Geometry
Coordinate Geometry – Algebra Meets Geometry
Coordinate geometry (also called analytical geometry) brings together algebra and geometry. By placing shapes on a numbered grid, we can use equations to describe, analyse, and solve geometric problems precisely.
The system is often called Cartesian geometry after René Descartes, whose Latinised name was Cartesius, but the French lawyer and mathematician Pierre de Fermat developed strikingly similar ideas at almost exactly the same time in the 1630s, in a manuscript that was not published until after his death; today the two are usually credited jointly as the founders of coordinate geometry. The (x, y) coordinate idea turned out to be one of the most quietly powerful tools ever invented in mathematics, since it lets purely algebraic equations be drawn as pictures, and purely visual shapes be manipulated with algebra instead of a ruler and compass. Every digital image relies on the same idea today: a computer screen is really just a giant coordinate grid of pixels, and every line, circle, or curve drawn in design software is calculated using exactly the formulas on this page.
The Coordinate System
Every point in a plane is described by an ordered pair (x, y). The x-axis runs horizontally; the y-axis runs vertically. They meet at the origin (0, 0). Positive x goes right, positive y goes up.
Key Formulas
| Formula | Expression | Use |
|---|---|---|
| Distance | d = √[(x₂−x₁)² + (y₂−y₁)²] | Length of a segment between two points |
| Midpoint | M = ((x₁+x₂)/2, (y₁+y₂)/2) | Centre point of a segment |
| Gradient (slope) | m = (y₂−y₁)/(x₂−x₁) | Steepness of a line |
| Equation of line | y = mx + c | Describes any straight line |
| Parallel lines | Same gradient m | Never meet |
| Perpendicular lines | m₁ × m₂ = −1 | Meet at 90° |
Where the Distance Formula Comes From
The distance formula isn't a separate rule to memorise — it's just Pythagoras' theorem in disguise. To find the distance between two points, build a right-angled triangle between them: one leg going straight across (the horizontal distance), one leg going straight up (the vertical distance), and the segment you want is the hypotenuse.
The horizontal leg has length |8 − 2| = 6 (the difference in x-values, x₂ − x₁). The vertical leg has length |11 − 3| = 8 (the difference in y-values, y₂ − y₁). By Pythagoras' theorem, the hypotenuse d satisfies d² = 6² + 8² = 36 + 64 = 100, so d = √100 = 10. Writing that in general symbols — using (x₂−x₁) and (y₂−y₁) as the two legs, for any pair of points — gives exactly the distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²].
Worked Examples
d = √[(8−2)² + (11−3)²] = √[36 + 64] = √100 = 10.
M = ((1+7)/2, (5+9)/2) = (4, 7). Midpoint: (4, 7).
y = mx + c. m = 2, c = 3 (y-intercept). Equation: y = 2x + 3.
Gradient: m = (9−5)/(4−2) = 2. Using y−5 = 2(x−2): y = 2x + 1. Equation: y = 2x + 1.
Key Takeaways
- Distance formula: d = √[(Δx)² + (Δy)²] — based on Pythagoras.
- Midpoint = average of the x-coordinates and average of the y-coordinates.
- Gradient = rise / run = (y₂−y₁)/(x₂−x₁).
- Perpendicular gradients multiply to −1.