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Coordinate Geometry

10 min read

Finding a point on a plane

Two number lines cross at right angles. The horizontal one is the x-axis, the vertical one is the y-axis, and they meet at the origin (0, 0). Every point has two coordinates : first how far across, then how far up or down.

The coordinate planeA grid with a horizontal x axis and a vertical y axis crossing at the origin, dividing the plane into four quadrants numbered anticlockwise from top right. Four points are plotted: A at 3 and 2, B at minus 2 and 3, C at minus 3 and minus 2, and D at 2 and minus 3.xyQuadrant 1 (+, +)Quadrant 2 (−, +)Quadrant 3 (−, −)Quadrant 4 (+, −)A (3, 2)B (−2, 3)C (−3, −2)D (2, −3)
Go along the x-axis first, then up or down.

Gradient: how steep is a line?

Gradient
A number that measures steepness: how far a line rises for each step it runs across.
Take any two points on the line.
Gradient of a straight lineA sloping line through two points. A right-angled triangle under the line shows the vertical rise of 2 and the horizontal run of 3, so the gradient is rise divided by run, which is 2 divided by 3.xyrun = 3 (across)rise = 2 (up)PQgradient = rise ÷ run = 2 ÷ 3
The triangle slides along the line. The gradient stays the same.
Worked exampleGradient between two points

Find the gradient of the line through and .

  1. Rise: change in y.
  2. Run: change in x.
  3. Divide.

Answer: The gradient is 3.

Drawing graphs of straight lines

m is the gradient. c is where the line crosses the y-axis.
  • gradient (steepness)
  • y-intercept (where the line meets the y-axis)
Try itChange a straight line

Change and . A bigger makes the line steeper; a negative makes it slope down; slides it up or down.

Check yourself Where is the point (0, 5)?
Check yourself What is the gradient of y = 3x − 4?

Key terms

Coordinates
A pair (x, y) giving a point's position.
Origin
The point (0, 0) where the axes cross.
Gradient
Rise divided by run; the steepness of a line.
Intercept
Where a line crosses an axis.

Summary

  • Coordinates (x, y): across first, then up or down.
  • Gradient = (y₂ − y₁) ÷ (x₂ − x₁).
  • A straight line is y = mx + c: m is the gradient, c the y-intercept.
Ratios, Profit and Loss Perimeters and Areas