Circle Theorems _WithProofs_.ppt
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Circle Theorems
Euclid of Alexandria
Circa 325 - 265 BC
O
The library of Alexandria wasthe foremost seat of learninin the world and functionedli!e a uni"ersity# The librarycontained 6$$ $$$manuscri%ts#
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diameter
Circumference
radius
c ho rd
&a'or (ement
&inor (ement
&inor Arc
&a'or Arc
&inor (ector
&a'or (ector
A )eminder about %arts of the Circle
T a n e n t
T a n e n t
Tanent
Parts
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o
Arc AB subtends anle x at the centre#
A B
xo
Arc AB subtends anle y at the circumference#
yo
Chord AB also subtends anle x at the centre#
Chord AB also subtends anle y at the circumference#
o
A
B
xo
yo
o
yo
xo
A
B
*ntroductory Terminoloy Term’gy
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Theorem +
&easure the anles at the centre and circumference and ma!e a con'ecture#
xo
yoxo
yo
xo
yo
xo
yo
xo
yo
xo
yo
xo
yo
xo
yo
oo o
o
o o o o
Th1
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The anle subtended at the centre of a circle ,by an arcor chord is twice the anle subtended at thecircumference by the same arc or chord# ,anle at centre-
2xo2xo 2x
o 2xo
2xo 2xo 2xo 2xo
Theorem +
&easure the anles at the centre and circumference and ma!e a con'ecture#
xo
xo
xoxo
xo xo xo xo
o oo o
o o o o
Anle x is subtended in the minor sement#
.atch for thisone later#
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o
A
B
/0o
xo
Exam%le 1uestions
+
ind the un!nown anles i"in reasons for your answers#
o
A
B
yo
2
35o
02o ,Anle at the centre#
$o,Anle at the centre
anle x 4
anle y 4
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,+/$ 2 x 02 4 6o ,*sos trianle7anle sum trianle#
0/o ,Anle at the centre
anle x 4
anle y 4
o
AB
02o
xo
Exam%le 1uestions
3
ind the un!nown anles i"in reasons for your answers#
o
A
B
%o
0
62o yo
8o
+20o ,Anle at the centre
,+/$ +2072 4 2/$ ,*sos trianle7anle sum trianle#
anle % 4
anle 8 4
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o 9iameter
$o
anle in a semi-circle$o anle in a semi-circle
2$o anle sum trianle
$o anle in a semi-circle
o
a
b
c
$o
d
3$o
e
ind the un!nownanles below statin areason#
anle a 4anle b 4
anle c 4
anle d 4
anle e 4 6$o
anle sum trianle
The anle in a semi-circle is a riht anle#Theorem 2
This is 'ust a s%ecial case of Theorem + and
is referred to as a theorem for con"enience#
Th2
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Anles subtended by an arc or chord inthe same sement are e8ual#Theorem 3
xo xo
xo
xo
xo
yo
yo
Th3
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3/o xo
yo
3$o
xo
yo
0$o
Anles subtended by an arc or chord inthe same sement are e8ual#
Theorem 3
ind the un!nown anles in each case
Anle x 4 anle y 4 3/o Anle x 4 3$o
Anle y 4 0$o
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The anle between a tanent and aradius is $o# ,Tan7rad
Theorem 0
o
Th4
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The anle between a tanent and aradius is $o# ,Tan7rad
Theorem 0
o
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+/$ ,$ : 36 4 50o Tan7rad and anle sum of trianle#
$o anle in a semi-circle
6$o anle sum trianle
anle x 4
anle y 4
anle ; 4
T
o
36oxo
yo
;o
3$o
A
B
*f OT is a radius and AB is atanent< find the un!nownanles< i"in reasons for youranswers#
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The Alternate (ement Theorem#Theorem 5
The anle between a tanent and a chord throuh the %oint ofcontact is e8ual to the anle subtended by that chord in thealternate sement#
xo
xo
yo
yo
05o ,Alt (e
6$o ,Alt (e
5o anle sum trianle
0 5 o
x o
y o
6 $ o
; o
ind the missin anles belowi"in reasons in each case#
anle x 4
anle y 4
anle ; 4 Th5
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Cyclic 1uadrilateral Theorem#Theorem 6
The o%%osite anles of a cyclic 8uadrilateral are su%%lementary#,They sum to +/$o
w
x
y
;
Anles x : w 4 +/$o
