%5E2%2B(y2-y1)%5E2.jpeg "(x2-x1)^2+(y2-y1)^2")
Understanding the Distance Formula: (x2 - x1)^2 + (y2 - y1)^2
The expression (x2 - x1)^2 + (y2 - y1)^2 is the core of the distance formula, which calculates the distance between two points in a two-dimensional plane. Let's break down how it works.
The Pythagorean Theorem
The distance formula is derived from the Pythagorean theorem, which states that in a right 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.
Imagine you have two points, (x1, y1) and (x2, y2). You can connect these points to form a right triangle, where:
- The horizontal side has length (x2 - x1).
- The vertical side has length (y2 - y1).
- The hypotenuse is the line segment connecting the two points, representing the distance we want to find.
Applying the Pythagorean Theorem
By applying the Pythagorean theorem, we can relate the sides of this triangle:
Distance^2 = (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]
Example
Let's say we have points (2, 3) and (5, 7). We can calculate the distance between them using the distance formula:
- Distance = √[(5 - 2)^2 + (7 - 3)^2]
- Distance = √[3^2 + 4^2]
- Distance = √(9 + 16)
- Distance = √25
- Distance = 5
Therefore, the distance between points (2, 3) and (5, 7) is 5 units.
Key Takeaways
The distance formula is a powerful tool for finding the distance between any two points in a plane. It's based on the fundamental principle of the Pythagorean theorem, making it a cornerstone of geometry and its applications in various fields.