Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Parallel & Perpendicular Lines

In one sentence. Same gradient means parallel; gradients multiplying to −1 mean perpendicular — and the constructions that follow. Higher Tier.

Coordinate Geometry — one of 2 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Pierre de Fermat. Learn Parallel & Perpendicular Lines, teach it, then defend it against Pierre de Fermat.

Where this sits in the specification

Specification 4MA1, 3.3 Graphs — parallel and perpendicular gradients (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 3.3 — Graphs — parallel and perpendicular gradients (Higher)

1 · Understand it

What it is

Parallel lines share a gradient (same climb, different start). Perpendicular lines have gradients multiplying to −1: the perpendicular gradient is the NEGATIVE RECIPROCAL — flip the fraction, flip the sign. Perpendicular to m = 2 is m = −1/2.

Why it works

Turning a right angle swaps the roles of across and up: a climb of 'right 1, up 2' rotates into 'up 1, left 2' — which as a gradient is −1/2. The flip is the swap of rise and run; the sign change is the quarter-turn. Multiply them: 2 × (−1/2) = −1, always.

When a question is asking for it

'Parallel to … through …' and 'perpendicular to … through …' equation-building; perpendicular bisectors (midpoint + perpendicular gradient); and tangent/radius questions in circle work, where the tangent is perpendicular to the radius.

How to hold it in your head

To write ANY line through a point: get the gradient (copied for parallel, negative-reciprocal for perpendicular), then y = mx + c with the point substituted to find c. It's the straight-lines recipe with one extra decision at the start.

2 · The mistakes that cost marks

  • Perpendicular gradient as just the negative: perpendicular to 2 is −1/2, not −2. Flip AND negate.
  • Perpendicular gradient as just the reciprocal: 1/2 fails too — the product must be −1, so the sign must change.
  • Rearranging sloppily: perpendicular to 2y = 6x + 1 needs the true gradient first (3), then −1/3.
  • For a perpendicular BISECTOR, using one of the endpoints instead of the midpoint.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Find the equation of the line perpendicular to y = 2x + 1 passing through (4, 3).

  1. Original gradient 2 → perpendicular gradient −1/2.
  2. y = −½x + c through (4, 3): 3 = −2 + c → c = 5.
  3. y = −½x + 5.

Answer: y = −½x + 5

P = (2, 5), Q = (6, 13). Where does the perpendicular bisector of PQ cross the y-axis?

  1. Midpoint: (4, 9). Gradient of PQ: 8/4 = 2 → bisector gradient −1/2.
  2. y = −½x + c through (4, 9): 9 = −2 + c → c = 11.
  3. Crosses the y-axis at (0, 11).

Answer: c = 11

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Write down the gradient of any line parallel to y = 4x − 7.
  2. Grade 7 — Write down the gradient of any line perpendicular to y = 2x + 1. (Type it like -1/2.)
  3. Grade 8 — Find the equation of the line parallel to y = 3x + 2 passing through (4, 5). (Type it like y=3x-7.)
  4. Grade 9 — Find the equation of the line perpendicular to y = 2x + 1 passing through (4, 3). (Type it like y=-0.5x+5.)
  5. 9+ — P = (2, 5) and Q = (6, 13). The perpendicular bisector of PQ crosses the y-axis at (0, c). Work out the value of c.

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Parallel & Perpendicular Lines, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Parallel & Perpendicular Lines works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Parallel & Perpendicular Lines against Pierre de Fermat — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

Completely free. No subscriptions, no paywalls, no account needed to start.

Carry on through the course