CBSE 9th Maths



The collection of all points in a plane which are at a fixed distance from a fixed point in the plane is called a circle.

Chord -

The line segment within the circle joining any 2 points on the circle is called the chord.


– A Chord passing through the centre of the circle is called the diameter. – The Diameter is 2 times the radius and it is the longest chord.


– The portion of a circle(curve) between 2 points is called an arc. – Among the two pieces made by an arc, the longer one is called a major arc and the shorter one is called a minor arc.


The perimeter of a circle is the distance covered by going around its boundary once. The perimeter of a circle has a special name: Circumference, which is π times the diameter which is given by the formula 2πr


A diameter of a circle divides it into two equal parts which an arc. Each of these two arcs is called a semi-circle.

Congruent Circles (Arc):

Two circles are said to be congruent if and only if either of them can be superposed on the other so as to cover exactly.

Cyclic Quadrilateral:

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle.

Common Chord:

The intersection point of two circles is the common chord of the circle.

Segment and Sector

– A circular segment is a region of a circle which is “cut off” from the rest of the circle by a secant or a chord. – Smaller region cut off by a chord is called minor segment and the bigger region is called major segment. –

-A sector is the portion of a circle enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector.

– For 2 equal arcs or for semicircles – both the segment and sector is called the semicircular region.

Tangent and Secant

line that touches the circle at exactly one point is called it’s tangent. A line that cuts a circle at two points is called a secant.

Circles and Their Chords

Theorem of equal chords subtending angles at the centre.

Proof: AB and CD are the 2 equal chords.

In Δ AOB and Δ COD

OB = OC [Radii]

OA = OD [Radii]

AB = CD [Given]

ΔAOB ≅ ΔCOD (SSS rule)

Hence, ∠AOB = ∠COD [CPCT]

Theorem of equal angles subtended by different chords.

– If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

Proof: In ΔAOB and ΔCOD

OB = OC [Radii] ∠AOB=∠COD [Given]

OA = OD [Radii]

ΔAOB ≅ ΔCOD (SAS rule)

Hence, AB=CD [CPCT]

Perpendicular from the centre to a chord bisects the chord.

Perpendicular from the centre of a circle to a chord bisects the chord.

Proof: PQ is a chord and ST is the perpendicular drawn from the centre.

From ΔORP and ΔORQ,

∠ORP=∠ORQ =900 OP = OQ(radii)

OR = OR(common)

Hence, ΔOCB ≅ ΔOCA (RHS rule)

Therefore AC = CB [CPCT]

A Line through the centre that bisects the chord is perpendicular to the chord.

– A line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Proof: OC drawn from the center to bisect chord AB.

From ΔOCA and ΔOCB,

OA = OB (Radii)

OC = OC (common)

AC = BC (Given)

Therefore, ΔOMA ≅ ΔOMB (SSS rule)

⇒∠OCA=∠OCB (C.P.C.T)

But, ∠OCA+∠OCB=1800

Hence, ∠OCA=∠OCB=900 ⇒OC⊥AB

Circle through 3 points

– There is one and only one circle passing through three given non collinear points. – A unique circle passes through 3 vertices of a triangle ABC called as the circum circle. The centre and radius are called the circum center and circum radius of this triangle, respectively.

Equal chords are at equal distances from the centre.

Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).


NCERT Solution TextBook
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