دليل المعلم - مصر - الصف الاول الثانوى 2013-2014
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دليل المعلم - مصر - الصف الاول الثانوى 2013-2014
Transcript of دليل المعلم - مصر - الصف الاول الثانوى 2013-2014
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((4( : ((5( : - 21. : . . . . . . . : . . . : q () . q q ( ) q ( ) ( ). . q q . q q : q . q q . q q . q q . q () . - 22. : q . q q () . q q ( 32%) q q () . q q ( 41%). q q (32%) (41%) q ( ) 22.3% 001 22.3% 3 . : / 60.0 40.0 01%21.0 80.0 02%42.0 61.0 04%60.0 40.0 01%60.0 40.0 01%/ 1 2 11 2121 2 11 21 2 11 260.0 40.0 01%06% 04% 001% 51 2 11 2 41 12 33 5111941 / 50.0 01%50.01.0 1.0 02%2.0 04%50.0 01%2.050.050.0 01%50.0 01%50.050.0 05%05% 001% 41 / 11211171 21 2 - 241 21 21 2741 23. / 540.0550.0011.022.0550.0550.0550.055%01%02%04%01%01%01%001%090.081.0540.0540.0540.054% 41 / 1 2 11 2131 2 11 21 2 11 21 2 11 2611 23 6111841 / 50.0 2.050.050.01.02.001%02%04%1.050.050.050.050.001%01%50.005%01%001%05% 41 / 1 2 1 2123243 6 1 2 1 21 1 2 1 21 1 2 1 21 7 741 .: q 5002 7 - 82 qq 1002 51 - 71 q q q 1002 q 8002 q - 24. - 25. - 26. IMdG1dhCG IMdG M Algebra, Relations and Functions : .Algebra, relationships and...functions).(.... : : . : . : . : . : . : . : . . . . . Complex NumberEquation Discriminant of the Equation Root of the Equation Coefficient of a TermImaginary Number Powers of a NumberSign of a functionInequality 27. .) - (: .) - (:.) - (: ) - (:. .) - (:.) - (: ----:www.phschool.com ). ).(algebra (( ) )2 -4C(. ). .( . : . - . . . . . . - - - . - - - - - . - - - - . . 28. 1-1 Solving Quadratic Equations in One Variable 1-1Solving Quadratic Equations in One Variable . .- M ; , :C++ =) =( .(. Equations, relations and functionsEquation:: ) :Factor Coefficient:C=++C=!C(. C - !+!C) :Function+ = )-Relation M:C!C(C- C++=C: .. Solving quadratic equation graphically: +1+- =.- =--- ---I --: -+----: ) (= - + + = - + + = - - = - = C + + = ! C )(.: : :M C + = SQdG GAGLEG : ) ! ( = dG : : : + + = : . 29. dG dG .=) (=+ : ? . )( .-: ------.+= =- } , {. = M: = : ) + () ( = = =++ = =-: }-,C = C = :=C{: = )- ( + ) ( = = )=- : (.=-= : = - F=) ( +) ( = + = )= :-Vertical line test(. :=+ F MM = - . )( = F. .-]-] .) ( F MM M: . .SQdG VY : C + + = : )( = )( = C + + . ) ( . ) = ( . }- { = - ]- ] . : : ) - ( + ) + ( - ) - ( + ) + ( ) - ( + ) + ( : + + = + - = + + = 30. M :M :-)(Table 7table-:: 6-)(x=-Mode)ALPHAEnD+4-?START 32(x) x)ALPHA==1Step=-.REPLAY:-:1)x f(x 4- 6 3- 0 4- 2- 1- 6- 0 6- 1 4- 2 0 3 6Mode.?.? 1= = ) (dGh jQdG 1 = ) + ( : + - = = = - .1 2 3 4 5 6 7 8 =[.-] :2/ . :=.: `==`=, ==MF`.+ =:` ) () (===M.: .:bdG d FGKEG f 2 = : - + + = : - - = = = - ) ( : 1 ) = ( . : = + : = 2 /. : = - : = M = , M M:: : = + + )( . ? ? . = = = = 31. :2: =-++?GeoGebra Graph)(/http://geogebra.org/coms)(/http://www.padowan.dk. ) (=++6 5) (=+ =+4 3 2 1--4312002143 : + - = - + = + + = 51-) (= )=-- + 5- ( =- =4 3= Geogebra Graph ) ( .2 1=8764523010 1 -) (= + =!) (=- C+-7 6 5=-C, =C `4 3= ,2== - )- ( ! )- ( - * * =1 5! -432100121*=!-= ! -I M :M .z: : . : : . M 32. .IU .---.,.., ,I II=}{=z=} { :1=) + + +..... + () + (:::d aVEG fCG )( = 1 )( = - + 2 )( = - + - 3 )( = - + .z 33. 1-2 1-2Complex Numbers Complex NumbersP N I/K K )- (=-.: =+ .+ = .MImaginary Number =- M:C,......:C = C,=!?, M , Imaginary numbers- =- =MComplex Number =! , )- (:,.U M F --I --- . )(. 34. SQdG GAGLEG Integer powers of i:dG ib:dG= = :=- *: : + = =- *- = === ++=+=-N=-1: SQdG VY =) -( *=)= *- =-(- *= *=)=--+( *== *-= *)*+=- ( *=-= * 1: : :M M = I : = ! - = -: --+ +C C + C. . :M : )-( ! - )-( = = . !. : - - ! * -= * =! * ! = ! = ! : + C C )( .M +Complex number*== **=- 35. =+CCdG dGC ) ( :. =C==C.