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The Coordinate Plane

A lesson on the Cartesian coordinate plane — plotting and reading points, understanding quadrants, calculating distances, and drawing basic shapes on the grid.

Last updated: 2 December 2025

★★☆☆☆ Level 2CoordinatesQuadrantOrdered Pair3 min read

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The Coordinate Plane

Two perpendicular number lines — the x-axis (horizontal) and y-axis (vertical) — form the coordinate plane. They cross at the origin (0, 0).

The coordinate plane with four quadrants and example points.
The coordinate plane with four quadrants and example points.

Ordered Pairs

Every point is written as (x,y)(x, y).

  • Move right for positive xx, left for negative xx
  • Move up for positive yy, down for negative yy

The Four Quadrants

QuadrantSignsMnemonic
I(+, +)top right
II(−, +)top left
III(−, −)bottom left
IV(+, −)bottom right

Plotting Points — Step by Step

  1. 1

    Find x

    Start at origin; move horizontally.
  2. 2

    Find y

    From that position, move vertically.
  3. 3

    Mark the point

    Draw and label the dot.

Distance Along a Horizontal or Vertical Line

Horizontal distance=∣x2−x1∣\text{Horizontal distance} = |x_2 - x_1|

Vertical distance=∣y2−y1∣\text{Vertical distance} = |y_2 - y_1|

Drawing Shapes

Plot vertices and connect them:

Rectangle with corners A(1,1)A(1,1), B(5,1)B(5,1), C(5,4)C(5,4), D(1,4)D(1,4):

Width = ∣5−1∣=4|5-1| = 4 units, Height = ∣4−1∣=3|4-1| = 3 units, Area = 12 square units.

Find the length of the segment from P(−3, 2) to Q(5, 2).

Both points have y=2y = 2 (horizontal segment).

PQ=∣5−(−3)∣=8 unitsPQ = |5 - (-3)| = 8 \text{ units}

Real world: GPS navigation

A GPS system models every location on Earth as a coordinate pair (longitude, latitude). Moving from point A(−3.2, 51.5) to B(−3.2, 52.0) is a vertical shift of ∣52.0−51.5∣=0.5°|52.0 - 51.5| = 0.5° north — a direct coordinate distance.

Real world: video-game level design

A game developer places a player spawn at (0, 0) and a treasure chest at (8, 6). The shortest straight-line path has horizontal distance 8 units and vertical distance 6 units. The grid tells the developer exactly how far apart the objects are.

Real world: weather maps

Meteorologists plot pressure systems on a coordinate grid. A high-pressure centre at (4, 3) moving to (7, 3) has shifted 3 units east (same y, different x), so the storm track is a horizontal displacement of ∣7−4∣=3|7 - 4| = 3 grid units.

Tip Always read the x-coordinate first (horizontal), then y (vertical). Think: "along the corridor, then up the stairs."

Practice Problems

Name the quadrant for (−4, −7).

Show answer
Quadrant III (both negative).

What is the distance between (2, 5) and (2, −3)?

Show answer
|5 − (−3)| = 8 units (vertical segment, same x).

Plot the points (0,3), (4,3), (4,0), (0,0) and name the shape.

Show answer
Rectangle (or square if 4×3 — it's a rectangle with width 4 and height 3).