CBSE Class 9 Maths Notes: Circles

Equal Chords and Subtended Angles

Definition: A chord is a line segment connecting two points on a circle. The angle subtended by a chord at the center of the circle is the angle formed by the radii drawn to the endpoints of the chord.

Core Principle: Equal chords of a circle subtend equal angles at the center. Conversely, if the angles subtended by two chords at the center are equal, then the chords are equal.

In simpler terms: If two chords have the same length, the angles they create at the circle’s center are also the same.

Perpendicular Bisector of a Chord

Core Principle: The perpendicular from the center of a circle to a chord bisects the chord. Conversely, the line joining the center of a circle to the midpoint of a chord is perpendicular to the chord.

In simpler terms: A line drawn from the circle’s center that meets a chord at a right angle will always cut the chord in half.

Equal Chords and Distance from the Center

Core Principle: Equal chords of a circle are equidistant from the center. Conversely, chords that are equidistant from the center are equal in length.

In simpler terms: Equal chords are the same distance away from the circle’s center.

Central Angle Theorem

Core Principle: The angle subtended by an arc at the center of a circle is double the angle subtended by it at any point on the remaining part of the circle.

In simpler terms: The central angle (the angle at the center) is twice the size of any angle formed by the same arc at any other point on the circle.

Formula: $\angle AOB = 2 \times \angle ACB$, where A and B are points on the circle, O is the center, and C is any other point on the circle on the major arc.

Angles in the Same Segment

Core Principle: Angles in the same segment of a circle are equal.

In simpler terms: All angles that are made by the same chord on the same side (or segment) of the chord are always the same size.

Condition for Concyclic Points

Definition: Points that lie on the same circle are called concyclic points.

Core Principle: If a line segment joining two points subtends equal angles at two other points lying on the same side of the line segment, then the four points are concyclic.

In simpler terms: If two angles made by a line segment on two different points are the same size, and both points are on the same side of the line segment, then all four points lie on a single circle.

Cyclic Quadrilateral Properties

Definition: A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle.

Core Principles:

• The sum of opposite angles in a cyclic quadrilateral is $180^\circ$ (supplementary).
• The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

In simpler terms: In a cyclic quadrilateral, the opposite angles always add up to 180 degrees, and the outside angle is equal to the inside opposite angle.