Anles y : ; 4 +/$o
8
%
r
s
Anles % : 8 4 +/$o
Anles r : s 4 +/$o
Th6
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+/$ /5 4 5o ,cyclic 8uad
+/$ ++$ 4 $o ,cyclic 8uad
Cyclic 1uadrilateral Theorem#Theorem 6
The o%%osite anles of a cyclic 8uadrilateral are su%%lementary#,They sum to +/$o
/5o
++$o
x
y
$o
+35o%
r
8
ind the missin
anles belowi"en reasons in
each case#
anle x 4
anle y 4
anle % 4
anle 8 4
anle r 4
+/$ +35 4 05o ,straiht line
+/$ $ 4 ++$o ,cyclic 8uad
+/$ 05 4 +35o ,cyclic 8uad
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Two Tanent Theorem#Theorem
rom any %oint outside a circle only two tanents can be drawn andthey are e8ual in lenth#
=
T
>1
)
=T 4 =1
=
T
>
1
)
=T 4 =1
Th7
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$o ,tan7rad
Two Tanent Theorem#Theorem
rom any %oint outside a circle only two tanents can be drawn andthey are e8ual in lenth#
= T
1Oxo
wo
/o
yo
;o
=1 and =T are tanents to a circle with centreO# ind the un!nown anles i"in reasons#
anle w 4
anle x 4
anle y 4
anle ; 4
$o ,tan7rad
0o ,anle at centre
36$o 2/ 4 /2o ,8uadrilateral
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$o ,tan7rad
Two Tanent Theorem#Theorem
rom any %oint outside a circle only two tanents can be drawn andthey are e8ual in lenth#
= T
1O
yo
5$o
xo
/$o
=1 and =T are tanents to a circle with centreO# ind the un!nown anles i"in reasons#
anle w 4
anle x 4
anle y 4
anle ; 4
+/$ +0$ 4 0$o ,anles sum tri
5$o ,isos trianle
5$o ,alt sewo
;o
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O
( T
3 cm
/ cm
ind lenth O(
O( 4 5 cm ,%ytha tri%le? 3
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Anle (OT 4 22o ,symmetry7conruenncy
ind anle x
O
( T
22o
xo
>
Anle x 4 +/$ ++2 4 6/o
,anle sum trianle
Chord Bisector Theorem#Theorem /
A line drawn %er%endicular to a chord and %assin throuh thecentre of a circle< bisects the chord##
O
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O(
T 65o
=
)
>
&ixed 1uestions
=T) is a tanent line to the circleat T# ind anles (>T< (OT< OT(and O(T#
Anle (>T 4
Anle (OT 4
Anle OT( 4
Anle O(T 4
65o ,Alt se
+3$o ,anle at centre
25o ,tan rad
25o ,isos trianleMixed
1
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22o ,cyclic 8uad
6/o ,tan rad
00o ,isos trianle
6/o ,alt se
Anle w 4
Anle x 4
Anle y 4
Anle ; 4
O
w
y
0/o
++$o
>
&ixed 1uestions
=) and =1 are tanents to the
circle# ind the missin anlesi"in reasons#
x;
=
1
)
Mixed Q 2
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@e was 0$ years old before he loo!ed in on eometry< which
ha%%ened accidentally# Bein in a entlemans library< EuclidsElements lay o%en and twas the 0 El libri +# @e read the%ro%osition# By od sayd he ,he would now and then swear anem%haticall Oath by way of em%hasis this is im%ossible (o hereads the 9emonstration of it which referred him bac! to
such a =ro%osition< which %ro%osition he read# That referredhim bac! to another which he also read# Et sic deince%s thatat last he was demonstrati"ely con"inced of the trueth# Thismade him in lo"e with eometry#
rom the life of Thomas @obbes in Dohn Aubreys Brief i"es< about +60
Thomas @obbes? =hiloso%her andscientist ,+5// +6
eometric =roofs
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FG@e studied and nearly mastered the (ix-boo!s of Euclid,eometry since he was a member of Conress# @e bean a
course of riid mental disci%line with the intent to im%ro"e hisfaculties< es%ecially his %owers of loic and lanuae#
@ence his fondness for Euclid< which he carried with him onthe circuit till he could demonstrate with ease all the
%ro%ositions in the six boo!sH often studyin far into the niht#(# =resident
,+/$ 65
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At the ae of twel"e * ex%erienced a second wonder of a totally differentnature? in a little boo! dealin with Euclidean %lane eometry< which came intomy hands at the beinnin of a school year# @ere were assertions as for
