CBSE Mathematics 2026–27 · Topic 11

Circles

Understand the idea, practise the reasoning and apply it in an investigation. These original notes follow the regular Class IX Mathematics syllabus; the Advanced course is separate.

What to learn

Be able to explain the key result, choose a method that fits the question, show reasoning clearly and verify the answer.

Core concepts

A chord joins two points on a circle. A perpendicular from the centre to a chord bisects it; equal chords are equidistant from the centre and subtend equal central angles.
An angle subtended by a diameter at the circumference is a right angle. Angles in the same segment are equal.
Opposite angles of a cyclic quadrilateral are supplementary. Identify the theorem’s conditions before applying it; a diagram is a guide, not proof.

How to choose and justify a circle theorem

A circle is the set of points at a fixed distance (the radius) from its centre. A chord joins two points on the circle; a diameter is a chord through the centre and is the longest chord. An arc is part of the circumference, a sector is bounded by two radii and an arc, and a segment is bounded by a chord and its arc.

To prove that the perpendicular from the centre to a chord bisects it, let OM be perpendicular to chord AB. The right triangles OMA and OMB have equal hypotenuses OA and OB (radii) and the common side OM. They are congruent by RHS, so AM=MB. This proof explains why the result holds; a drawing alone cannot establish it.

For a fixed circle, longer chords lie closer to the centre. If a chord has length c and its perpendicular distance from the centre is d in a circle of radius r, the right-triangle relation gives (c/2)²+d²=r². This lets you compare chord length and distance without measuring the diagram.

An arc subtends twice as large an angle at the centre as at a point on the remaining circumference. Therefore angles standing on the same chord in the same segment are equal, while a diameter subtends 180° at the centre and 90° at the circumference. Four points are concyclic when they lie on one circle; a quadrilateral is cyclic exactly when a pair of opposite angles is supplementary.

Worked reasoning: In cyclic quadrilateral ABCD, if ∠A=2x+8° and ∠C=3x−3°, then (2x+8)+(3x−3)=180, so 5x=175 and x=35. The angles are 78° and 102°. Name the cyclic condition before using the supplementary-angle rule.

Check: If a theorem needs parallel lines, a right angle, equal radii, or concyclic points, identify that condition in the question or prove it first.

Original CBSE-style solved examples

Attempt each calculation before reading the solution.

Example · cyclic quadrilateral

Opposite angles are (2x+10)° and (3x−20)°. They sum to 180°: 5x−10=180, so x=38. The angles are 86° and 94°.

Example · chord bisector

A circle has radius 13 cm. A perpendicular from the centre meets a chord 5 cm from the centre. The half-chord is √(13²−5²)=√144=12 cm, so the full chord is 24 cm.

Practice questions with answers

1. A cyclic quadrilateral has one angle of 104°. Find its opposite angle.

180°−104°=76°.

2. A diameter subtends an angle at a point on the circle. Find it.

90°.

Practical investigation

Draw a circle and a chord. Join the centre to the chord midpoint, construct a perpendicular and measure. Repeat with a second chord; record what stays true.

Keep a record of materials, steps, observations and conclusion.

Self-check

Can I state the definition or theorem precisely?
Can I explain why the method applies?
Did I check signs, units, labels and the final result?

Use your prescribed NCERT textbook and teacher’s sequence. These are original study notes, not copied textbook answers.