=2=+ = + +==-II= =! + M- 2: =+=+ +=Equality of two complex numbers=+E:=C=E3:+)(-I= +=- = =. :. :+C F=- = = M = = .............. = = - = .............. = .............. = - = .............. = .............. = = .............. = .............. - - : + + - + - + dG dG :aVEG jQJ : + M + 36. -1 23 ! : ) = + (+ +)= ++ ( Operations on complex numbers: :4 ! ) ) + ! (+) + ( (() =) = ) + ( + )- + ( = : = ) ++ +) = -(: ) - C( + ) - E( = : = C = E dG dG = = - = = : . )( . 4: ) ) (() () + () + ) ( ( (+= ) + ( + )- + () - C( = ) - E((= (+F) = + C = + E :3() = ) M :) - C( = -) - E((+) + (= =+F 37. +C) Conjugate NumbersC() +(=) ( ) =:+aVEG jQJ( = : ) - ( - ) - ( ) + ( + ) - ( ) + ( ) - () + ( )- ( =:.?.? :5) + +() - ( =dG dG+ +=+ + ) -* - -=(=+=++-:=+F4F M : + C - C : = C + C =- 5 + : - -- - + -5 - - + M + + M 38. dGh jQdG :6 (.6 = + a=`=) -(+) + (= - 6 + - + = ) + ( + 1:) - ( = + : ) - ( = ]) - ([ = ) - + ( = ) - - ( = )- ( = - = - :bdG d FGKEG f + = . jQdGh fCG c fCG J U U jdGddG AGOCG M MUM MU,I M M,UM MM M.= ) + ( + )- + ( + = * + + + +)M M,UMU. 39. 1-3 Determining the Types of Roots of a Quadratic Equation 1-3 )Determining The Type of the Roots of(a Quadratic Equation () ? C: -Discriminant +- C+C=+ -!CC- C-C - C. ., , CRoot,- CDiscriminant1 - MI+- =+-: -+ ==: =C= =C=- M:=- - C= = - * )- ( = a-. = = - C = - * * =a. M-I --- )(.SQdG GAGLEG :dG . 40. SQdG VY -=C: ==-a. : .MM) C = ) ( . . . C . :M C . + C C . dG dG )( . ) ( 1 =) (=: =) + (= ) (2.+ =, =C a a .== , = C :GOTQEGM( C = =C- *- *-=-= ` `= - !C= - )- ( !*: + = )- ( * * = =-- C=!- : - + = + + = - - = - + = = C = = ) - C( = - * * = C = - = ) - C( = - * * . ) ( = C = - = ) - C( = - * = = C = - = - ) - C ( = - * * - 41. : a = - ! -C C = ! ?. 2.+ =3+ ) (:+ =: = C ) ( * * = - = - = ) () + ( = =- =M + + .=:++ =-+ =:- -:=-: : + - : 3+.+ = + . = ) + ( + ) - ( = = ) + ( ) - ( = : - + = :1. =- ,- ,:+. = ?)?(?.dG dG M - = . - - + -- . : : - - = . ) (. : - + = : = C = - = ) - C( = - * * = - = - M2 = = -dGh jQdG = = = = - .bdG d FGKEG fCG 1 C : - ) + (C + = C . 2 - + - = . dG1 - + = . 2 : - - = 42. 1-4 The Relation Between Two Roots of the Second Degree Equation and the Coefficients of its Terms The Relation Between Two Roots of the Second Degree Equation and the Coefficients of its Terms1-4 M+ = + = + = * = ? ? M, Sum and product of two roots ,C -- C+CSum of Two Roots+ --CProduct of Two Roots)C= (=C=C 1=C --===C: +=-++)C(:+:: =C- C= + = - M =+:=- = -=CC= - =- = - =-SQdG VY : . I : - - ? C . ? ) C . ( dG dG )(. SQdG GAGLEG = C + = - = = C + = - dG = C = 1 + = - = - + = = . : + - = + = - = - 43. 1 =+:= ) : () + (=2+ =-.1 ) + () + ( = ) - () - (= , = , =C`C=:=-= 2 + - + + - 3 } -{ = - += }- {- C!C -!=`=!=} +!= {- 2 3 + - =+-= )( .. .-3) + (+=C-*I C:C=C= ,, adG dG=C+`= - a= - = 2 = : + + = }- -{ 3 - = - = + - = } - {. 4 a = ) + ( - + = =+ a*= ` = ) + ( ) - ( = - = =C` ) + () - (=C ` C ` + =C= 4) + (-=+. M Forming the quadratic equation whose roots are known,I::C -)-(C+C `++ +CC==Ca+ =-`+=!C:,=CC) + (+=1 + - + = . 2 : - - - - + - - + 4, = ( -) + = + a : `a= )- ( = -:) + (+==5 ++: --* - -=- + + =- - -*=+=*-=+ +== -== =-a+= :`+) + (+3 + + ] : -+=[=+ = 5 --:+ - 44. gQL Y e HdG dOdG jJ :J : ) - () - ( = : - ) + ( + = dG dG:(.) - ( )iNCG HJ dOe eH HJ dOe jJForming a quadratic equation from the roots of another equation 6 =. =C5 + - = : + = : + =M + =) + ( -`= - : + =- - ==-+ = , = = ) ( * )- (=-+ =) + ( = + = + = a=)` = )- ( =(+ =aVEG jQJ : - ]: + = [ + - ]: - + = [ + - ]: - + = [ - ]: + + = [-`: ) I+ =:M(+=++ = 6: = + 1: 2+-- =-+. : = C + + dGh jQdG C : 1 - + = : )( = - - - = : )( = ) - ( + = - = + + = : )( = ) - ( 2 : - + = )- ( ) ( :bdG d FGKEG f : - + = + = ) + ( - + ) - ( = ) + ( - M + + = + ) + ( - + = = dG dG6 + - = + + = - - = dG1 : C + + . 2 - - = 45. 1-5 1-5 Sign of a Function :Sign of a Function .?