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Circles

10 Circles

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.

Radius(r):

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
    the radius of the circle.

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.

For semi circle angle formed by arcs at centre is  180 .

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

To prove :

AOB = COD

Proof :

In AOB  and  COD

AB = CD …..(Given)

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

BO = DO ……………(  Radii of same circle)

AOB  COD ................(SSS rule) So, AOB = COD .................(CPCT)

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.

AOB = COD

To prove :

AOB = COD

Proof :

In AOB  and  COD AOB = COD ..............(Given)

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

BO = DO ……………(  Radii of same circle)

AOB  COD ................(SOS rule)  AB = CD .................(CPCT)

Theorem 3

Perpendicular from centre to a chord bisects the chord.

Given:

O is centre of the circle.

AB is chord.

OM  AB

To prove :

OM bisects AB  i.e.  AM = BM

Proof :

In right AOM and BOM

AO = BO   ……….(Radii)

OM = OM   ………..(Common)

AMO = BMO AOM = BOM  . (RH S rule)

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,

 AM = BM

To prove :

OM  AB

Proof:

 In  AOM = BOM

AO = BO ………(Radii)

OM = OM…….   (Common_

AM = BM………(Given)

AOM = BOM   (SSS  rule) So AMO = BMO (CPCT)

But

AMO + BMO = 180  (Linear pair) 2 AMO = 180  AMO = 900

 

So   OM  AB

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

OM  AB ,  ON  CD

To Prove:

ON = OM

Proof:

N, M are midpoints of  AB and CD  (Perpendicular from centre to a chord bisects the   the chord)
 AN = 12 AB
CM = 12 CD

But, AB = CD

 AN = CM ...(1)
Inright OCM and OAN

AN = CM …(From 1)

OA = OC …(Radii)

OCM  OAN .....(R.H.S rule)
 ON = OM .....(CPCT)

Theorem 7 : Converse:

Chords equidistant from centre are equal.

Given:

AB and CD are two chords.

OM  AB ,  ON  CD

ON = OM (As chords are equidiastantfrom the centre)

To prove:

AB = CD

Proof:

 Inright  OCMand OAN

ON = OM  …(Given)

OA = OC  …(Radii)

OCM  OAN ...... (R H S rule)

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