Learning Objectives
- Understand basic terms related to circles
- Learn theorems about chords, arcs, and angles
- Understand the angle subtended by a chord at the centre and at a point on the circle
- Learn properties of cyclic quadrilaterals
Key Concepts
Basic Terms
A circle is the collection of all points in a plane that are at a fixed distance (radius) from a fixed point (centre). A chord is a line segment joining two points on the circle. The diameter is the longest chord, passing through the centre.
Theorems on Chords
Theorem 1: Equal chords of a circle subtend equal angles at the centre.
Converse: If the angles subtended by two chords at the centre are equal, then the chords are equal.
Theorem 2: The perpendicular from the centre of a circle to a chord bisects the chord.
Converse: The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
Theorem 3: Equal chords of a circle are equidistant from the centre.
Converse: Chords equidistant from the centre are equal.
Arc and Angle Subtended
Theorem: The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Corollary: Angles in the same segment of a circle are equal.
Theorem: The angle in a semicircle is a right angle (90°).
Cyclic Quadrilateral
A cyclic quadrilateral is a quadrilateral whose all four vertices lie on a circle.
Theorem: The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
Converse: If the sum of a pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.
Summary
A circle is defined by its centre and radius. Equal chords subtend equal angles and are equidistant from the centre. The perpendicular from the centre bisects a chord. The angle at the centre is twice the angle at the circumference for the same arc. In a cyclic quadrilateral, opposite angles are supplementary.
Important Terms
- Chord: A line segment whose endpoints lie on the circle
- Arc: A continuous piece of a circle between two points
- Segment: Region between a chord and its arc
- Sector: Region between two radii and the arc
- Cyclic Quadrilateral: A quadrilateral with all vertices on a circle
Quick Revision
- Perpendicular from centre bisects the chord
- Equal chords → equal distance from centre → equal angles at centre
- Angle at centre = 2 × angle at circumference (same arc)
- Angle in a semicircle = 90°
- Cyclic quadrilateral: opposite angles sum to 180°