Circles - Maths project

A power point presentation on the topic Circles for Maths class.

1. MATHS PROJECT
TOPIC
CIRCLES
M
.RAMKI
‘ IX’’

2. INTRODUCTION
Youmayhavecomeacrossmanyobjectindailylife,
whichareroundinshape,suchaswheelsofa
vehicle,clocks,coins,buttonsofashirt,etc.Ina
clock,thesecond’shandgoesroundthedialofthe
clockrapidlyanditstipmovesinaroundpath.This
pathtracedbythetipofthesecond’shandiscalled
aCIRCLE.

3. Circles and Its Related Terms
Definition:
Thecollectionofallthepointsinaplane,whichareatafixed
distancefromafixedpointintheplain,iscalledacircle.
Thefixedpointiscalledthecentreofthe
circleandthefixeddistanceiscalledthe
radiusofthecircle.

4. Acircledividestheplainonwhichitliesinto
threeparts.Theyare:
(i)Insidethecircle,whichisalsocalledthe
interiorofthecircle.
(ii)Thecircle.
(iii)Outsidethecircle,whichisalsocalledthe
exteriorofthecircle.
Thecircleanditsinteriormakeupthecircular
region.

5. The upcoming activities will make you
know more about the TERMS and
DEFINITIONS related to CIRCLES.

6. Activity---------------------------------1
TaketwopointsP&Qonacircle,
thenthelinesegmentPQiscalled
achordofacircle.Thechord,which
passesthroughthecentreofthe
circle,iscalledadiameterofthe
circle.Asinthecaseofradius,the
word‘diameter’isalsousedintwo
senses,ie.,asalinesegmentand
alsoasitslength.Inthefigure
aside,AOBisadiameterofthe
circle.
o
Figure
A
P
B
Q

7. Activity--------------------------2
Apieceofacirclebetween
twopointsiscalledanarc.
Lookatthepiecesofthe
circlebetweentwopointsP
&Qinfigure.Thelonger
pieceiscalledthemajorarcPQ
andtheshorterpieceis
calledtheminorarcPQ.When
P&Qareendsofadiameter,
thenbotharcsareequaland
eachiscalledasemicircle.
.
R
Major arc PQ
Minor arc PQ
P Q
Figure

8. Thelengthofthecompletecircleiscalledits
circumference.Theregionbetweenachord
andeitherofitsarcsiscalledasegmentof
thecircularregionofthecircle.Thereare
twotypesofsegmentsalso,whicharethe
majorsegmentandtheminorsegment(fig1).
Theregionbetweenanarcandthetworadii,
joiningthecentretotheendpointsofthe
arciscalledasector.Likesegments,the
minorarccorrespondstotheminorsector
andthemajorarccorrespondstothemajor
sector.Infig2,theregionOPQistheminor
sectorandremainingpartofthecircular
regionisthemajorsector.Whentwoarcsare
equal,ie.,eachisasemicircle,thenboth
segmentsandbothsectorsbecomethesame
andeachisknownasasemicircularregion.
P Q
Figure 1
P Q
Figure 2
Major
segment
Minor
segment
Major sector
O
Minor sector

9. AngleSubtendedbyaChordata
Point
TakealinesegmentPQandapointRnot
onthelinecontainingPQ.JoinPRand
QR.Then<PRQiscalledtheangle
subtendedbythelinesegmentPQatthe
pointR.<POQistheanglesubtendedby
thechordPQatthecentreO,<PRQand
<PSQarerespectivelytheangles
subtendedbyPQatpointsR&Sonthe
majorandminorarcsPQ.
.R
. O
P Q
.
S
Figure

10. THEOREMS OFCIRCLES
SOME OF THE THEOREMSWHICH YOU
WILL BE SEEING IN THE FOLLOWING
SLIDES WILL MAKE YOUUNDERSTAND
MORE ABOUT CIRCLES.

11. Theorem -1
Equal chords of a circle
subtend equal angles at the
centre.

12. Theorem -2
If the angle subtended by
the chords of a circle at
the centre are equal, then
the chords are equal.

13. Theorem -3
The perpendicular from the
centre of a circle to a
chord bisects the chord.

14. Theorem -4
The line drawn through
the centre of a circle
to bisect a chord is
perpendicular to the chord.

15. Theorem -5
There is one and only one
circle passing through three
given non-collinear points.

16. Theorem -6
Equal chords of a circle
(or of congruent circles ) are
equidistant from the centre.

17. Theorem -7
Chords equidistant from
the centre of a circle
are equal in length.

18. Theorem -8
The angle subtended by
an arc at the centre is
double the angle subtended
by it at any point on the
remaining part of the
circle.

19. Theorem -9
Angles in the same
segment of a circle are
equal.

20. Theorem -10
If a line segment joining two
points subtends equal angles at
two other points lying on the
same side of the line
containing the line segment,
the four points lie on a
circle, ie., they are concyclic.

21. Theorem -11
The sum of either pair
of opposite angles of a
cyclic quadrilateral is 180°.

22. Theorem -12
If the sum of a pair of
opposite angles of a
quadrilateral is 180°, the
quadrilateral is cyclic.

23. A JOKE ON CIRCLES
THERE IS A JOKE ON CIRCLES WHICH IS
VERY INTERESTING.
I THINK YOU ALL WILL
LIKE IT
AND
ENJOY IT.

24. HAVE YOU ALL SEEN A BILLIARDS GAME, THE BALLUSED
IN IT IS CIRCLEIN SHAPE. I THINK YOU ALL KNOW THIS,
BUT HAVE YOU SEEN A HEN PLAYING IT WITHOUT A STICK.
DO YOU WANT TO SEE IT ?
COMELET’STAKEALOOKATIT.