() :,() ) ) (=,(Sign of a function HdG dGdG IQTEG :hCGConstant Function,M).I! (.Linear Function Quadratic Function , I 1: ` ) (=- MI: ) a ( ) a(`, MI) (=First: The sign of the Constant Function) (=I --I --- )(.SQdG GAGLEG dG . 46. SQdG VY :1: : 1 . 2 . 3 . ) (=) (=-(dG dGdG) dhCG LQdG dGO IQTEG :kfK ) (=Second: Sign of the Linear Function!+) (==-. 2) (=::) (=: ) (= == ) (=-:: ) (= )( . = 2) (=-.dG dG M1 : :HdG dGdG IQTEG :dK.Third: Sign of the Quadratic Function) (=C: : 1 .+C C=+++!C: :2 .3) (= = : ) - () + (= :-: )() ( ]-[) (= = )( = [ ]}-3) (= { + + + + + + - - - - - - - -: : )( = - )( = - .+. 47. : C:: :. : :C:CI())4) (=.+) C( = )- ( * *= =- + = )I( 4) (=:: C. :=C!) (==.:C () ) (=:C (!) ) (==I(++ + - - - - - - - - - - - ++ + - : - )( = = - = - : )( = )- - ()- + ( = : )( = ) - () + ( = : )( = ) - () + ( ` )( = 3 - + = ) - () - ( = )( = = )( = != M+ + ++ + - - - - - + + ++ + : = )( = * - = )( = * - = )( = - --- - - --- - = :)( = - + = :)( = ) - () - ( = :)( = ) - () - ( - ] [ [- ]-4 = - * - * - = - = )( + + ++ + + + ++ + = )( = - = )( = - --- -- --- -- --- -5 - a - - = ` + + = ) + ( = = - )( = = -} - { )( dG dG2 )( = = - )(- )(--- - -- - -- - -- --- --- - M-- 48. dGh jQdG 5) (=.+1 )( = = )( )( C)( = )- ( * *==+ = :) ( = =: (). =!=) (= 5 )( = = ) (=.)( I6+ - =-)( ) = ( - ) * * - ( -) = ( C= )( = = }- {--++:=)- ( - * *)( [- ] )( - ]- [`=-=- + =-`-+=-)I+(?I+ - = + - =I )( = = }- {1: )( [- ]) (= ) (= --) (= - ) (= ++ ) (= ) (=-+)( - ]- [ = - * * = )( = = - = - )( = - * - * - ` )([- -] = - )(= - * - * ` )([- ] = )( = * - * ` )([ ] = )( = * * ` )([ ])( - }-{ = - * * )( = bdG d FGKEG fCG 1 : . : )( = )-()-()+( . dG . : 2 )( = : )( = ) - () + ( )( = - )( = )( = + - : )( = - + )( = - + + + + - - - - - + + + + + - - - 49. 1-6 1-6Quadratic Inequalities :HdG jdGQuadratic InequalitiesQuadratic Inequalities ? : ) (= M.: - ] M: 1] I- -IInequality- ] [[- - [-,::M:: ) (=--I M --- )(.SQdG GAGLEG dG )( . 50. GOTQEG :M) (=--) (= : . =` ) - () + ( = =- = +++++:M - - .: [-- ] [ +++++ ] .1: +: ) + ( .( + ) H2 ( + ) H ( + ) a + H + ` ` M - - + + + + - - - = -: - = : - = : - [- ] dG dG1 - ]- [ [- ]+: }- - { ) (=++ :: :+ H+ = + ) + () + ( =: . : - = : ) - () + ( = : = }- { + . - + : ]- - [ + + [ - -] : ) + () - ( G + + - = + + - = ) + () - ( = }- { ++ + - - - - - - - - - - + + = - : )- + ()- - ( = : ) + () - ( = : ) + () - ( [- -[ ] ] - [- ] 51. dG dG 2: ) + ( + ) + (G+G2 : - [ - ] ]- -[ 1?2? :3)+ ( ( + )a `)( + )a ) ( +) (`dGh jQdG ( ( )+ +` `-+`-` )+ () ( = } {1+ + + : :-++ =+ + + -- - - - + + +} { - --) (=-+ +++[ 4:) (= :-+-]] ) + (2 3[- ) + (4 . . !. : + + = ) + () + ( [- ]bdG d FGKEG f M - - = . dG ) + ( + M 52. -2 Similarity Similarity : . : ( ...) . : ( : ( ). : ( ). : ( ...) ...) : ( ...) . . . : : . : . : . : . : . . : . : . : ............ Ratio Proportion Measure of an Angle Length Area Cross Product Extreme Mean Similar Polygons Similar Triangles Corresponding Sides Congruent Angles Regular Polygon Quadrilateral Pentagon Postulate/Axiom Perimeter Area of polygon Chord Secant Tangent DiameterCommon External Tangent Common Internal Tangent Concentric Circles ( ) Similarity Ratio 53. (2 - 1): . (2 - 2): . (2 - 3): . (2 - 4): . - - - - - . . . 04 fractals . : ............ : ............ : ..... . 61 . - - - - . - ( ) - - - - - - - . - - - - - . - . 54. 2 Similarity of Polygons 2Similarity of Polygons . C E .//// C E . . ./ : c - Cc Cc c c c/ - / . . . . .CC/ / / / / / C E E C .C E CE Similar PolygonsSimilar Triangles Congruent Angles Regular Polygon QuadrilateralSimilar polygons .Corresponding Sides ./Pentagon :1- : :cC c / /C c/ / c / c/ / cEc / E c : / / / / C = = / C E = /E C E CE/ / /C/ / /E C E 2- (+) . 43 - - - - - - - - - - . C + E : c / C c c / c c / c c / E c - - - . 