Coordinate Geometry
Coordinate geometry bridges algebra and geometry by placing shapes on a numbered grid. It lets you describe geometric objects using equations and calculate distances, midpoints, and slopes using arithmetic.
In this lesson
1 The Cartesian Plane
The Cartesian plane is a flat surface defined by two perpendicular number lines: the x-axis (horizontal) and y-axis (vertical). Their intersection is the origin (0, 0). Every point on the plane has a unique address written as (x, y) — an ordered pair.
The plane is divided into four quadrants: Quadrant I (x>0, y>0, top right), Quadrant II (x<0, y>0, top left), Quadrant III (x<0, y<0, bottom left), Quadrant IV (x>0, y<0, bottom right).
The Cartesian plane is named after René Descartes, the French mathematician who invented coordinate geometry in 1637. His insight — that every point in a plane can be described by two numbers — unified algebra and geometry and made calculus possible.
2 The Distance Formula
The distance between two points (x₁, y₁) and (x₂, y₂) is: d = √((x₂−x₁)² + (y₂−y₁)²)
This is the Pythagorean theorem applied to coordinates. The horizontal distance (x₂−x₁) and vertical distance (y₂−y₁) form the legs of a right triangle, and d is the hypotenuse.
53 The Midpoint Formula
The midpoint between (x₁, y₁) and (x₂, y₂) is: M = ((x₁+x₂)/2, (y₁+y₂)/2)
You're simply averaging the x-coordinates and averaging the y-coordinates. The result is the point exactly halfway between the two given points.
4 Slope and Line Equations
Slope measures steepness: m = (y₂−y₁)/(x₂−x₁) = rise/run. Positive slope goes up left to right. Negative slope goes down. Zero slope is horizontal. Undefined slope is vertical.
A line's equation in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis).
Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: if one has slope 2, the perpendicular has slope −½. Their slopes multiply to −1.
5 Applications
GPS and mapping: every location on Earth is described by two coordinates (latitude and longitude). Distance calculations between cities use the coordinate distance formula adapted for a sphere. Google Maps calculates routes using coordinate geometry at massive scale.
Computer graphics: every pixel on a screen has coordinates. Drawing lines, shapes, and animations all rely on coordinate geometry. Game physics engines calculate collisions using distance formulas between object coordinates.
Try the Slope Calculator
Find slope and the full line equation from any two coordinate points.
Practice Problems
📚 Further Reading & Resources
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