Circle Theorems & Reasons (Basic Properties of Circles)
Your ultimate cheat sheet for circle geometry theorems. Learn to spot hidden angles, chords, and tangents, and master the exact abbreviations needed to secure full steps marks on the DSE.
DSE Circle Reasons: Full List of Abbreviations
Write the reason in brackets after every step of a circle-geometry answer, e.g. ∠ABC = 50° (∠s in the same segment). The textbook form ∠ at centre twice ∠ at ⊙ce and the short form ∠ at centre = 2∠ at circum. are both accepted.
| Property | Reason to write |
|---|---|
| The perpendicular from the centre to a chord bisects the chord | line from centre ⊥ chord bisects chord |
| The line joining the centre to the mid-point of a chord is perpendicular to the chord | line joining centre to mid-pt. of chord ⊥ chord |
| Equal chords are equidistant from the centre | equal chords, equidistant from centre |
| Chords equidistant from the centre are equal | chords equidistant from centre are equal |
| Equal angles at the centre stand on equal chords and equal arcs | equal ∠s, equal chords / equal ∠s, equal arcs |
| Arcs are proportional to the angles at the centre | arcs prop. to ∠s at centre |
| The angle at the centre is twice the angle at the circumference | ∠ at centre twice ∠ at ⊙ce |
| Angles in the same segment are equal | ∠s in the same segment |
| The angle in a semicircle is 90° | ∠ in semi-circle |
| Opposite angles of a cyclic quadrilateral add up to 180° | opp. ∠s, cyclic quad. |
| The exterior angle of a cyclic quadrilateral equals the interior opposite angle | ext. ∠, cyclic quad. |
| A tangent is perpendicular to the radius at the point of contact | tangent ⊥ radius |
| Two tangents from an external point are equal in length | tangent properties |
| The angle between a tangent and a chord equals the angle in the alternate segment | ∠ in alt. segment |
| Proving four points are concyclic | converse of ∠s in the same segment / opp. ∠s supp. / ext. ∠ = int. opp. ∠ |
The Core Concept
How to solve Circle Geometry: Circle geometry questions are like visual puzzles. The examiners hide angles by overlapping triangles, chords, and tangents. Your goal is to systematically scan the diagram for three families of properties: Chord Properties, Angle Properties, and Tangent Properties.
DSE Basic Properties of Circles Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Chords & Arcs | • Perpendicular line from center to a chord bisects the chord. • Equal chords are equidistant from the center. | • line from centre ⟂ chord bisects chord • equal chords, equidistant from centre |
| 2. Angles & Arcs | • Angle at the center is twice the angle at the circumference. • Angles subtended by the same arc are equal. • Angle in a semicircle is 90°. | • ∠ at centre = 2 ∠ at circum. • ∠s in the same segment • ∠ in semicircle |
| 3. Cyclic Quads | • Opposite angles of a cyclic quadrilateral add up to 180°. • Exterior angle of a cyclic quad equals the interior opposite angle. | • opp. ∠s, cyclic quad. • ext. ∠, cyclic quad. |
| 4. Tangent Basics | • A tangent is perpendicular to the radius at the point of contact. • Tangents drawn from an external point are equal in length. | • tangent ⟂ radius • tangent properties |
| 5. Alternate Segment | The angle between a tangent and a chord equals the angle subtended by the chord in the alternate segment. | • ∠ in alt. segment |
💡 Exam Traps & Secrets
⚠️ The Alternate Segment Blind SpotThe angle in alternate segment (∠ in alt. segment) is the single most missed theorem in Section B. To spot it: (1) Find the tangent line and locate the point of contact. (2) Trace the chord starting from that point to form a triangle inside the circle. (3) The angle between the tangent and the chord is equal to the "opposite interior" corner of that triangle.
⚠️ The "Forget the Reasons" TrapIn Paper 1 Section A2/B, even if your calculations are 100% correct, you will lose up to 3 to 4 marks if you do not write the official geometric reasons (the abbreviations in the table above) next to your steps.
⚠️ The Secret Isosceles TriangleIf you are completely stuck on a circle angle question, look for the center O and draw lines to the vertices on the circumference. Because all radii of a circle are equal in length (
OA = OB = OC = radius), you will instantly create isosceles triangles. You can then use base ∠s, isos. Δ to find the missing angles.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Pythagoras theorem + line from centre ⟂ chord)In a circle, chord AB is 8 cm long and the radius is 5 cm. Find the perpendicular distance from the center O to the chord.
(opp. ∠s, cyclic quad + ∠ at centre = 2∠ at circum.)Given a circle with center O and a cyclic quad ABCD where ∠ADC = 110°, find the reflex angle ∠AOC.
(tangent properties + angle sum of polygon)Two tangents TP and TQ are drawn from an external point T to a circle with center O. If ∠PTQ = 50°, find ∠OPQ.
Basic Properties of Circles Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
Frequently Asked Questions
The three most tested theorems: (1) inscribed angle = ½ central angle (∠ at centre = 2∠ at circum.), (2) angles in the same segment are equal (∠s in the same segment), (3) angle in a semicircle = 90° (∠ in semicircle). Tangent-radius perpendicularity (tangent ⟂ radius) and the alternate segment theorem (∠ in alt. segment) also appear regularly in Section B.
State each theorem as you use it, using the exact official DSE abbreviations from the cheat sheet above. For example: "∠ABC = 50° (∠s in the same segment)" or "TP = TQ (tangent properties)". The examiner looks for these exact phrases — omitting them costs method marks even if your final answer is correct.
Draw all radii from the center to relevant points on the circumference. This creates isosceles triangles (radii are equal). Then look for: angle at centre, angles in same segment, cyclic quadrilaterals, and tangent-chord angles. Connect the center to the point of tangency to instantly create a right angle (tangent ⟂ radius).