exam%le< the intersection of the 3 altitudes of a trianle in one %oint< whichthouh by no means e"ident< could ne"ertheless be %ro"ed with such certaintythat any doubt a%%eared to be out of the 8uestion# This lucidity andcertainty< made an indescribable im%ression u%on me#
or exam%le * remember that an uncle told me the =ythaorean Theorem before the holy eometry boo!let had come into my hands# After much
effort * succeeded in I%ro"inJ this theorem on the basis of similarity oftrianles# or anyone who ex%eriences Kthese feelinsL for the first time< it ismar"ellous enouh that man is ca%able at all to reach such a deree ofcertainty and %urity in %ure thin!in as the ree!s showed us for the firsttime to be %ossible in eometry# rom %% -++ in the o%enin autobiora%hical s!etch of AlbertEinstein? =hiloso%her (cientist< edited by =aul Arthur#(chill%< %ublished +5+
Albert Einstein
E =
mc2
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MExtend AO to 9
MAO 4 BO 4 CO ,radii of same circle
MTrianle AOB is isosceles,base anles e8ual
9
α
αMTrianle AOC is isosceles,base anles e8ual
β
βMAnle AOB 4 +/$ - 2α ,anle sum trianle
MAnle AOC 4 +/$ - 2β ,anle sum trianle
MAnle COB 4 36$ ,AOB : AOC,Ns at %oint
MAnle COB 4 36$ ,+/$ - 2α : +/$ - 2β
MAnle COB 4 2α : 2β 4 2,α: β 4 2 x N CAB
To %ro"e that anle COB 4 2 x anle CAB
1E9
To =ro"e that the anle subtended by an arc or chord at thecentre of a circle is twice the anle subtended at thecircumference by the same arc or chord#
O
C
B
A
Theorem + and 2 Proof 1/2
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O
To =ro"e that anles subtended by an arc or chord in the samesement are e8ual#
C
B
A
Theorem 3
9
To %ro"e that anle CAB 4 anle B9C
M.ith centre of circle O draw linesOB and OC#
MAnle COB 4 2 x anle CAB ,Theorem +#
MAnle COB 4 2 x anle B9C ,Theorem +#
M2 x anle CAB 4 2 x anle B9C
MAnle CAB 4 anle B9C
1E9
Proof 3
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To %ro"e that the anle between a tanent and a radius drawn tothe %oint of contact is a riht anle#
Pote that this %roof is i"en %rimarily for your interest and com%leteness#
9emonstration of the %roof is beyond the C(E course but is well worthloo!in at# The %roofs u% to now ha"e been deducti"e %roofs# That is theystart with a %remise< ,a statement to be %ro"en followed by a chain ofdeducti"e reasonin that leads to the desired conclusion#
The ty%e of %roof that follows is a little different and is !nown as I)educto
ad absurdumJ *t was first ex%loited with reat success by ancient ree!mathematicians# The idea is to assume that the %remise is not true and thena%%ly a deducti"e arument that leads to an absurd or contradictorystatement# The contradictory nature of the statement means that the InottrueJ %remise is false and so the %remise is %ro"en true#
T o p r o v e “ A ”
A i s p r o v e n
A s s u m e “ n o t A ”
“ n o t A ” f a l s eC o n t ra d i c t o r y s t a t e m e n t
C h a i n o f d e d u c t i v e r e a s o n i n g
1 23
4
Proof 4
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To %ro"e that OT is %er%endicular to AB
MAssume that OT is not %er%endicular to ABMThen there must be a %oint< 9 say< on AB suchthat O9 is %er%endicular to AB#
9
CM(ince O9T is a riht anle then anle OT9 isacute ,anle sum of a trianle#
MBut the reater anle is o%%osite the reaterside therefore OT is reater than O9#
MBut OT 4 OC ,radii of the same circletherefore OC is also reater than O9< the%art reater than the whole which is
im%ossible#MTherefore O9 is not %er%endicular to AB#
MBy a similar arument neither is any otherstraiht line exce%t OT#
MTherefore OT is %er%endicular to AB#
1E9
To %ro"e that the anle between a tanent and a radius drawn tothe %oint of contact is a riht anle#
T o p ro v e “ A ”
A i s p r o v e n
A s s u m e “ n o t A ”
“ n o t A ” f a l s eC o n t r a d i c t o r y s t a t e m e n t
C h a i n o f d e d u c t i v e r e a s o n i n g
1 23
4
O
A
BT
Theorem 0
