Geometry › Coordinates

Coordinates

Pin down any point on a flat surface with two numbers. Descartes' big idea — and the bridge between geometry and algebra.

Coordinate plotter

Drag a vertex. The (x, y) follows your finger.

Coordinates

A(0, 5)

B(-4, -3)

C(4, -3)

x then y

First how far across, then how far up. Right and up are positive.

The Cartesian grid

Two perpendicular number lines — the x-axis (horizontal) and the y-axis (vertical). They meet at the origin, written (0, 0). Every point on the plane gets two numbers: how far across, then how far up.

How to read (x, y)

x first, then y. Always.
Positive x → right. Negative x → left.
Positive y → up. Negative y → down.
The grid splits into four quadrants, numbered I, II, III, IV starting top-right and going counterclockwise.

Why it's a big deal

Once shapes have coordinates, every geometric question becomes algebra. The line through (0, 0) and (1, 1) is y = x. The circle of radius 1 at the origin is x² + y² = 1. Geometry and algebra finally speak the same language.

Distance between points

For two points (x₁, y₁) and (x₂, y₂):

distance = √((x₂ − x₁)² + (y₂ − y₁)²)

That's Pythagoras, with sides Δx and Δy.

Midpoint

The midpoint of two points is just the average of each coordinate:

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)