AB = \sqrt(1-0)^2 + (0-1)^2 + (0-0)^2 = \sqrt1 + 1 = \sqrt2 - Groen Casting
Understanding the Distance Formula: Deriving AB = √2 Using the Coordinate Plane
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.
Understanding the Context
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:
$$
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$
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Key Insights
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:
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$$
AB = \sqrt{(0 - 1)^2 + (1 - 0)^2}
$$
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:
