Cyclic Quadrilaterals - Circles Class 9 Maths Chapter 10

Circles: Cyclic Quadrilaterals, Angle Properties, and Arc Theorems

Circles are among the most fundamental shapes in geometry, appearing in everything from wheels and clocks to planetary orbits and architectural domes. This chapter in CBSE Class 9 Mathematics explores the properties of angles in circles — the angles subtended by arcs at the centre and at the circumference, the angles in the same segment, and the special properties of cyclic quadrilaterals. Understanding these properties allows students to solve a wide range of geometric problems and prove elegant theorems.

A cyclic quadrilateral is a four-sided polygon whose vertices all lie on the circumference of a single circle. The most important property of a cyclic quadrilateral is that its opposite angles are supplementary — they add up to 180°. This can be proved using the theorem that the angle subtended by an arc at the centre is twice the angle subtended at any point on the remaining part of the circle. Let ABCD be a cyclic quadrilateral inscribed in a circle with centre O. The arc ADC subtends angle AOC at the centre and angle ABC at the circumference. By the central angle theorem, angle AOC = 2 × angle ABC. Similarly, the arc ABC subtends angle AOC (on the other side, measuring 360° − AOC) and angle ADC at the circumference. Therefore, angle ABC + angle ADC = (1/2) × angle AOC + (1/2) × (360° − angle AOC) = 180°. This result extends to: if one side of a cyclic quadrilateral is extended, the exterior angle equals the interior opposite angle.

Several important circle theorems are covered in this chapter. The angle at the centre theorem: the angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle. Angles in the same segment: all angles subtended by the same arc (or chord) at points on the same segment are equal. This follows directly from the central angle theorem since they all subtend the same central angle. Equal chords subtend equal angles at the centre and at the circumference. The angle in a semicircle is always a right angle (90°) — this is because the diameter subtends 180° at the centre, so the angle at the circumference is 180°/2 = 90°. This is known as Thales' theorem and provides a quick way to construct right-angled triangles inscribed in a semicircle. For chords of a circle, the perpendicular from the centre to a chord bisects the chord, and conversely, the line joining the centre to the midpoint of a chord is perpendicular to it. If two chords intersect inside a circle, the products of the segments of each chord are equal (the intersecting chords theorem). These properties are used in board examinations to prove that given points lie on a circle, to find unknown angles, to establish the cyclic nature of a quadrilateral, and to solve problems involving arc lengths and chord lengths.

• A cyclic quadrilateral has all vertices on a circle; opposite angles are supplementary (∠A + ∠C = 180°).
• Angle at centre theorem: angle subtended by an arc at the centre is twice the angle subtended at the circumference.
• Angles in the same segment (subtended by the same arc) are equal; angle in a semicircle is always 90° (Thales' theorem).
• Perpendicular from centre to chord bisects the chord; if chords intersect, products of their segments are equal.
• Exterior angle of a cyclic quadrilateral equals the interior opposite angle — useful for proving geometric relationships.