Distance Between Two Points

In the realm of mathematics, understanding the distance between two points is crucial for various applications, from geometry and coordinate systems to real-world scenarios like navigation and mapping. The Distance Formula, a powerful tool derived from the Pythagorean Theorem, provides a precise way to calculate this distance on a Cartesian Plane.

The Pythagorean Theorem: A Foundation

Before diving into the Distance Formula, let's revisit the Pythagorean Theorem, which forms its basis. This fundamental 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 (legs). Mathematically, it's expressed as:

a² + b² = c²

where:

• a and b represent the lengths of the legs
• c represents the length of the hypotenuse

Deriving the Distance Formula

Imagine two points, A (x₁, y₁) and B (x₂, y₂), on a Cartesian Plane. We can connect these points with a straight line segment. This line segment forms the hypotenuse of a right-angled triangle, where the legs are parallel to the x-axis and y-axis.

The horizontal leg has a length of |x₂ - x₁| (absolute value to ensure a positive length), and the vertical leg has a length of |y₂ - y₁|.

Now, applying the Pythagorean Theorem to this triangle, we get:

Distance² = (x₂ - x₁)² + (y₂ - y₁)²

To find the distance itself, we take the square root of both sides:

Distance = √((x₂ - x₁)² + (y₂ - y₁)² )

This is the Distance Formula.

Applying the Distance Formula

Let's illustrate the Distance Formula with an example. Consider two points: A (2, 3) and B (5, 7).

Using the Distance Formula, we can calculate the distance between them:

Distance = √((5 - 2)² + (7 - 3)²)

Distance = √(3² + 4²)

Distance = √(9 + 16)

Distance = √25

Distance = 5

Therefore, the distance between points A and B is 5 units.

Conclusion

The Distance Formula is a powerful tool that allows us to calculate the distance between any two points on a Cartesian Plane. It's a fundamental concept in geometry and has wide applications in various fields. By understanding the derivation and application of the Distance Formula, you can confidently tackle problems involving distances in coordinate systems.