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Step 2: Place on number line
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Step 1: Solve (both cases)
:;!.<!.9!
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Examples
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1) Equal Arcs = Equal Angles
•
•
2) Equal Chords = Equal Angles
IJ81-!1+>9!98K<,0?!,J81-!106-,9!
1<!<=,!>,0<+,!7;!<=,!>.+>-,!
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IJ81-!>=7+?9!98K<,0?!,J81-!
106-,9!1<!<=,!>,0<+,!7;!<=,!>.+>-,!
$70L,+9,!.9!<+8,!
D%97$)C2(6$2#'$&)
2) L in same segment … 3) L in semi-circle = 90o
1) Angle at centre = Twice circumference
M06-,!1<!<=,!>,0<+,!.9!<N.>,!<=,!106-,!1<!<=,!>.+>8/;,+,0>,!
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1) Line from centre bisects chord
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$70L,+9,!.9!<+8,!
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2) Equal chords = equal distances
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;+7/!<=,!>,0<+,!
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3) Intercepting chords
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1) Cyclic Quads – Opp. Angles supplementary
•
•
•
R@@79.<,!106-,9!.0!>2>-.>!J81?+.-1<,+1-9!1+,!
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2) Exterior Angle of a Cyclic Quad
•
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7
IG<,+.7+!106-,!7;!1!>2>-.>!J81?+.-1<,+1-!.9!,J81-!<7!<=,!
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1) Tangents form 90o to centre
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2) Line passes through contact
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•
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3) Angle in the alternate segment
S=,0!>.+>-,9!<78>=T!1!-.0,!;+7/!
<=,.+!>,0<+,9!@199,9!<=+786=!<=,.+!
>70<1><!@7.0<!
M06-,!.0!<=,!1-<,+01<,!9,6/,0<!
•
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5) Square length of tangent
4) Tangents from exterior point
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Example 1
•
•
Example 3
%=,+,!1+,!Q!+71?9!;+7/!M!<7!)T!10?!E!+71?9!;+7/!)!<7!$F!
"7N!/102!1++106,/,0<9!7;!@1<=9!1+,!@799.K-,_!
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Example 2
•
•
"7N!/102!1++106,/,0<9!>10!K,!/1?,!7;!3!K7G,9!
9,-,><,?!;+7/!Q!K7G,9!
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5F %=,!3!K729!1+,!<76,<=,+!
o 3#$0*!*1$!4-5,!%&6$!0+!&+7&8&790%:!*1$+!
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Example 1
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1B D!'.6.<9!1+,!89,?!
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Example 1
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Example 1
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1B "7N!/102!1++106,/,0<9!1+,!<=,+,_!
H"-,,&4%$!0+,<$#,IK9'4$#!-;!=%02$,!
W!3ZE!G!54E!W!5`D`Z444!
KB *+7K1K.-.<2!<=1<!<=,2!>70<1.0!10!.?,0<.>1-!-,<<,+!10?!?.6.<!
3%#$2%,7)B$6$#'#'(%)
Example 1
M++106,/,0<9!89.06!-,<<,+9!7;!a/1//1-b!
c.0?!@,+/8<1<.709T!10?!<=,0!?.L.?,!K2!<=,!;1><7+.1-!7;!<=,!1/780<!7;!<./,9!1!+,@,<.<.70!7>>8+9!
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Example 1
•
•
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!",.67$&)
o S129!Q!6.+-9!d!D!K729!>10!K,!1++106,?!W!Q\!G!D\!
o S129!Q!6.+-9!d!Q!K729!>10!K,!1++106,?!1+780?!1!<1K-,!W!Q\!G!Q\!
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W!54$Z!G!D$3!d!54$D!G!D$E!d!54$Q!G!D$Q!d!54$D!G!D$D!W!D`H4!
Notation
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Example 1
Example 3
:0!=7N!/102!N129!>10!Q!K,!9,-,><,?!;+7/!54!.;O!
1B %=,!7-?,9<!@,+970!.9!.0!,1>=!9,<!
L$792*!-+$!;#-'!*1$!MNOP"!EQR(!0+7!*1$!?/OPK3!
E(S(T3QKM!
W!H$E!W!YQ!
KB %=,!7-?,9<!@,+970!.9!07<!.0>-8?,?!
L$792*!-+$!;#-'!*1$!MNOP"!EQR(!
W!H$Q!W!53Z!
>B *+7K1K.-.<2!<=1<!<=,!7-?,9<!.9!.0>-8?,?!
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Example 2
Q!e,0!10?!E!)729!1+,!9,-,><,?!;+7/!Y!e,0!10?!D!)729F!:0!
=7N!/102!N129!>10!<=,2!K,!1++106,?_!
>&+7!2-'4&+0*&-+,!*1$+!'9%*&=%5!45!*1$!0##0+A$'$+*!0'-9+*!
Y
$Q!G!D$E!G!`\!W!ED3Y444!
Example 4
"7N!/102!9,-,><.709!7;!D!K77f9!>10!K,!/1?,T!.;!102!08/K,+!
>10!K,!<1f,0!1<!1!<./,_!
?77!$021!=-,,&4%5!2-'4&+0*&-+!
D
$5!d!D$3!d!D$E!d!D$Q!d!D$D!]!RX!]!3D![!5!W!E5!
Example 5
M!>-199!7;!5D!>70<1.09!D!@+,;,><9F!"7N!/102!6+78@9!7;!Y!>10!
K,!;7+/,?!N.<=O!
,8 <)C2$:$/#&)
>&+7!2-'4&+0*&-+,!;-#!4-*1!A#-9=,!0+7!'9%*&=%5!
