Vikasana - CET 2012

CIRCLE

Vikasana - CET 2012

The Collection of allThe Collection of all points in a plane whichpoints in a plane which are at fixed distance from a fixed point is constant

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Th fi d i t i ll dThe fixed point is called the centre and fixedthe centre and fixed distance is called radius. The plural of radius is “ dii”“radii”

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A circle divides the plane in toA circle divides the plane in to three parts. They are i) interior circle (ii) circle (iii) Exterior circlecircle

Exterior

Interior

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Chord: A chord of a circle is aChord: A chord of a circle is a line segment joining at two points on the circle.In this figure PQ RS and AOBIn this figure PQ, RS and AOB are the chords

P Q

B

SR

A O

Vikasana - CET 2012SR

A diameter is a chord of a circle passing through the centrepassing through the centre .Diameter is the longest chord Di t 2 diDiameter = 2 radius

BAO

BA

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Arc of a circle: A continuous piece of aA continuous piece of a circle is called an arc of acircle is called an arc of a circle

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We observe there are twoWe observe there are two pieces one longer which is called major arc and othercalled major arc and other smaller is called the minor arc

QP

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Semi circle: A diameter of a circle divides it into two equalcircle divides it into two equal arcs. E h f th t i ll dEach of these two arcs is called semicircle

OA B

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The length of complete circle is calledcircle is called circumference which iscircumference which is equal to 2πr

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Segment: The region between a g gchord and a arc is called segment. The segment containing the minorThe segment containing the minor arc is called minor segment. The segment containing the major arcsegment containing the major arc is called major segment

Majorsegment

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Sector of a circle: The region enclosed by an arc of a circle and its two boundaryof a circle and its two boundary radii is called sector

rrr

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Cyclic Quadrilateral :Cyclic Quadrilateral : A quadrilateral ABCD is said to be c clic if all its

B

said to be cyclic if all its vertices lie on a circle. A

C

Points lying on a circle are said to be concyclic

D

are said to be concyclic.

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Congruency of circleCongruency of circle

Two circles arc congruent if and only if they have equaland only if they have equal radii

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Theorem: The perpendicularTheorem: The perpendicular from the centre of a circle bi t th h dbisect the chord

OL ⊥r AB AThen LA = LB O

LThe converse is true

B

L

Vikasana - CET 2012B

Equal chords of circle subtendEqual chords of circle subtend equal angles at the centre.

AB CDAB = CD Then ∠AOB = ∠COD

and converse is trueB

Cand converse is true A

O

Vikasana - CET 2012A

D

There is only one circle passing through three given non collinear points. Apoints.

This circle is called BThis circle is called the circum circle. The centre and

di ll d i t

BC

radius are called circum centre and circum radius.

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Equal chords of a circle areEqual chords of a circle are equidistance from the centre If

AB = CDAB = CD Then OL = OM Th i t

A CThe converse is true O

L M

DBVikasana - CET 2012

DB

If two chords of a circle are equal, then their corresponding arcs arecorresponding arcs are congruent and conversely if two arcs

are congruent, then theirare congruent, then their corresponding chords are equal.

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The angle subtendedA The angle subtended by an arc at thecentre is double the

Ocentre is double the angle subtended by P Q

it at any point on the remaining part of the circle i.e. p

∠POQ = 2∠PAQ

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Angles in the same segment ofAngles in the same segment of a circle are equal

A B

∠PAQ = ∠PBQ A B

P Q

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Angle in a semicircle is aAngle in a semicircle is a right angle

APB AQB ARB 900∠APB = ∠AQB =∠ARB =900

P Q

A BR

A

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The sum of either pair of opposite angles of a cyclic Quadric lateral is g y Q1800, and the converse is true∠A +∠C=1800 ∠B +∠D=1800∠A +∠C=180 ∠B +∠D=180

D

CA

Vikasana - CET 2012B

A tangent to a circle is a lineA tangent to a circle is a line that intersect the circle at onl one point There is onlonly one point. There is only one tangent at a point of gcircle.

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If the line intersect a circle in two distinct points, then it is p ,called “secant”, of the circle The tangent is acircle. The tangent is a

special case of the secant when the two end points of itsthe two end points of its corresponding chord coincide

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The tangent at any point of aThe tangent at any point of a circle is perpendicular to the

di th h th i t fradius through the point of contact. OP ⊥ AB

O

A BP

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There is no tangent to a circleThere is no tangent to a circle passing through a point lying inside the circle.

P

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There is one and only oneThere is one and only one tangent to a circle passing through a point lying on the circlecircle

P

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There are exactly two tangentsThere are exactly two tangents to a circle through a point lying

t id th i l Th l thout side the circle. The length PT is called length of tangent g g

T1

PP

Vikasana - CET 2012T2

The length of tangents drawn from an external point to afrom an external point to a circle are equal

T1

PP

Vikasana - CET 2012T2

Area of the circle = πr2

When the degree measure of theWhen the degree measure of the angle at the centre is θ, then area of the sector 2θof the sector 2r

360θ

= ×π

O θ

Vikasana - CET 2012BA

We know thatWe know thatCircumference = 2πrL th f f t fLength of an arc of sector of angle θ is 2 rθ

× πg0 2 r

360× π

θ

O

Vikasana - CET 2012BA

Vikasana - CET 2012