Basic properties of circles
The circle theorems on chords, angles, cyclic quadrilaterals and tangents, and how to give the right reason for each step. It is the longest senior unit and one students often find hard because of the many facts.
What you will learn
- 11.1 Chords and arcs:
- 11.2 Angles:
- 11.3 Cyclic quadrilaterals: opposite angles add to 180°; the exterior angle equals the interior opposite angle.
- NF 11.4 Tests for concyclic points and cyclic quadrilaterals.
- NF 11.5 Tangents:
- NF 11.6 Geometric proofs with circle properties (junior geometry may be used too).
You need first: Geometry and data basics, Similar triangles
Key terms
| English | 中文 |
|---|---|
| centre, radius, diameter | 圓心，半徑，直徑 |
| chord, arc, segment | 弦，弧，弓形 |
| circumference | 圓周 |
| angle at the centre / at the circumference | 圓心角／圓周角 |
| angle in a semicircle | 半圓上的圓周角 |
| cyclic quadrilateral | 圓內接四邊形 |
| concyclic points | 共圓點 |
| tangent, point of contact | 切線，切點 |
| angle in the alternate segment | 交錯弓形的圓周角 |
Standard steps
Find an angle
- Mark the centre, radii, and any right angles on the diagram.
- Radii are equal, so look for isosceles triangles.
- Look for one of the "big five" shapes:
| Shape you see | Fact | Reason (short form) |
|---|---|---|
| an angle at O and one at the circumference on the same arc | centre = 2 × circumference | ∠ at centre twice ∠ at ⊙ce |
| two angles standing on the same arc | equal | ∠s in the same segment |
| a diameter with a triangle on it | 90° | ∠ in semi-circle |
| four points on the circle | opposite angles add to 180° | opp. ∠s, cyclic quad. |
| a tangent touching the circle | 90° to the radius | tangent ⊥ radius |
- Write one new angle per line, with the reason in brackets.
Reason short forms differ slightly between textbooks. Use the student's own book's version.
Chord problems
Draw the perpendicular from O to the chord. It bisects the chord, and gives a right-angled triangle for Pythagoras: r² = d² + (half-chord)².
Proofs (NF)
- Write "Given" facts you will use.
- Each line: a statement, then the reason in brackets.
- To prove four points are concyclic, use one of the tests: converse of ∠s in the same segment, opposite angles supplementary, or exterior angle = interior opposite angle.
- End with the exact statement you were asked to prove.
Worked examples
Let M be the foot of the perpendicular from O to AB.
AM = 16 ÷ 2 = 8 cm (line from centre ⊥ chord bisects chord)
OM = √(10² − 8²) = 6 cm (Pythagoras' theorem)2x + (x + 30°) = 180° (opp. ∠s, cyclic quad.)
3x = 150°
x = 50°
∠A = 100°∠ACB = 130° ÷ 2 = 65° (∠ at centre twice ∠ at ⊙ce).
Common mistakes
- Using "angle at the centre = 2 × angle at the circumference" when the two angles stand on different arcs.
- Calling any four-sided figure in a circle "cyclic" when a vertex is not on the circle.
- Forgetting that radii are equal (the isosceles triangle is often the key).
- Using a tangent property when the line is not stated or proven to be a tangent.
- Proving the converse of what was asked.
How the DSE tests it
- Section A: angle-chasing items in Paper 2 (some of the slowest questions; see Curriculum and exam); chord and Pythagoras questions.
- Section B: prove concyclic points or a tangent, then use it in coordinate geometry.
Questions you might not ask in class
How am I supposed to remember all these reasons?
Don't memorise a list. Learn the five shapes above; each one comes with its reason. Most questions use only these.
How do I know where to start?
Start from what's given and mark it. Then look for radii (isosceles triangles) and the five shapes. Each new angle often unlocks the next.
What exactly is a 'segment'?
The region between a chord and its arc. "Angles in the same segment" means the angles' vertices are on the same side of the chord.