Circles: Angles, Chords and Cyclic Quadrilaterals
What the exam tests
Circle questions in Kerala SSLC test angle relationships at the centre and on the circumference, properties of chords, and cyclic quadrilaterals. You must identify which arc an angle subtends before applying a theorem.
Core rules
- The angle at the centre is twice the inscribed angle standing on the same arc.
- Angles in the same segment are equal.
- The angle in a semicircle is 90°; this applies only when the chord is a diameter.
- Opposite angles of a cyclic quadrilateral are supplementary.
- A perpendicular from the centre to a chord bisects the chord.
- Equal chords are equidistant from the centre and subtend equal angles at the centre.
Worked example
A circle has centre O and ∠AOB = 110°, where A and B are on the circle. A point C lies on the major arc AB. The inscribed angle ∠ACB stands on the same minor arc AB as the central angle, so ∠ACB = 110°/2 = 55°. If C were on the minor arc instead, it would stand on the major arc, and the angle would be 180° − 55° = 125°.
Traps
Do not halve the wrong angle when the points lie on different arcs. A cyclic quadrilateral uses opposite angles, not adjacent angles. Do not assume a chord is a diameter just because it passes near the centre. Equal chords give equal arcs and equal central angles, but not necessarily equal inscribed angles unless they stand on the same arc. Keep the distinction between an angle at the centre and an angle at the circumference.