1 C = E : C + E . C E . . 51 8 6 C( + 2) a C + EEEC C : = = = = + 2 = = 51 = 21 6 8 = 21 = 3 8 2 51 = 3 # = 01 6+ 2 = 3 # = 7 2 2 . 9 C 5 8 8 58 621 E E1 69 21 - - 12 21 6 c70 CE82 C9 086C27 E1 43 93 02 12 ().03 C C = = = EE = ( ) ! 0 C E = CE21 - ( ) - - - - - .E/ /Similarity Ratio ./E ( ) : /Ec Ec //011c 61 53 55. . 1- .2 1 / 2- ( 1 + 2) () ( 3 / 4) . 3- 1 + 3 2 + 3 : 1 + 2q q .1243q 1 53 q 1 . 3 + 4 1234- . 012 : 9 C + 9 E E = 8 = 9 10 = E 9 C = 18. 9 C .9 E C9 a + 9 E C C C + + C 9 C = = = = ` E E E + + E 9 E : C = = 81 = C 8 9 01 72 ` C = 8 * 18 = 42 = 9 * 3 = 72 30 = 3 * 10 = C 72 : ( ) 1 + 2 = ( ) 263 8 C1 - q : q : q q q q . q q : 1- . 2- . () q 2 63 q . q 43 q C /C E/ / /E : . q : q : 1- . 2- . (+) = 2 ./ / / / / / / / C = = = E C = E E C E C - 13 56. : q q C = ! 0 0 ! E 2 : C + E c( X ) E C C 195 = E .8C58c07c8 E 511c6 C E C = 5 = 8 : = 60 = 41E: = E * C * = . + C E EC: = = = E = E C C = 41 = 1 2 = 5 = 5 2 5 2q 3 q ( 1 2). : 1101 3 4 ( ).4 5 = 8= 41` = 7 = 48 C E = 60 5 = 8C= 60` = 3 = 84 C E:ESimilarity ratio of two polygons 1 2 1 :1 2 1 01 2 1 =1 2 .73 - : 1 3 2 3.1 1 3 2 3.2 1 + 2 1 + 2 .3 = 3 = 1 1 3 6 2 = 9 = 3 2 3 6 2 a 1 + 3 2 + ` 1 + 2 = 3 = 1 1 2 9 33 01 51 42. ( - E - ) : 9 +9 C C 9 +9 9 E C +9 ( ) = : 1 1 001 ....... . 57.1 5.0 52.1 q q 1 1 2 ( 1 2.) 9 9+ C C 9 +9 C The Golden Ratio 61 : 9 4 : 3 .8323 - - 57. : .51 * 42 = 63 468 2. 3 3 1 2 4 5 1 2 4 C E () (1) C E: C + E 11 : = 1 + 5 = 1 - 50 2 2 -1 8161C 8161 : 1 . (0711 - 0521) (2541 - 9151) : 11235831... 1 1 . . 5 8 31 ............. . 3 5 8 247 21 5 491 . 8 021 : - - ... . : - 2 6.2 93c 5 : 81 : - - - .CE1 C = # 1 = -1 ` 2 - - 1 = E21 : 247 - 7161 .05c5.7 C + E : : C E . . () . 21 8. 5 93 . : 2. : C + E = 4 5 C + E = 5 8 . = 3 C + E 4 c(X ) = 09 3.6 = E Cc = 591 * 8 = 6 62. : 01 * 42 = 42 491. . 5 4 09. . 2 = 05 = 3 = 56 5 = 08 = 4832 491 = 8161 - 21 = 52 = 0052. - 33 58. 01 8 7 5 . . . 5 .43 - 6 . . cc c c c cc 7 . . 8 . . 9 : : : . . . . (12) . 02 12 . M : : () 01 23 5 47. . 6 .5 + + . . : : C E )0 1-( C (2 0) (2 3) )3 1-( E (4 0) (9 0) C E . M 59. 2 Similarity of Triangles . . 2 Postulate/AxiomSimilarity of Triangles (006 .) . . . . 1- 9 C : c( X c50 = )Cc(X) = 07 C c = 4 05c2- 9 E : c( X c50 = )Ec(X) = 07 E c = 5 07c4 C 4- . C C E = = E . : . - . .)postulate (or axiom . : cE c / C c / c 9 C + 9 E C () - - - .E . . E 52cC54c08c55 c55c C56c56c75 cC07c C - ( ) - - - - . - - - -E E C52c03c . : .04 : . 3- : C E 07c 1- . ( ) . 04 94 22 52 ().2- : ( ). 3- ( ). - 14 - 53 60. C C E C E // 12 = E C = 3 C = 4 E = 24. 9 E C + 9 C : E C 3 E a // C . ` E C c / C c E C C E Cc a / Cc C Ec / c C ` 9 E C + 9 C 24 21()( ) ( ) E C 9 a + 9 C E E` C = C = : C C EC 3 24 = 4 = 12 + E C 5 E(4 5) ( + 4) (2 + 3) C( + 5) 1 .24 - q : q . . : 9 C + 9 E 9 + 9 . 9 C + 9 E ( ) 9 C + 9 E 9 C + 9 9 = 11 (9 E C + 9 C ) (9 C + 9 C ) (9 E C + 9 C ) 63CC 2 3 . - E 3 q 14 q 1 . 2 9 C + 9 E C . C 2 q E q . 4 3)12 + E C(3 = E C = 4 * 24 4 * 24 = 33 =36 + E C 36 = E C = 654 q q 04 . = 52 1E . .CEEE C E // C C E : : 9 E C + 9 C . 2 : C C E E // C C E // C . : 9 E C + 9 E E` 9 E C + 9 C E a // ` 9 E + 9 C E a // C (1) (2) : 9 E C + 9 E (1) (2) ( ) C : C C E E // C C E . . E E = = : . 2 E . : C = E C C 9 E C 9 C . E C c(X) = c( X C) = 09 c c ( ) (1) ` 9 E C 9 + C (2) 9 C E + 9 C a ` 9 E C E9 + C + 9 C - CE34 61. q . q C CEC= C E E E.C: 9 C : = E C c90 = )Cc( X : ( E = 2)CE * E : 9 C = E C c90 = )Cc( X a ` 9 E C E 9 + C EC : E =E CE()q q .Eq q 2 34 3 4 44. ( E = 2)CE * E : 4 81 CE(2+6 8 1) E 4 5 . C C C EC= : ( C)2 = * E ( C)2 = * EC ` =E C C44 (): ( C)2 = * E E3 6Indirect measarement . . C E 621 = ci ` = ci C E c( X) = c( X) = 09c Cc( X ) = Ec( X ) = (09 - c)i C ` 9 C + 9 E : E = C = 57 57 . - Eci ci = 2 31 = 3 31 = 6 = 3 2 q . q 2 = - + - 4 C 2C C2 + + = 0 0 ! C51 : 6 - 21 51 . . = . 