Vikasana - CET 2012kea.kar.nic.in/vikasana/bridge/maths/chap_10_ppt.pdf · called major arcmajor...
Transcript of Vikasana - CET 2012kea.kar.nic.in/vikasana/bridge/maths/chap_10_ppt.pdf · called major arcmajor...
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
Vikasana - CET 2012
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”
Vikasana - CET 2012
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
Vikasana - CET 2012
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
Vikasana - CET 2012
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
Vikasana - CET 2012
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
Vikasana - CET 2012
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
Vikasana - CET 2012
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
Vikasana - CET 2012 minor
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
Vikasana - CET 2012
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.
Vikasana - CET 2012
Congruency of circleCongruency of circle
Two circles arc congruent if and only if they have equaland only if they have equal radii
Vikasana - CET 2012
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.
Vikasana - CET 2012
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.
Vikasana - CET 2012
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
Vikasana - CET 2012
Angles in the same segment ofAngles in the same segment of a circle are equal
A B
∠PAQ = ∠PBQ A B
P Q
Vikasana - CET 2012
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
Vikasana - CET 2012
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.
Vikasana - CET 2012
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
Vikasana - CET 2012
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