The Coordinate Plane

(x, y) — over, then up. Every point gets an address.

The idea

The Cartesian coordinate system is the natural method for specifying the location of points in the plane.

Choose a point $O$, the origin, and draw two perpendicular number lines through it, each with $0$ at $O$: the horizontal $x$-axis, positive to the right, and the vertical $y$-axis, positive upward.

Given any point $P$, drop a perpendicular from $P$ to each axis. The one meeting the $x$-axis lands at a number $a$, and the one meeting the $y$-axis lands at a number $b$. We write $P = (a, b)$. The order matters, so $(5, 1)$ and $(1, 5)$ name different points.

The axes cut the plane into four quadrants, numbered I to IV counterclockwise from the upper right, and the signs of $a$ and $b$ decide which one holds $P$. A point with $a = 0$ or $b = 0$ lies on an axis, and so in no quadrant.

Ways to work on it

• Walkthrough. Ordered pairs, axes, and the four quadrants.
• Practice. Place signed points in their quadrants.
• Hardest. Complete an axis-parallel rectangle from three vertices.

Not sure where to start? Take the ten-question placement test.