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To %ro"e that the anle between a tanent and a chord throuh the%oint of contact is e8ual to the anle subtended by the chord inthe alternate sement#
Theorem 5
A
T
B
C
9
O
To %ro"e that anle BT9 4 anle TC9
M.ith centre of circle O< draw straiht linesO9 and OT#
Met anle 9TB be denoted by α#
α
MThen anle 9TO 4 $ - α ,Theorem 0 tan7rad
$ - α
MAlso anle T9O 4 $ - α ,*sos trianle
$ - α
MTherefore anle TO9 4 +/$ ,$ - α : $ - α4 2α ,anle sum trianle
2α
MAnle TC9 4 α ,Theorem + anle at thecentre
α
MAnle BT9 4 anle TC9 1E9
Proof 5
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To %ro"e that anles A : C and B : 9 4 +/$$
M9raw straiht lines AC and B9
MChord 9C subtends e8ual anles α ,same sementα
α
MChord A9 subtends e8ual anles β ,same sement
β
β
MChord AB subtends e8ual anles γ ,same sement
γ
γ MChord BC subtends e8ual anles δ ,same sement
δ
δ
M2,α : β : γ : δ 4 36$o ,Anle sum 8uadrilateral
∀α : β : γ : δ 4 +/$o
Anles A : C and B : 9 4 +/$$ 1E9
A
B
9
C
To %ro"e that the o%%osite anles of a cyclic 8uadrilateral aresu%%lementary ,(um to +/$o#
Theorem 6
α β γ δ
al%ha beta amma delta
Proof 6
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To %ro"e that A= 4 B=#
M.ith centre of circle at O< draw straihtlines OA and OB#
To %ro"e that the two tanents drawn from a %oint outside a circleare of e8ual lenth#
Theorem
O
A
B
=
MOA 4 OB ,radii of the same circle
MAnle =AO 4 =BO 4 $o ,tanent radius#
M9raw straiht line O=#
M*n trianles OB= and OA=< OA 4 OB and O=is common to both#
MTrianles OB= and OA= are conruent ,)@(
MTherefore A= 4 B=# 1E9
Proof 7
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To %ro"e that a line< drawn %er%endicular to a chord and %assinthrouh the centre of a circle< bisects the chord#
Theorem /
O
A B C
To %ro"e that AB 4 BC#
Mrom centre O draw straiht lines OA and OC#
M*n trianles OAB and OCB< OC 4 OA ,radii of samecircle and OB is common to both#
MAnle OBA 4 anle OBC ,anles on straiht line
MTrianles OAB and OCB are conruent ,)@(
MTherefore AB 4 BC 1E9
Proof 8
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Worksheet 1
=arts of the Circle
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Worksheet 2
xo
yo
o
xo
yoo
xo yo
o
xo
yoo
xo
yoo
xo
yo
o
Th1&easure the anle subtended at the centre ,y and the anle subtended at thecircumference ,x in each case and ma!e a con'ecture about their relationshi%#
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To =ro"e that the anle subtended by an arc or chord at thecentre of a circle is twice the anle subtended at thecircumference by the same arc or chord#
O
C
B
A
Theorem + and 2
Worksheet 3
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To =ro"e that anles subtended by an arc or chord in the samesement are e8ual#
A
Theorem 3
O
C
B
9
Worksheet 4
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To %ro"e that the anle between a tanent and a radius drawn tothe %oint of contact is a riht anle#
T o p ro v e “ A ”
A i s p r o v e n
A s s u m e “ n o t A ”
“ n o t A ” f a l s eC o n t r a d i c t o r y s t a t e m e n t
C h a i n o f d e d u c t i v e r e a s o n i n g
1 23
4
O
A
BT
Theorem 0 Worksheet 5
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To %ro"e that the anle between a tanent and a chord throuh the%oint of contact is e8ual to the anle subtended by the chord inthe alternate sement#
Theorem 5
A
T
B
C
9
O
Worksheet 6 h h l f l l l
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A
B
9
C
To %ro"e that the o%%osite anles of a cyclic 8uadrilateral aresu%%lementary ,(um to +/$o#
Theorem 6α β χ δ
Al%ha Beta Chi delta
Worksheet 7
h h d f d l
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Worksheet 8
To %ro"e that the two tanents drawn from a %oint outside a circleare of e8ual lenth#
Theorem
O
A
B
=
T h li d di l h d d i
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To %ro"e that a line< drawn %er%endicular to a chord and %assinthrouh the centre of a circle< bisects the chord#
Theorem /
O
A B C