6 ` 51 = 21` = 4 (3 6 )2 = ( + 3) 2 + 3 - 45 = 0 ( - 6)( + 9) = 0 2 = 3 * 6 ( ) C 34` = 21 2 = 4(4 + 9) = 9(9 + 4) E 4(2 + 1) = 63 - 9 2 = 81 * 8 = 441 3 4 .: ( C)2 = * E 9 C 9 + E C E ` C =E 9 C : = E C c90 = )Cc( X a () ` 9 C 9 + E C :651 21 q q : = 54 - 73 62. : 1 6 0 6 .C : C E E = : 9 C + 9 E ` = 581 08 = 040406 +1 08 4 0 = 04 6 8 = 61 4 5521 3 04 8 :4 21 6 . 841 = 521 1 841 q q 5 54 . C = ECE : C C = E // C .` = 821 a // ` = 03 C = = C C C C a = E () C = ` = C E C C C E a = = E` = 23 ( (1))` 9 C + 9 C (1)() (2) (1) (2) : = = C Eq PPT q . 1 : .64()( ) - : . :58121 C :81 C = 21 = 4 = 81 + 6 = 4 81 3 9 3 C = 81 = 4 531 3 C C = = C 7 . : = C a ` E C CE C E = = ` a E CE CE = = (1) (2)` C E CE ` 9 = 9 C E : c(X ) = E Cc(X) c(X ) = Ec(X )C 83(1) (2) - 6` 9 C + 9 C 9 a + 9 ` C c( X ) = c( X ) : C c EC C : C { = E} = E = E C C ` = E a = E C E C ` E = E a = E (1) (2) : C = = C E E 9 C + 9 E C // E( )(1) ( )(2)E` C c(X ) = c(X E)C ` C //E C C E E: C` // E C ` C // 31 9 C + 9 Cc 9q q 6 7 74.( ) 9 C / 9 E ` 9 E + 9 C C 9 a + 9 C ` 9 C + 9 E = = E C E C E // E C = : C // - E74 63. 2 . C C : E E c / C c = EEC: 9 C + 9 E : C C = E // C a // + 1 = - 3`=4 E E 6 : = + 4 = 1 2`=8C` 9 C + 9 C () (1) C C C = C C C E a = E C C ` = E C = E C() C = E ()` 9 C / 9 E ( ) 9 C + 9 E (1) (2) : 9 C + 9 E (2) . C C = 8 C = 01 = 21 C C = 2 E 4 = E. 9 E + 9 C E . C E . (1) (2) ` 9 E + 9 C E = 1 C 2842016(2)4 E() ` E = 1 C E = 1 * 01 = 5 2 2 - C C a = 8 C = 2 ` = 6 E C : (1) Ec / Cc 1 = 4 = E = 6 = 1 21 2 8 2 C E ` = C 8 q 01 11 21 q 42 . q (2) 84 q 8 9 . C9 a + 9 E ` c / c C E = a = 1 2 = 1 2 C ` C = C c a / c C = ` 9 C + 9 E : C C E = = E` C * E = C * E ( )C( ) 8 9 . : : : E C9 a + 9 C ( 2) 9 C 9 + E ` = 57 E E ` = 51 ` = C = 1 3 C a = 2 = 1 1 = 3 = E 9 3 3 6 C C a = C = E C `=3 ` 9 E + 9 C ECE = ` 9 E C + 9 C E: C E E C = E C = 1 = 8 a C C 2 ` = 821 5 8 ` 9 E C + 9 C - 93 64. : E :C C ` =( + 4) C E(2)()E .E C0 E 4 60 0 8 50 21 C - 8 : 47 6 = E C 3 2 = E 57 C E Cc(X ) = c(X c90 = )E = 6. : 12 22 32 . - 8 E5 94 C (52) . : 9 C + 9 E.04 C E : C * E = C * E 9 C + 9 E E E = 4 5 C6 (+1) C 7 (1) C (1) (2) 9 C 9 + E C01 C 2 C C E c C( a)2 = * E C = 21 = 3 :3 57 c c 5 4 4 E9E3 ) = 4 2 = E. E C .E6( - C E ( C)2 = * E : 9 C 9 + E CC :E . E 4 . C C : 9 E C + 9 3 C { = E}C35 3 :` c(X E) = c(X C) c E / c C c a E C E ` C E . 01 8 7 5 . . . 5 . 65. 2 2The Relation Between the Area of two Similar Polygons () . () .The Relation Between The Area Of TwoCSimilar Polygons C . 1- : 9 + 9 C .2- C 3- C E C // /E E E/ /E E 9 /E E + 9 4- : 9 + 9 C 1 3Perimeter Area Area of a Polygon Corresponding Sides9 /E E + 9 C 44 =1 63 9639 + 9 /E E5- ( ) :: . 3EC - 2 E = E = {} C 9 a + 9 E ` c(X) = c(X) C = = C E C E :( ): C = C E E (2) C9(W ) = 1 * C = * C 2 E9(W ) 1 * C E 2 - ( ) - - - - . (1) (2) : 2 2 2 C9(W ) C C C = E * E = E l l = b l = b b CE . E9(W ) : C9(W ) = C l C 2b = C E E E E9(W ) C9(W ) : E9(W )C =Elb 2 - - - .. . :EC1- 9 C + 9 E . 2 C9(W ) =C l b E9(W ) E .2- 9 C + 9 E EC C Cc E Ec .2 C9(W ) = C l b E9(W ) E . - : . - - - .c(X) = c(X) = 09c(Xc) = c(X)` 9 C + 9 E : (1)E . W : C = C = {} . ? . : . 2 2 C9(W ) C = E l l = b l = bb CE : E9(W ) : .: 9 C + 9 E 05 05 75 62 72 ().15 - 14 66. C : C C E EC E = 3 E // C . 4q : q 2 . . E 9 C =4872. : 9 E C. E .() 2()9 ` E C 9(W) = 487 * 94 = 441 E C 9(W) = ` 3 j 7 487 a E = 9 C - 9 E C 2 ` E = 487 - 441 = 046 22 q . q : C c E 2q 05 q (3)3C8 C 9(W ) = 84 : 9(W )E2101 E2 4 : 9 09 . 9 C + 9 E 2 ` C9(W ) = C l 4 = b C = 2 E 3 9 E9(W ) E 9 C a 9 EC = E = 21 3 ` 9 C 9 E (9 09 C ) = 2 ` 9 C = 06 3q (3) q . q : q 9 E C: E a // ` 9 E C + 9 C E) E ` C 9(W ) = C l b C C 9(W25 - 1- q 25 q . 