# CBSE 9th Maths

## Circles

### Circle:

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.

### Diameter

– 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.

### Arc

– 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.

### Circumference

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

### Semi-circle:

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.

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

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]

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,

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,

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).