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.
1. 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.
2. 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 |
Click the graph to explore it interactively →
💡 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.