Circles

Board: GSEB Class: 9

Chapter Overview

A circle is the set of all points equidistant from a fixed point (center). A chord is a line segment joining two points on the circle. Equal chords subtend equal angles at the center. The perpendicular from the center to a chord bisects the chord. The angle subtended by an arc at the center is twice the angle subtended at any point on the remaining part of the circle. Angles in the same segment are equal. The angle in a semicircle is 90°. A cyclic quadrilateral has all vertices on the circle, and opposite angles sum to 180°.

Topics Covered

Circle Definition
Chords and Arcs
Equal Chords Theorems
Perpendicular from Center
Angle at Center Theorem
Angles in Same Segment
Angle in Semicircle
Cyclic Quadrilateral

Key Formulas

Equal chords => equal angles at center

Perpendicular from center bisects chord

Angle at center = 2 x angle at circumference

Angles in same segment are equal

Angle in semicircle = 90°

Cyclic quadrilateral: A + C = 180°, B + D = 180°

Real-World Applications

Applications: Wheel mechanics, satellite orbits, clock design, gear systems, architectural arches.

Study Tips

Tip: Practice theorem proofs step by step

Tip: Draw accurate circle diagrams for each theorem

Tip: Learn cyclic quadrilateral properties well