 # Circles

## 10 Circles

#### Circle :

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

#### Centre:

The fixed point is called centre of the circle.

The fixed distance is called the radius of the circle.

#### Interior region of circle:

It is the part of plane which is within the circle.

•  The distance between any point in interior region and centre is less than

#### Exterior region of the circle:

The part of plane which is out side the circle.

• The distance between any point in exterior region and centre is
greater than the radius of the circle.

#### Chord :

The line segment joining any two points of the circle is called chord.

#### Diameter(d) :

The diameter is chord passing through the centre.
It is the longest chord of the circle.
Diameter = 2 X r

#### Arc :

A piece of circle between any two points is called arc.

#### Central angle :

The angle formed at centre by joining end points of Chord /arc and centre is called centre angle of arc/chord.

#### Minor arc :

If central angle of an arc is acute ,the arc is called minor arc.

#### Major arc :

If central angle of an arc is obtuse, the arc is called major arc.

#### Semicircles :

The diameter divides part of circles in two equal arcs, called semicircles.

#### Segment :

The interior part of circle, the region between chord and arc is called segment.

#### Major segment :

The region between major arc and chord is called major segment.

#### Minor segment :

The region between minor arc and chord is called minor segment.

#### Sector :

The region between two radii and an arc is called sector.

#### Major Sector :

The region between two radii and an arc is called sector.

#### Minor Sector :

The region between minor arc and radii is called minor sector.

#### Theorem 1

Equal chords of a circle subtend equal angles at the centres.

#### Given :

O is centre of the circle.
Chord AB = chord CD

#### Proof :

AB = CD …..(Given)

AO = CO …………….(  Radii of same circle)

BO = DO ……………(  Radii of same circle)

#### Theorem 2

If angles subtended by the chord of a circle at the centre are equal, then the Chords are equal.

#### Given :

O is centre of the circle.

#### Proof :

AO = CO …………….(  Radii of same circle)

BO = DO ……………(  Radii of same circle)

#### Theorem 3

Perpendicular from centre to a chord bisects the chord.

Given:

O is centre of the circle.

AB is chord.

#### To prove :

OM bisects AB  i.e.  AM = BM

#### Proof :

OM = OM   ………..(Common)

So, AM = BM …………….. (CSCT)

#### Theorem 4

The line joining centre and midpoint of chord is perpendicular to chord.

#### Given:

O is centre of the circle.

AB  is chord.

M  is midpoint Of  AB,

#### Proof:

OM = OM…….   (Common_

AM = BM………(Given)

But

#### Theorem 5

There is one and only one circle passing through three non-collinear points.

Remark: There is a unique circle passing through vertices of the triangle.

#### Theorem 6

Equal chords are equidistant from centre.

#### Given:

AB and CD are two chords.

AB = CD

ON = OM

#### Proof:

But, AB = CD

AN = CM …(From 1)

#### Theorem 7 : Converse:

Chords equidistant from centre are equal.

#### Given:

AB and CD are two chords.

ON = OM (As chords are equidiastantfrom the centre)

#### To prove:

AB = CD

Proof:

ON = OM  …(Given)