2- . . C (9 C ) = 1 2 C * * 99 C E Cc(X ) = Ec(X ) : C : C = 8 = 4 = 21 = 4 E 01 5 51 5 9 C + 9 ` 9 C + 9 E : 4 2 C ` (9 C ) = ( 5 ) c / c = = (9 E ) ` (9 C ) = 1 C * 2 (9 ) 1 * 2* C 2 = ( )= C * 24( = )( ) - 84 = 61 (9 E ) 52 * 2 ` (9 E ) = 526184 = 57 67. q q : 2 C E C9(W ) = 3 E9(W ) 4 54 3 . . = 82 . C 1 01. C 2 C9(W ) = 46 = =101 * 015 9C = 46 = ` 510 1 10 j *2 = 46 * 01 * 01 * 015 * 015 - 046E22 E . - . The ratio between the area of two similar polygons .1- (1) (2).2- (1) C . - .E /= q 3 35 q Google Earth : - ... . / (1) 62.C35 C9 a + 9 E (9 C ) = 3 (9 E ) 4 3 ` (9 C ) = 21 (9 E ) ` (9 C ) (9 E ) 3 54 3 (9 E ) = 2 ` (9 E ) = 09 3 a = 82 = 2q 35 q : . q (4) q : C 2 ( C) /) = / = ( / ( C ) C 2/ 2 . = 41 3 55 . 4 . ( C )E 9 C + 9 E / / / / = 1 ( C 9 ) E C = = C = C + + C E E E + + E C 1 = E 9 C /C/ / 3 /E 9 E - 34 68. C = 2 C E = 2 /C/ 5 /C/ / /E/ 53- (2) . E C a ` = 1 61 C/ / C a + ` = 21 = 3 61 4 W 9 W1W ` 16 = 2) 3 ( = 1W 2 2 4 531 9 16 = 2W 2 240 = 16 * 135 = 2W 9 4 5 55 65 : : = 3 : = 3 4 C C 4 = 3 C = 4 3: 4 ! 0 = 3 C = 4 65 . 25 = 2) 5 ( = 1W 9 3 W 2 W- W ` 19 - 25 = 2 W 9 2 ` 23 = 61 9 2W = 81 . = 23 + 81 = 05 . q 6 7 65 75 q .44 /C (2)./E E//: ( ) = ) = - 2 . ( //CC . 4/E E ////C////: C E + /C/ / /E C 2 ( W C E) = /C l/ b :C//( W C E )/ : C C C /C E C/ E /C// a C E + /C/ / /E ` (). : 2 E 2 E C9(W) C9(W )E C 9(W ) / / / = l/ b /E / / / = l/ / b /E /C9(W/) C9(W ) E C 9(W ) E E C ( ) a/ / = / /E = /E/ = / /C45= /E l/ b E2 - E C9(W) C9(W )E C9(W ) = = ` /C9(W/ /) /C9(W/ /E /C9(W )/E/)= /C l/ b C2 C9(W ) + C9(W E C9(W + )E) ( )://(): . a = 52 2 ` = 004 E / / // ` 9 C/ / + 9 C C c( X/ c(X = )/E) ` / // /E E 9 C/ C 9 + /E E2 - / C c(X/ /) = c(X) = 1 4 EC = /C l/ b2 /C9(W/ /) + C9(W E C9(W + ) E ) C 2 ( W C E) = b / / l : C ( W /C/ / /E/) ////// C 2 ( C) /C l/ /C( = b )2/ 2 C + E /C/ / C/ /E/ = 1 : C 3 C E ( W C )E / / / / /C/ / E ( W C ) E/ C E /C/ / /E/ 4 : 52 C E : / C/ / C /C/ / /E 1 : 4 522 . . 21 61 = 5312. . C E : c40 = )Cc(X = 3 C 16 = E. 4 : : c(X) : : (W C ( W : )E ) a C + E ` c(X = )Cc(X) c(X) = 04( c ) ` C = 4 a= 3 C 3 4 C E = 61 ` 4 = = 3 *461 = 21 3( )( ) (W C ( W : )E ) = ( C)2 : ( ) 2 = 612 : 9 61 : 9( ) 2 - C = 4 = 3 !055 69. 3 : 4. 5222 . a = 3 : 4 ` = 3 : 4 2 = 9 2 = 61 522 ` 9 + 61 = 522 = 9 + 61 = 9 2 ` = 9 * 9 = 18 2 ` = 61 * 9 = 441 C E . . 2 (W C (W : )E ) = ( )2 : ( ) ()() (1) = ( W C )E ` ( W )= l b (2) (1) (2) : ( W C (W : )E ) = ( )2 : ( )65E a C + E 2C 22 3121 C` ( ): ( ): ( ) 2 = (5)2: (31)2: (21) ( ) = 0012 ( ) = 001 * 961 = 0272 52 ( ) = 001 * 441 = 6752 52 q q : - - - . : C = {} C 6 7 75 . CE a CE ` 9 E + 9 ( C )E // C // = 3 C 5 CC . C = 52 = 9 = 21. L c 9 9 C (1) . E9 a + 9 ( ) E ( ) = 52 ( ) = 52 ( ) 961 ( ) 441 - 5 = ( E ) ( ) C C : 5 : 3 23 . a C + E ` 9 C + 9 9 E + 9 ` 9 + 9 (1) (2) : ( C )E E = = E = ( ) 2(2) - 54 70. H 1 2 : = H C = 2 H = 2 H a C E C E : 2 (W C (W : )E ) = ( ( : 2)E ) C C C C : M N . (W (W + )M (W = )N ) a + M ( W C( )M) ` ( W ) = ( C)22= c(X a) = 09c( C)2 + ( ) ( C)2M2N(1)2( C)2 + ( )2 = ( C)( W )M2O( W ( )N ) 2 a + N ` ( W ) = ( C) 2 ( W ( W )M )N ( C)2 ( ) 2 ( W a ( W + )O C( = )O)2 + ( C)(2)( W )N (1) (2) + ( W ) = 1 ( W ) (W (W + )M (W = )N ) ( ) = ( )2 = 2 1 = 2H 4 2 2H C ( C )EC C C C = 5 = 31 C C C . 0012 . : C E ( ) ` 9 E + 9 E CC a = 3 2C 9 E + 9 E CE E 9( W ) 2 9( W )E C75 - 3 C C =5= a C C + E9 a + 9 C E C = ECC C = 3 E = {} 2: 9 E 9 E C c(X E) = = Cc(X )E c(X) = )Cc(X E` (9 E ) = ( E )2 = ( C )2 = 52 9 C C (9 )E C : C CE C E E . C = 54 5 EC C = 4 : (9 E ) (9 C E) (9 C ) (9 C E) : 0212 0722 . 2 3 0942 05229 7 61 61 7: 5 : 0472. .64 - 62 72 . 71. 