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Distance Formula in Mathematics

How to Calculate the Space Between Two Points

The Distance Formula is an essential tool in coordinate geometry. It allows us to calculate the precise distance between any two points on a Cartesian plane, even if the line connecting them is diagonal.

It is widely used in fields ranging from navigation (GPS systems) to video game development (calculating if a character has hit an object).

1. The Pythagorean Connection

The Distance Formula is not a new discovery; it is actually just the Pythagorean Theorem (a² + b² = c²) in disguise.

[Image of distance formula derived from Pythagorean theorem]

When you draw a line between two points on a graph, you can form a right-angled triangle where:

  • The horizontal distance is the change in x (run).
  • The vertical distance is the change in y (rise).
  • The hypotenuse is the straight-line distance you want to find.

2. The Formula

Given two points, Point A (x₁, y₁) and Point B (x₂, y₂), the distance d between them is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

In simple terms:

  1. Subtract the x-coordinates and square the result.
  2. Subtract the y-coordinates and square the result.
  3. Add these two numbers together.
  4. Take the square root of the total.

3. Step-by-Step Example

Let's find the distance between two points: A(2, 3) and B(6, 6).

Step 1: Identify coordinates

x₁ = 2, y₁ = 3
x₂ = 6, y₂ = 6

Step 2: Plug into the formula

d = √[(6 - 2)² + (6 - 3)²]

Step 3: Solve the parentheses

d = √[(4)² + (3)²]

Step 4: Square the numbers

d = √[16 + 9]

Step 5: Add and find the square root

d = √25
d = 5 units

4. Handling Negative Coordinates

Students often make mistakes when negative numbers are involved. Be careful with double negatives (subtracting a negative makes it positive).

Example: Distance between (-4, -1) and (1, 2).

d = √[(1 - (-4))² + (2 - (-1))²]
d = √[(1 + 4)² + (2 + 1)²]
d = √[(5)² + (3)²]
d = √[25 + 9]
d = √34 ≈ 5.83 units

5. Distance in 3D Space

Coordinate geometry isn't limited to flat paper. In 3D space (using an x, y, and z-axis), the formula simply expands to include the third dimension.

d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
[Image of distance formula in 3D space]

Conclusion

Mastering the Distance Formula bridges the gap between basic arithmetic and spatial geometry. It provides a reliable method to quantify "how far" something is, regardless of direction.