W!54$Z!G!D$3!W!3544!
Z!e,0!/7L,?!.0<7!6+78@9!7;!ET!3T!10?!5F!:0!=7N!/102!N129!
>78-?!<=,9,!6+78@9!K,!9,-,><,?_!
L$*#02*!*1$!=#$8&-9,!MNOP"!EQR(!+9'4$#!;#-'!*1$!29##$+*!
?/OPK3!E(S(T3QKM!+9'4$#!
W!Z$E!G!E$3!G!5$5!W!Z4!
H'%(.',7)C2(N,N'7'#K)
:;!@!.9!<=,!@+7K1K.-.<2!7;!98>>,99!10?!J!.9!<=,!@+7K1K.-.<2!7;!
;1.-8+,!;7+!10!,L,0<T!<=,!@+7K1K.-.<2!7;!+!98>>,99,9!.0!0!
.0?,@,0?,0<!,L,0<9!.9!6.L,0!K2O!!
!
Example 1
M!>1+!199,/K-2!@-10<!=19!1!>,+<1.0!/1>=.0,!N.<=!10!1L,+16,!
@+7K1K.-.<2!7;!4F5!7;!K+,1f.06!?7N0F!:;!<=,!199,/K-2!@-10<!
=19!Y!7;!<=,9,!/1>=.0,9T!N=1<!.9!<=,!@+7K1K.-.<2T!>7++,><!<7!E!
?,>./1-!@-1>,9T!<=1<!1<!-,19<!Z!N.--!K,!.0!677?!N7+f.06!7+?,+!
1<!102!70,!<./,_!
*AK+7f,0B!W!4F5!]!*AN7+f.06B!W!4FH!
*A1<!-,19<!ZB!W!*AZB!d!*A`B!d!*AYB!
Example 2
M!98@,+/1+f,<!?.9@-12!>70<1.09!D!?.;;,+,0<!K+10?9!7;!<7/1<7!
@19<,T!.0>-8?.06!%7/%7/!K+10?F!RL,+!<=,!@,+.7?!7;!1!N,,fT!
Y4!@,7@-,!+10?7/-2!K82!<7/1<7!@19<,F!
5,8)W2$,#$&#)%1.N$2)(:)6$(67$)#()N1K)#-$)5AA8)N2,%=X)
c.0?!<=,!a6+,1<,9<!>7C,;;.>.,0<b!/,<=7?!
5N8)0'%=)#-$)62(N,N'7'#K)(:)#-'&)%1.N$2)(:)6$(67$)N1K'%9)#-$)
AA)N2,%=R)/(22$/#)#()<)=$/'.,7)67,/$&G)
Example 3
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<,00.9!.9!5UE!"7N!/102!61/,9!9=78-?!K,!@-12,?!98>=!<=1<!
<=,!@+7K1K-2!7;!N.00.06!1<!-,19<!70,!61/,!.9!4FH_!
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Example 5
%2@.9<!=19!<7!>7++,><!5!7;!Y44!N7+?9F!:;!1!@16,!>70<1.09!344!
N7+?9T!;.0?!<=,!@+7K!F7;!/7+,!<=10!70,!>7++,><.70!@,+!@16,F!
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Example 4
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<./,9F!%=.9!@+7K1K.-.<2!.9!<N.>,!<=1<!7;!54!+7--9!N.<=!10!,L,0!
08/K,+!Q!<./,9F!S=1<!.9!<=,!@+7K1K-2!<=1<!10!,L,0!08/K,+!
N.--!1@@,1+!.0!1!9.06-,!<+.1-_!
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*+,-./.01+2!"#$!!"#$%&'(%)*)+,#-$.,#'/&!3454!
*16,!Y!
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3%'45)56*%+!-("!789'6&5#6!:!;*9<')*95=&!
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Definition
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Case 1: Equal powers
Case 3: Larger on top
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h,+7B!
Case 2: Larger on bottom
!
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Graph
Horizontal Assymptotes (Find the limit [above])
#.0>,!<=,!@7N,+!.9!-1+6,+!70!<=,!K7<<7/T!<=,!-./.<!.9!2!W!4!
!
Symmetry (Odd, Even, Etc)
Y-Intercept (Let x = 0)
;ACGB!W!;AGB!!
!
FOF!c80><.70!.9!,L,0!
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Other
X-Intercepts (Let y = 0)
•
!
•
&01K-,!<7!97-L,!FOF!^7!iC:0<,+>,@<9!
!
Vertical Assymptotes (Restrictions on Domain)
M--!+,1-!G!1+,!@799.K-,!.0!<=.9!,J81<.70!
FOF!^7!L,+<.>1-!1992/@<7<,9!
S=,0!.0!?78K<T!98K!.0!@7.0<9!.0<7!<=,!6+1@=!<7!
?,<,+/.0,!<=,!9=1@,!
%1f,!,1>=!;1><7+!10?!6+1@=!1!a68.?,!6+1@=b!A9,,!,G<3B!
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3%'45)56*%+!-("!789'6&5#6!:!;*9<')*95=&!
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Z#-$2)!",.67$&)
Example 1
Example 3
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IL,0!c80><.70!
k,+<F!M99!!!3T!C3!
"7+F!M99!!!4!
iC:0<,+>,@<!W!^7!
(C:0<,+>,@<!W!Cj!!
l8.?,!6+1@=!.9!.0!K-8,!A2!W!G3![!QB!
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Example 4
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5!
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Example 4
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Graphs of Inverse Functions
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Example 2
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Example 1
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Example 2
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General
Proof
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