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Ganita Manjari • Class 9 • Ch-5 Circles ⬅Chapter Hub
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GANITA MANJARI • CLASS 9 • CHAPTER 5
01

I’m Up and Down,
and Round and Round

An exploration of circles — definitions, symmetries, chords, central angles and concyclicity

Designed with Dedication & Love For Mathematics

Akash Srivastva, TGT (Mathematics)

Ganita Manjari • Class 9 • Chapter 5 Akash Srivastva | www.akashmaths.online
INTRODUCTION • CIRCLES IN NATURE
02

Origins of Circular Shapes

Nature

Raindrop ripples, sunflower seed heads, full moon, solar eclipse

History & Art

Gudahandi cave paintings (Odisha) show early circular motifs

Think & Reflect

List 5 circular objects from daily life. What common property do all of them share?

Book p.92 Akash Srivastva | www.akashmaths.online
5.1 DEFINITIONS
03

What is a Circle?

A circle is the set of all points in a plane that are equidistant from a fixed point (the centre). It is also called the locus of points equidistant from a given point.
Centre
Fixed point
Radius
Distance from centre
Chord
Line joining two points

A chord that passes through the centre is called a diameter.

Book p.93 Akash Srivastva | www.akashmaths.online
5.2 SYMMETRIES OF A CIRCLE
04

Perfect Symmetry

Rotational Symmetry

A circle looks the same when rotated by any angle about its centre (infinite order).

Reflection Symmetry

Every diameter is a line of reflection symmetry. Folding along any diameter gives exact halves.

Think & Reflect

How many lines of reflection symmetry does a regular hexagon have? What about a square?

Book p.94 Akash Srivastva | www.akashmaths.online
5.3 HOW MANY CIRCLES?
05

Circles through Two Points

Infinitely many circles can pass through any two distinct points A and B. Their centres lie on the perpendicular bisector of AB.
The smallest circle through A and B has AB as diameter (radius = ½ AB).
Book p.94–95 Akash Srivastva | www.akashmaths.online
5.3 CIRCUMCIRCLE OF A TRIANGLE
06

Three Non-collinear Points

Exactly one unique circle passes through any three non-collinear points. This is the circumcircle of the triangle formed by them.
Acute
Centre inside
Right
Centre at midpoint of hypotenuse
Obtuse
Centre outside
Book p.95 Akash Srivastva | www.akashmaths.online
5.4 EQUAL CHORDS & CENTRAL ANGLES
07

Theorems 2 & 3

Theorem: Equal chords of a circle subtend equal angles at the centre.
Converse: Chords that subtend equal angles at the centre are equal.

Proved using SAS congruence of the two triangles formed by radii and the chords.

Book p.96 Akash Srivastva | www.akashmaths.online
5.5 PERPENDICULAR FROM CENTRE TO A CHORD
08

Theorems 4 & 5

The line from the centre to the midpoint of a chord is perpendicular to the chord.
Converse: The perpendicular from the centre to a chord bisects the chord.

Proved by RHS congruence of the two right triangles formed.

Book p.97 Akash Srivastva | www.akashmaths.online
5.6 DISTANCE OF CHORDS FROM CENTRE
09

Equal Chords are Equidistant

Chords of equal length are equidistant from the centre (and conversely).
Chord length formula: L = 2√(r² − d²)
where d = perpendicular distance from centre.

Longer chords are closer to the centre. Diameter is the longest chord (d = 0).

Book p.98 Akash Srivastva | www.akashmaths.online
5.7 ARCS OF A CIRCLE
10

Minor Arc, Major Arc & Semicircle

Minor Arc
Central angle < 180°
Major Arc
Central angle > 180°
Semicircle
Central angle = 180°

Two points on a circle divide it into two arcs.

Book p.99 Akash Srivastva | www.akashmaths.online
5.7 CENTRAL ANGLE THEOREM
11

Theorem 9

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.

∠AOB = 2 × ∠ACB

(where O is centre and C is on the circumference)

Book p.100 Akash Srivastva | www.akashmaths.online
5.7 ANGLE IN A SEMICIRCLE
12

Corollary

The angle subtended by a diameter at any point on the circle is always a right angle (90°).

Think & Reflect

If one side of a triangle is the diameter of its circumcircle, what type of triangle is it always?

Book p.100–101 Akash Srivastva | www.akashmaths.online
5.7 ANGLES IN THE SAME SEGMENT
13

Corollary

Angles subtended by the same arc at any points on the remaining part of the circle are equal.

All such angles equal half the central angle of that arc.

Book p.101 Akash Srivastva | www.akashmaths.online
5.8 CONCYCLIC POINTS
14

When do points lie on a circle?

Points are called concyclic if they all lie on the circumference of the same circle.
Any three non-collinear points are always concyclic. Four or more points are concyclic only if they satisfy special angle conditions.
Book p.102 Akash Srivastva | www.akashmaths.online
5.8 CYCLIC QUADRILATERAL
15

Theorem 11

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

Think & Reflect

Can a parallelogram be cyclic? Only if it is a rectangle (or square). Why?

Book p.102–103 Akash Srivastva | www.akashmaths.online
5.8 CONVERSE THEOREM
16

Theorem 12

If the sum of a pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.

This gives a practical test to check whether four points lie on a circle.

Book p.103 Akash Srivastva | www.akashmaths.online
REAL-WORLD APPLICATIONS
17

Where do we use these ideas?

Finding the centre of a broken circular object (using perpendicular bisectors of two chords)
Designing athletic tracks and curved lanes
Architecture of circular arches and domes
Navigation and surveying problems
Applications Akash Srivastva | www.akashmaths.online
KEY TAKEAWAYS
18

Master Summary

Circle = locus of points equidistant from centre
Infinite rotational + reflection symmetry
Equal chords ↔ equal central angles
Perpendicular from centre bisects chord
Angle at centre = 2 × angle at circumference
Angle in a semicircle = 90°
Angles in the same segment are equal
Opposite angles of cyclic quad sum to 180°
Chapter 5 Summary Akash Srivastva | www.akashmaths.online
GANITA MANJARI • CLASS 9 • CHAPTER 5
19

Thank You

Circles teach us that beauty and symmetry can be found in perfect balance.

“The circle is the most perfect of all geometric figures.”

— Euclid

Designed with Dedication & Love For Mathematics

Akash Srivastva, TGT (Mathematics)

Ganita Manjari • Class 9 Akash Srivastva | www.akashmaths.online
01 / 19