THE COLLECTION OF ALL THE POINTS IN A PLANE , WHICH ARE AT A FIXED

DISTANCE FROM A FIXED POINT IN A

PLANE, IS CALLED A CIRCLE

Know all about a Circle

Parts of a circle

Click icon to add picture

O

A B

C

DLine OB and OA are the radii of the circle

AB and CD are chords of the circle

CF is also the chord of the cirle known as DIAMETER

Diameter is the longest ----------------- of the circle

F

Area in green part is known as major sector

Area in minor part is known as -----------------

And the arc comprised in these sectors are respectively known as

Major arc

Minor arc.

Angles made in circle : the angles lying anywhere ON the the circle made by chords is known as SUBTENDED angle ( line AC is the chord)

Angle ABC is subtended angle in circle with centre o

Angle DOE is the central angle as it is making angle at the centre.

Major segment , minor segment and Semicircles

A segment is any region in a circle separated by a chord

Portion in green region is known as the Major segment

Portion in purple color is known as minor segment

What is the segment separated by a diameter known as??

Quick recap of all the terms From the figure aside name the following :

1. Points in the interior of the circle

2. Diameter of the circle

3.Radius of the circle

4.Subtended angle in the circle

5.Central angle in the circle

6.Major sector

7.Minor sector

8.Semicricle

A

BO

CD

Equal chords of a circle subtend equal angles at the centre

Equal chords of a circle subtend equal angles at the centre

Click icon to add pictureGiven: Chord AB = chord DC

To Prove:

angle AOB= angle DOC

Proof:

In Triangle ABC and triangle DOC

AB=DC given

AO=OC radii of same circle

BO=OD radii of same circle

Triangle AOB= Triangle DOC

angle AOB= angle DOC (C.P.C.T)

Hence proved…….

O

A

BC

O

D

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

Click icon to add pictureGiven :

Angle AOB= angle COD

To prove:

chord AB= Chord CD

Proof:

In triangle AOB and triangle COD

Angle AOB= angle COD (given )

AO=OC radii of same circle

BO=OD radii of same circle

Triangle AOB= Triangle DOC

chord AB= Chord CD

A B`

C

D

O

The perpendicular from the centre of the circle bisects the chord.

Click icon to add pictureGiven :

OD perpendicular AB

To prove:AD=DB

Proof:

In triangle AOD and triangle DOB

OA=OB radius

OD=OD common side

Angle ODA=angle ODB (90 degrees.)

Triangle AOD=ODB

(R-H-S test)

AD=DB ( C.P.C.T)

O

A BD

The line drawn through the centre of a circle to bisect the chord is perpendicular to the chord

Click icon to add pictureGiven : AD=DB

To prove: OD perpendicular AB

Proof:

In triangle AOD and triangle DOB

OA=OB radius

OD=OD common side

AD=DB given

triangle AOD = triangle DOB S-S-S test

Angle ODB=OAD (C.P.C.T)

Angle ODB+angle OAD=180 linear pair

Angle ODB= ½ angleADB

Angle ODB=90

O

A BD

Circle through 1,2,3, points

On a sheet of paper try drawing circle through one point

Two points Three pointsWhat do you see?

Answers

Many circles can be drawn from one pointMany circles can be drawn from two points But one and only one circle can be drawn

from three points.

The length of the perpendicular from a point to a line is the (shortest) distance of the line from the centre

Click icon to add pictureTry naming them and proving it.

OD is perpendicular to the line

Others are all hypotenuse

In a right angle triangle hypotenuse is the longest side…

So ………………………………………….

O

D

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

Click icon to add pictureGiven: AB=CD

To prove: OF=OE

Draw OF perpendicular to OE

OOOA

B

C

E

D

O

F

Pick statements in proper order to prove the theorem and match the reasons

Statements AF=FB

AF=1/2AB CE=ED

CE=1/2CD CE=AF

Chord AF=chord CEOA =OCOB=ODIn triangles AOF and OCETriangles congruent byAngle F= Angle EOF=OE

Reasons Radii of same circleC.P.C.TGivenRadii of same circle S-S-S test S-A-S testEach 90 degrees

Chords Equidistant from the centre of a circle are equal in length

(converse of the earlier theorem)

Try proving this…………………..

Have fun

Concentric circles : Circle with same centre are

known as concentric circ

ooo

Angles Subtended by an Arc of a chord.

Click icon to add pictureThe angels subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle

Angle . AMB is half of angle AOB

Angle AOB= angle of arc ACB

Angle AMB= ½ of arc AMB

o

A B

C

M

Angles in the same segment of a circle are equal

Click icon to add pictureAngles ADB

ACB

AEB

All lie in arc AMB

Hence all are equal to ½ arc AMB

So angle

ADB =ACB=AEB=1/2 arc AMB

AB

C

D

E

M

Cyclic Quadrilaterals

A Quadrilateral whose 4 corners are on sides of the circle is known as cyclic Quadrilateral

Properties of Cyclic Quadrilateral

1. the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees

If the sum of opposite angles of a quadrilateral is 180 degrees its cyclic quadrilateral.