The Distance Formula: Understanding (x2  x1)² + (y2  y1)²
The expression (x2  x1)² + (y2  y1)² is a fundamental formula in mathematics, representing the distance between two points in a twodimensional plane. This formula is known as the distance formula and is derived from the Pythagorean theorem.
Understanding the Formula
Let's break down the formula:
 (x1, y1) and (x2, y2) represent the coordinates of two points in the Cartesian plane.
 (x2  x1) represents the difference in the xcoordinates of the two points. This essentially calculates the horizontal distance between the points.
 (y2  y1) represents the difference in the ycoordinates of the two points. This calculates the vertical distance between the points.
 Squaring the horizontal and vertical distances gives us the squared lengths of the sides of a right triangle formed by connecting the two points and drawing a horizontal and vertical line.
 Adding the squared distances represents applying the Pythagorean theorem, where the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. The hypotenuse in this case is the line connecting the two points, which is the distance we want to find.
 Taking the square root of the entire expression finally gives us the distance between the two points.
Application in RealWorld Scenarios
The distance formula is crucial in various realworld applications:
 Navigation: GPS systems utilize the distance formula to calculate the distance between two locations on a map.
 Engineering: Architects and engineers use the distance formula to calculate the length of structures, such as bridges and buildings.
 Computer Science: The formula is used in various algorithms, including pathfinding and collision detection in games and simulations.
 Physics: The distance formula is used in various physics calculations, such as calculating the distance traveled by an object or the magnitude of a force.
Examples

Find the distance between the points (2, 3) and (5, 7).
 x1 = 2, y1 = 3, x2 = 5, y2 = 7
 Distance = √((52)² + (73)²)
 Distance = √(3² + 4²)
 Distance = √(9 + 16)
 Distance = √25 = 5

Find the distance between the points (1, 4) and (3, 2).
 x1 = 1, y1 = 4, x2 = 3, y2 = 2
 Distance = √((3  (1))² + (2  4)²)
 Distance = √(4² + (6)²)
 Distance = √(16 + 36)
 Distance = √52 = 2√13
Conclusion
The distance formula, represented by (x2  x1)² + (y2  y1)², is a fundamental concept in mathematics with wideranging applications in various fields. It allows us to calculate the distance between two points in a plane, which is essential for solving numerous realworld problems.