2 2Applications of Similarity in the circle . . . . Chord Secant Tangent Diameter Common External Tangent CEEEC (2) (1) C (3) (1): * C * E (2): C * E C C * C (3): C * E C ( C)2 C E : * C = * E C Concentric CirclesApplications of Similarity in The Cricle . .Common Internal TangentE (1)CE (2) : E C : E C = - ( ) . . : . . E C 85 ` * C = * E . . - - - - - - . - ( ) - - - . - - - - . 85 66 82 13 (). - 74 72. 9 q q . C a 4 = C 34 : C { = E}C = 4 = 9 4 = E 3 E` 4 = C = 3 ! 0 ( ) C a { = E} ` * C = * E : 4 * 3 = 9 * 4 212 = 63 2 = 3 = 3 = 3 3 ( ) E C ( ) . 85326CE8 2 3 3 C51-94 2 : C { = E} C = 5 9 = E 3 = E. 3 E = . C a { = E} ` * = C * E ( ) : ( + 5) = 3 (3 + 9) 2 + 5 - 63 = ( - 4) ( + 9) = ` = 4 = -9 ` = 4. - E9 5 C95 (1) C a { = E} (2) 95 06 . ` * = C * E : ` = 64 5(5 + ) = 4 * 21 : C a { = E} a }C{ = E ` C * C = C * E C ` C * = * E * 3 = 3 * 9 `=9 3 * 6 = 2 ` E = {}C 2 2 * 3 = 42 C * C = C * E C ` = ( + 5) = 2 * 6 (51 - ) * = 9 * 4 2 + 5 - 21 = 2 ` - 51 + 63 = 0 ( - 3)( - 21) = 0 ` = 3 = 21 = -5 ! 52 - 4 * 1 * -21= -5 ! 37 84 - 22 ` = -5 + 37 771 2 = -5 - 37 2`=3 73. 2 4 E5C28C( ) C32E 465E . 1 : C E ` ( = 2)C * E q q (1 06) 3CE : C C E . 4 = E 5 = E C5 4 E 3 06 . a C ` ( = 2)C * E () ( 36 = )5 + 4(4 = 2)C ` 6 = C C . ( ) E CC0621 4 3 988 5 5C - :C 9 9 + E C 4 5 16 26 .EE c( X = )C c( X) E C . C C C = 51 C = 21. C E 4 = E C C C = 5. E . C 5 21 C * E C a = 4 * 51 = 06 C * C = 5 * 21 = 06 ` C * E C = C * C a E = { C * E C }C = C * C ` E E 514E( ) 89202 3 E5 4 ( = 2)C * C C - 4 2401E42 0163 5 C E . C C E C C = 56 = 21 = 21 = 01 C E ( C )E * C = * E : C E . * C = * E E C = : 16 E C 2 . q q . .16 : C a { = E} C * = 5 * 02 = 001 * 100 = 10 * 10 = E ` C * = * E ` C E . C a { = E} * E = 4 * 9 * 12 * 3 = C ` * E = * C ` C E . - 94 74. C a { = E} * 216 = 6 * 36 = C * E = 82 * 27 = 6102 ` C E C C = 8 C = 4 C E C E 12 = E. C E C a * 64 = )12 + 4( 4 = E C C ( C)2 = (8)2 = 46 ` ( C)2 = C * E C ` C E . C a * C(36 = 9 * 4 = E C)2 = 63 ` E . 60546 CC9 6 : : . . .4 = H C a E ` * C = * E 72 * 72 = 9 * (2)9 - H 281 = 9 - H 45 = H 54 . 9 - 7 5 6 36 .E6 4 E8 q q ( - - - ).CEEq 6 26 36 q .6C 6E26 : 4 21 C E C C = 01 () 36 = 64 - 10 = E C 2 C( a) = 63 C * 36 = 10 * 3.6 = E C ` ( C)2 = C * E C E . C * E C a = 6 * 31 = 87 ( C)2 = (9)2 = 18 ` E :845 C45 9245 2E - : C a { = E} (1) ` * C = E * a = E C ` C = = 6 4 - H2 = E (1) ` 4 (265 = H 6 * 6 = )4 - H : = 56 6 * 6 = 634 * (56 * 52) = 63 . 75. : a C E C E ` * = C * E = 9 (9 + 7) = 441 8 : 8 . 8 081 . 0471 . 8 C 8 8 : C . ( ) C = 081 3480 = 1740 * 2 = E ( ) 3660 = 180 + 3480 = E C : ( C)2 = C * ( E C ) = 081 * 0663 = 008856 C - 218 218 . C E ( a )2 = * C : : 4 . 4 21 : 4 . 2 .4 2 C E36 97E` E : - * = C * E ` ( )2 = * E a ` ( )2 = * 144 = C = 21.: C ( E ) C C C a C 2 ` C * C( = E C) 2 `3=H 2(2 + 2 )4( = )H : = 3 : ( C)2 = (4)2 = 61 * E C 16 = )3 * 2 + 2(2 = E ` C 4 E - E : C. C C ( ) E E = 9 7 = E 3 5C15 76. 8 C5 3E ( ). EC5351 E E C/ / / / / / /C / E CE C/C C + E /C/ / /EC ( - - ) . . () ( ): () () C /C C 2/2- 3-E*52 . C E /C/ / /E . / / 1- : E / c -/Cc Cc c / c c/ - Ec Ec Homothetic Transformation || = 0 || = - 0 | = /C| * CW( )/C( = )C: /C C .( ) C C: 101/CCCE 5 C C = 31. C E C E = 7 C E CC /C /C /C C C460 /C C/C C /C C /C - :W( 2)( )/C( = ) CW( 2) () = (/) /C / /C/ // C . 2 ( W )/C( = ) C ( W) = ( ) /( ) 2 04 3 8 2 57.4 C = 912 = E C : 82 13.C( ): /C/ C ///: 9 /C/ 9 CC = C = c C ` 9 /C/ = 9 C : // / / / : C = = C = : /C/ // C C C : . . . . : | | = // C ==2 C// C == 1 2 C/E//C/C//E/CC /C/ / /E C E = 2 : /C/ = 2 C/ / = 2 / 2 = /E C E 2 = /C /E,E /C/ / C = 1 : 2 /C/ = 1 C/ / = 1 2 2 / 1 = /C C 2 - 25 - 56 77. = - 10 2/C C /C C /C / / / / // /C/ = |- 1 | C = 1 C 2 2 / / = |- 1 | = 1 2 2C / | 1 -| = /C 1 = C C 2 2 : 9 C 9 E C 9 9 C E 9 C 9 C 9 9 C 9 C E 9 C 2 C E 2 3 . /C/ / /E C E . ( W C )E EEC 23 : = 2 * . 5422. : 2 2 2 081 5 089Pantograph : 4 - - - - . : 1- . 