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 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° |
Worked Examples
Find the distance between A(2, 3) and B(8, 11).
d = √[(8−2)² + (11−3)²] = √[36 + 64] = √100 = 10.
Find the midpoint of P(1, 5) and Q(7, 9).
M = ((1+7)/2, (5+9)/2) = (4, 7). Midpoint: (4, 7).
Find the equation of the line through (0, 3) with gradient 2.
y = mx + c. m = 2, c = 3 (y-intercept). Equation: y = 2x + 3.
Find the equation of the line through (2, 5) and (4, 9).
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.
Practice Questions
- Find the distance between C(3, −1) and D(7, 2).
- Find the midpoint of (5, −4) and (1, 8).
- A line has gradient 3 and passes through (0, −2). Write its equation.
- Find the gradient of the perpendicular to the line y = 4x + 1.
- Write the equation of the line through (3, 7) perpendicular to y = 3x − 5.