AB = \sqrt{(1-0)^2 + (0-1)^2 + (0-0)^2} = \sqrt{1 + 1} = \sqrt{2}

February 17, 2026

$AB = \sqrt{(1-0)^2 + (0-1)^2 + (0-0)^2} = \sqrt{1 + 1} = \sqrt{2}$

Understanding the Distance Formula: Deriving AB = √2 Using the Coordinate Plane

In mathematics, especially in geometry and coordinate systems, calculating distances between points is a fundamental skill. One elegant example involves finding the distance between two points, A and B, using the 2D coordinate plane. This article explores how the distance formula works, with a clear step-by-step derivation of the distance formula AB = √[(1−0)² + (0−1)² + (0−0)²] = √2.

What is the Distance Between Two Points?

When two points are defined on a coordinate plane by their ordered pairs — for example, A = (1, 0) and B = (0, 1) — the distance formula allows us to compute how far apart they are. This formula comes directly from the Pythagorean theorem.

Given two points A = (x₁, y₁) and B = (x₂, y₂), the distance AB is calculated as:

$$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

In our case,

A = (1, 0) → $x_1 = 1$, $y_1 = 0$- B = (0, 1) → $x_2 = 0$, $y_2 = 1$

Applying the Formula to Points A(1, 0) and B(0, 1)

Substitute these coordinates into the distance formula:

$$AB = \sqrt{(0 - 1)^2 + (1 - 0)^2}$$

Simplify the differences inside the parentheses:

$$AB = \sqrt{(-1)^2 + (1)^2}$$

Now calculate the squares:

$$AB = \sqrt{1 + 1} = \sqrt{2}$$

Why This Formula Works: The Pythagorean Theorem in 2D

The distance formula is nothing more than an application of the Pythagorean theorem in a coordinate system. If we visualize the points A(1, 0) and B(0, 1), connecting them forms a right triangle with legs along the x-axis and y-axis.

The horizontal leg has length $ |1 - 0| = 1 $- The vertical leg has length $ |0 - 1| = 1 $

Then, the distance AB becomes the hypotenuse:

$$\ ext{Hypotenuse} = \sqrt{(\ ext{leg}_1)^2 + (\ ext{leg}_2)^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$$

This geometric interpretation confirms the algebraic result.

Practical Uses of the Distance Formula

Understanding this formula is essential for:

Solving geometry problems involving point-to-point distance- Programming and computer graphics, where spatial measurements matter- Physics, especially when calculating displacement or trajectory components- Any field relying on spatial relationships and coordinate-based analysis

Summary

The distance between two points in a coordinate plane is computed using:

$$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

For points A(1, 0) and B(0, 1), plugging in values gives:

$$AB = \sqrt{(0 - 1)^2 + (1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2}$$

This simple yet powerful result shows how basic geometry and algebra combine to solve real-world spatial problems. Whether in math class, engineering, or computer science, mastering the distance formula is indispensable.

Keywords: distance formula, coordinate geometry, distance between points formula, √2 calculation, Pythagorean theorem 2D, solving distance in plane, AB distance derivation, math tutorial distance formula

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