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AS Level · Pure Mathematics 1 WMA11

Equation of a Straight Line

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Edexcel IAL Maths · Pure 1

Equation of a Straight Line

Coordinate Geometry · Line Equations · Parallel & Perpendicular · Modelling

The Big Idea: Any straight line can be fully described by two things — its gradient (steepness) and a point it passes through — and understanding how gradients relate tells you whether lines are parallel, perpendicular, or the same line.

Chapter Overview

Distance Formula
Midpoint Formula
Gradient Formula
y = mx + c
y − y₁ = m(x − x₁)
ax + by + c = 0
Parallel Lines
Perpendicular Lines
Collinear Lines
Real-Life Modelling

1 · Basic Coordinate Geometry

1.1 — Length of a Line Segment (Distance Formula)

If you have two points on a coordinate grid — say A(x₁, y₁) and B(x₂, y₂) — and you want the straight-line distance between them, picture the invisible right-angled triangle formed by moving horizontally then vertically from A to B. The distance is just the hypotenuse of that triangle, found using Pythagoras' theorem.

Analogy: Think of your city as a grid. You want to travel from block (2, 3) to block (8, 11). You move 6 blocks east and 8 blocks north. The diagonal as-the-crow-flies distance is √(6² + 8²) = √100 = 10 blocks.
Distance Formula
d = √[ (x₂ − x₁)² + (y₂ − y₁)² ]
Square the difference in x, square the difference in y, add them, then square-root.
💡Pro tip: Work with d² for as long as possible. Only take the square root when you need a final answer. This avoids messy surds mid-calculation and prevents rounding errors.
Worked Example
The line segment PQ is the diameter of a circle with P(−2, −5) and Q(8, 7). Find (i) the centre of the circle, and (ii) the length of the diameter in the form a√b.
Centre = midpoint M: M = ( (−2+8)/2 , (−5+7)/2 ) = (6/2, 2/2) = (3, 1)
Diameter = distance PQ: d² = (8−(−2))² + (7−(−5))² = 10² + 12² = 100 + 144 = 244
Simplify: d = √244 = √(4 × 61) = 2√61
Practice Question
Find the exact length of the line segment AB where A = (−3, 4) and B = (5, −2). Give your answer in the form k√n.

1.2 — Midpoint of a Line Segment

The midpoint M of a line segment is simply the average of the two x-coordinates and the average of the two y-coordinates. You're finding the "middle" in both directions simultaneously.

Midpoint Formula
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
Add both x-values, halve. Add both y-values, halve. That's your midpoint.
✅Key use: If a chord of a circle has endpoints A and B, the centre of the circle lies on the perpendicular bisector — which passes through the midpoint of AB. This combination (midpoint + perpendicular gradient) is an extremely common exam pattern.
Practice Question
M is the midpoint of line segment PQ. P = (1, −3) and M = (4, 2). Find the coordinates of Q.

1.3 — Gradient of a Line Segment

Gradient measures how steeply a line rises or falls. It's the change in y for every 1 unit of movement in x. A gradient of 3 means: "for every step right, you go 3 steps up." A gradient of −½ means: "for every 2 steps right, you go 1 step down."

Gradient Formula
m = (y₂ − y₁) / (x₂ − x₁)
Rise ÷ Run — change in y divided by change in x. Order of points doesn't matter, but be consistent.
🔵 Reading gradients:
Practice Question
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Also in the full note
  • 2 · Equation of a Straight Line
  • 3 · Parallel & Perpendicular Gradients
  • 4 · Modelling with Straight Lines
  • What to Memorise
  • Concepts Checklist
  • Exam Tips & Common Mistakes
  • 2.1 — The Three Forms You Must Know
  • 2.2 — How to Find the Equation of a Line (Step by Step)
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More Pure Mathematics 1 topics

Laws of Indices & Surds
Laws of Indices
Basic Coordinate Geometry
Linear Inequalities
Translations
Quadratic Graphs
Expanding Brackets
Basic Trigonometry
Linear Simultaneous Equations using Elimination
Sketching Polynomials