Circle Theorems _WithProofs_.ppt

8/9/2019 Circle Theorems _WithProofs_.ppt

1/43

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#


2/43

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


3/43

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


4/43

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


5/43

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#


6/43

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


7/43

,+/$ 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


8/43

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


9/43

Anles subtended by an arc or chord inthe same sement are e8ual#Theorem 3

xo xo

xo

xo

xo

yo

yo

Th3


10/43

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


11/43

The anle between a tanent and aradius is $o# ,Tan7rad

Theorem 0

o

Th4


12/43

The anle between a tanent and aradius is $o# ,Tan7rad

Theorem 0

o


13/43

+/$ ,$ : 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#


14/43

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


15/43

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


16/43

+/$ /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


17/43

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


18/43

$o ,tan7rad



= 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


19/43

$o ,tan7rad



= 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


20/43

O

( T

3 cm

/ cm

ind lenth O(

O( 4 5 cm ,%ytha tri%le? 3


21/43

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


22/43

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


23/43

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


24/43

@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


25/43

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


26/43

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


27/43

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


28/43

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


29/43

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


30/43

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


T o p ro v e “ A ”

A i s p r o v e n


“ 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


1 23

4

O

A

BT

Theorem 0


31/43

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


32/43

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


33/43

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


34/43

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


35/43

Worksheet 1

=arts of the Circle


36/43

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%#


37/43

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


38/43

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


39/43


T o p ro v e “ A ”

A i s p r o v e n


“ 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


1 23

4

O

A

BT

Theorem 0 Worksheet 5


40/43

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


41/43

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


42/43

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


43/43

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

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Circle Theorems _WithProofs_.ppt