ICSE Class 10 Maths: Circles & Tangents — Complete Notes 2026

Basic Circle Terminology

• Circle: locus of points equidistant from centre; chord, arc, sector, segment definitions
• Minor arc < semicircle; major arc > semicircle; diameter = longest chord
• Concentric circles: same centre, different radii; equal circles: equal radii

Chord Properties

• Perpendicular from centre to chord bisects the chord (and vice versa)
• Equal chords are equidistant from the centre; converse also true
• Angle subtended by chord at centre = 2 × angle at any point on major arc

Angle Properties of Circles

• Angles in the same segment are equal (equal arcs subtend equal angles at circumference)
• Angle in a semicircle = 90° (angle subtended by diameter)
• Exterior angle of a cyclic quadrilateral = interior opposite angle

Cyclic Quadrilateral

• All four vertices on the circle; opposite angles are supplementary (sum = 180°)
• Converse: if opposite angles sum to 180°, the quadrilateral is cyclic
• ICSE standard proof: angles in same segment / opposite angles of cyclic quad

Tangents to a Circle

• Tangent is perpendicular to radius at point of contact
• From an external point, two tangents are equal in length: TA = TB
• Tangent-chord angle = angle in alternate segment (Tangent-Chord theorem)

Constructions Related to Circles

• Constructing tangent at a point on the circle (perpendicular to radius)
• Constructing tangents from an external point (bisect line joining centre to external point)
• Constructing circumscribed and inscribed circles of triangles

ICSE Exam — Circles

• Prove: angles in the same segment are equal
• In cyclic quadrilateral ABCD, angle A = 75°, find angle C
• TA and TB are tangents from T; angle ATB = 60°, find angle AOB