Coordinate Pair Calculator
Use this calculator to find the distance, midpoint, and slope between two given coordinate points (x1, y1) and (x2, y2).
Point 1 (P1)
Point 2 (P2)
Results:
Understanding Coordinate Pairs and Their Calculations
In mathematics, a coordinate pair, also known as an ordered pair, is a set of two numbers used to locate a point on a two-dimensional plane, typically the Cartesian coordinate system. These pairs are written in the form (x, y), where 'x' represents the horizontal position (abscissa) and 'y' represents the vertical position (ordinate).
The Cartesian Coordinate System
The Cartesian system consists of two perpendicular number lines: the horizontal x-axis and the vertical y-axis. Their intersection point is called the origin (0,0). Any point on this plane can be uniquely identified by its coordinate pair, indicating its distance and direction from the origin along each axis.
Key Calculations with Coordinate Pairs
Coordinate pairs are fundamental for various geometric and algebraic calculations. Our calculator helps you determine three essential properties between two points:
1. Distance Between Two Points
The distance formula is derived from the Pythagorean theorem. If you have two points, P1(x1, y1) and P2(x2, y2), the distance (d) between them is calculated as:
d = √((x2 - x1)² + (y2 - y1)²)
This formula essentially finds the length of the hypotenuse of a right-angled triangle formed by the two points and their projected differences along the x and y axes.
Example: Let P1 = (1, 2) and P2 = (4, 6).
d = √((4 - 1)² + (6 - 2)²)
d = √((3)² + (4)²)
d = √(9 + 16)
d = √25
d = 5
2. Midpoint of a Line Segment
The midpoint is the exact center of the line segment connecting two points. It's found by averaging the x-coordinates and averaging the y-coordinates of the two points:
Midpoint (Mx, My) = ((x1 + x2)/2, (y1 + y2)/2)
Example: Using P1 = (1, 2) and P2 = (4, 6).
Mx = (1 + 4)/2 = 5/2 = 2.5
My = (2 + 6)/2 = 8/2 = 4
Midpoint = (2.5, 4)
3. Slope of a Line
The slope (m) of a line describes its steepness and direction. It's the ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run) between two points:
m = (y2 - y1) / (x2 - x1)
A positive slope indicates an upward trend from left to right, a negative slope indicates a downward trend, a zero slope means a horizontal line, and an undefined slope means a vertical line (where x1 = x2).
Example: Using P1 = (1, 2) and P2 = (4, 6).
m = (6 - 2) / (4 - 1)
m = 4 / 3
m ≈ 1.3333
Example of Undefined Slope: Let P1 = (2, 3) and P2 = (2, 7).
m = (7 - 3) / (2 - 2)
m = 4 / 0
The slope is undefined, indicating a vertical line.
This calculator provides a quick and accurate way to perform these fundamental calculations, which are crucial in geometry, physics, engineering, and computer graphics.