CBSE Class 9 Maths — Circles

Chapter Overview and Weightage

Circles is a geometry chapter in Class 9 Maths that consistently carries 5–8 marks in CBSE annual exams. Questions range from 1-mark definition/theorem identification to 4-mark proof questions. This chapter is also foundational for Class 10 Circles (tangents) and Class 11/12 coordinate geometry.

Key Concepts You Must Know

Basic Terms

Circle: Locus of all points in a plane equidistant from a fixed point (centre). The fixed distance is the radius (r).

Chord: A line segment joining any two points on the circle. The diameter is the longest chord.

Arc: A part of the circle between two points. A minor arc is smaller than a semicircle; a major arc is larger.

Sector: Region between two radii and the arc between them.

Segment: Region between a chord and the arc it cuts off. A minor segment is below the chord (minor arc side); a major segment is above.

Cyclic quadrilateral: A quadrilateral whose all four vertices lie on a circle.

Priority of Concepts for Exam

1. Equal chords and their distances from the centre
2. Angle subtended by an arc at centre vs circumference
3. Angles in the same segment
4. Cyclic quadrilateral properties
5. Angle in a semicircle

Important Theorems (All Provable in Exams)

Understanding Theorem 3 (Most Important)

If arc AB subtends angle AOB at centre O and angle ACB at any point C on the major arc:

Three cases:

• When C is on the major arc (arc AB is minor)
• When C is on the minor arc (arc AB is major)
• When O lies on AC (reflex angle at centre case)

All three cases lead to the same $2\theta$ relationship.

Cyclic Quadrilateral Theorem

If ABCD is a cyclic quadrilateral (all vertices on circle):

Converse: If the sum of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.

Solved Previous Year Questions

Question 1: O is the centre of a circle and AB is a chord. If OM ⊥ AB and OM = 4 cm, AB = 6 cm, find the radius of the circle.

Solution:

Since OM ⊥ AB and O is the centre, OM bisects AB (perpendicular from centre bisects chord).

$AM = \dfrac{AB}{2} = \dfrac{6}{2} = 3\ \text{cm}$

In right triangle OMA: $OA^2 = OM^2 + AM^2 = 4^2 + 3^2 = 16 + 9 = 25$

$OA = 5\ \text{cm}$

Radius = 5 cm

Question 2: In the figure, O is the centre of a circle. If ∠AOC = 100°, find ∠ABC (where B is on the major arc).

Solution:

The reflex angle ∠AOC = 360° – 100° = 260° (taking the major arc side).

By Theorem 3 (angle at centre = 2 × angle at circumference):

∠ABC = 130°

Alternatively: ∠ABC = 180° – (∠AOC / 2) = 180° – 50° = 130°.

Question 3: ABCD is a cyclic quadrilateral in which AC and BD are diameters. If ∠DAC = 35°, find ∠DBC.

Solution:

Since AB is a chord and both ∠DAC and ∠DBC are angles subtended by arc DC on the same side:

$\angle DAC = \angle DBC$ (angles in the same segment, both subtend arc DC)

∠DBC = 35°

Difficulty Distribution

Expert Strategy

The most efficient approach for this chapter:

1. Master the three main theorems (angle-arc relationship, angles in same segment, cyclic quadrilateral). These cover 80% of questions.

2. Always draw a diagram. Circle geometry problems are nearly impossible to solve without a figure. Even if the question doesn’t ask you to draw, always sketch.

3. Name angles systematically. Use letters to name each angle and write the theorem name when applying it. Examiners award marks for citing the theorem.

4. For proofs: Memorise the structure (given → to prove → construction → proof). The perpendicular from centre bisecting chord proof uses congruence (RHS criterion). Know it word for word.

Common Traps