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

Circle Properties & Tangents

In one sentence. Radii, chords and tangents, and the right angles they secretly generate — the foundation the circle theorems build on.

Circles & Circle Theorems — one of 2 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Ibn al-Haytham. Learn Circle Properties & Tangents, teach it, then defend it against Ibn al-Haytham.

Where this sits in the specification

Specification 4MA1, 4.6 Circle properties. 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.

  • 4.6 — Circle properties

1 · Understand it

What it is

The working parts of a circle: radius (centre to edge), chord (edge to edge), tangent (touches at exactly one point), arc, sector, segment. Three power facts: a tangent is perpendicular to the radius at the contact point; two tangents from an external point are equal in length; the perpendicular from the centre to a chord bisects the chord.

Why it works

The tangent–radius right angle exists because the radius is the shortest path from centre to tangent line — and shortest paths meet lines at right angles. Two tangents from a point are equal because the two triangles formed (centre, external point, contact points) are mirror images. The chord fact is isosceles-triangle symmetry: two radii to the chord's ends make an isosceles triangle, and its axis of symmetry bisects the base.

When a question is asking for it

Any diagram with a tangent → mark the 90° immediately. Any chord with the centre involved → drop the perpendicular and halve the chord. These two reflexes, plus Pythagoras, solve most non-theorem circle questions.

How to hold it in your head

Two radii always make an isosceles triangle — spot the 'radius pair, equal pair' pattern everywhere. And a wheel on a road is the tangent picture: the road touches at one point, and the spoke to that point stands square to the road.

2 · The mistakes that cost marks

  • Marking the tangent angle 90° to the CHORD instead of to the radius at the contact point.
  • Treating any line from the centre to a chord as its bisector. Only the PERPENDICULAR one bisects.
  • Forgetting the two-tangent equality — the symmetric kite in tangent diagrams goes unnoticed.
  • Using the full chord instead of its half in the Pythagoras step.

3 · Worked examples

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

A circle has radius 13 cm. A chord has length 24 cm. How far is the chord from the centre?

  1. Perpendicular from the centre bisects the chord: half-chord = 12.
  2. Right triangle: radius 13 (hypotenuse), half-chord 12, distance d.
  3. d = √(13² − 12²) = √25 = 5 cm.

Answer: 5 cm

Two radii OA and OB make an angle of 48° at the centre. Find angle OAB.

  1. OA = OB (radii) → triangle OAB is isosceles.
  2. Base angles: (180 − 48)/2.

Answer: 66°

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 — A tangent touches a circle at point P. What is the angle between the tangent and the radius OP, in degrees? (answer in °)
  2. Grade 7 — From an external point P, two tangents touch a circle at A and B. Given PA = 9 cm, write down the length of PB. (answer in cm)
  3. Grade 8 — O is the centre of a circle, and A and B lie on it. Angle AOB = 48°. Work out angle OAB. (answer in °)
  4. Grade 9 — A chord of a circle has length 16 cm. The perpendicular distance from the centre of the circle to the chord is 15 cm. Work out the radius of the circle. (answer in cm)
  5. 9+ — A circle centred at the origin passes through the point P(3, 4). Work out the gradient of the TANGENT to the circle at P. (Type it like -3/4.)

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 Circle Properties & Tangents, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Circle Properties & Tangents 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 Circle Properties & Tangents against Ibn al-Haytham — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 9 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.

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