2- C = C 3- E . 66 46 . :C( W /C/ / )/E : EC 1 2 - 1 2 1 C 2 2- C 1- C 2 3 2 1 7 2 6 7 - Two Similar Polygons ()Similarity Ratio /C/ / + /E C E /C/ / /E C E / / / / / / / / C = = = E C = E ! 0 E EC C Golden Rectangle .Golden Ratio 8161 : 1 : : . (1): . () () 52 . : q ( )Dynamic Geometry q () Edit numberical edit 52 (). q . q 52. (2): . 1: . 2: .C 5.2C The relation between the area of two similar polygons 3: . : . 4: . - () .76 - 35 78. - The Triangle Proportionality Thearems3 The Triangle Proportionality Theorems : : ( ). : . . : : ( ...) . : ( ..) . : ( ...) . ( ).). :( ()..). : . . : ( . : . Ratio Proportion Parallel Midpoint Median Transversal BisectorInterior Bisector Exterior Bisector Perpendicular 79. ( ) (3 - 1): .(3 - 2): .(3 - 3): . - - - - - . (003 .) : : . (--) (). . . 21 . - - - - - . - ( ) - - - - - - . - - - - - . - - - . 80. 3 3Parallel Lines and Proportional PartsParallel Lines and Proportional Parts . . . .C1- C C E E // C .2- : E E C C 3-E C E C . E E . C E C E Parallel Midpoint Median 1Transversal . C: C E // C E C : E = : E a // ` 9 C + 9 E C ( ) : C = C (1) C EC : C E a C . . . . . . . 07 - - - . ` C = E + E C C = C + (1) (2) :E (2) E + E C = C + C EC E E C C + = + : C E C E C C 1+ +1= E EC C E = ` EC C : C = E C ( ) E - . - ( ) - - - - . - - - . q (1) q 27 . . q (1) 17 q 07 87 83 04 . - (). : (17) -2 6 = 5 = 21 3 242 = 01 03 + 4 = 51 32 = 8 = 2 2 2 + 4 - 54 = 0 . ( - 5) ( + 9) = 0 = 5 1 2 3 65 - 81. ` E + E C = C + E C EC : E a = C C : E = C 61 21 : // C = 61 = 21. C = 42 . = 12 C. C C a // ` = 42 * ` = 216142 = 81. : 61 = 21 C C a // ` = * + : 612121 = C ` 21C = 822112 = 94. : E // . ( ) C C E635 -201 022C42EE 51 3 +4 C C C C C E : = E ( ). C :E C C EC EC C = C = E C E17 - . (1) 17 . (2) 27 . : C a { = E} C // E ` C = : }C{ = E E C C // // . E C = 6 C = 5 12 = E C = 4. C E . a // E }C{ = E 21 C : 6 = 5 a // E 01 21 = 4 E5621 C 4E EC ` = C E ` E = 84 C E C ` C = C 9 C :E` C = 01 : E // C C { = E} : C = 8 = 9 = 21. . E : C = 6 = 9 18 = E. EC . 1 . CE CE C E C EC : C E C C E . E = E // : 9 E C + 9 C .27 - - E Cc / c . E 8 = 9 21 E 135 = E C a { = E} C // E` = C E = 6 81 51 C = 27 .. ( ) . C = E C aC E E ` C = E C C C Cc a EC C ` 9 E C + 9 C E Cc( X) = Cc( X ) C( ) ` E // - 75 82. (3) (4) 37 (3) (4).EC a E EC ` E 3 = E C 7` E // C = E C a 6 C = 24 = 7 42 6 9 C :E a // E C(1)C a // E (2) ` = E (1) (2) = C 1 5 10C82 20 E 64 CC 9 C : a // C C ` = C = Ca E` // E 9 E C: C C a // E ` = E C (1) (2) : = C E 9 C :EE(1)(2)E37 - C E . // E C C // E . : // C : C C E E C .E ` // C : ( - - - ) . (5) .: CC9 E // E E = 1 E 2 9 C E // E E CE: E . = 1 E 2 // = : . : // . C : C C E E // C E // C ( )2 = * .E : . C C = E = E C ` E // E C a { = E} C // E C 06 = 54 ` = C 54 + 501 C E * ` C = 06 54051 = 002 .4785E83 // E E C . // . E a = C - C E C C // 9 C :` E 9E` E // E6 3 E // . C C = 6 = 3 41 7E ` = E C 9 :E 42 E C ! E C a C( )24 = 82 = 7 E C 02 5 E C9 a + 9 C ()` E = E C = 1 C 3 5 = 1 ` = 51 3 C = 9 = 3 51 5 E 01 5 C = E C a E52 - 61 = 3 = 3 = 1 C = 4 = 1 8 2 6 2 C = E // .5 8 a E C C`=EC : 3 = 6 = E CE3 : C C . : E // . C4 - C54 06 501 E 83. : . .33 Cq 57 q .103E39: . . q 1 2 q . . ( ) : 1- 1 // 2 // 3 // 4 / C /CE/ / /E . 2- : C C / / / / / C/ /E/ E C/ /CC/ / 3 /EE 2 2.........................14Talis' Theorem .: 1 // 2 // 3 // 4 / : C : : /C = E/ : / / : / E / : C // / 2 3 // / 3 4 . / // /C C a / C // /C / ` C / /C : C = /C EC // C //E9 C 9 C E // C34 7 / / C = C // = C E = C E 3 + 2 2 + 4 = 51 21 = = 5 11 4 ( ) = . + 1 = 9 6 5 2 = 01 6 (5) 5 4 - 1 = 3 3 - 2 2 (6) 57 C / / E C / /C E E / / = E/ E / CE CE C / / = C/ = .. . / C C (8) E // C57 - (5)1 2/q q 57 67 7 8 9= /C / / = EC C /01 + 02 = 21 + 8 =6 = 55 4 5 + 5 = 45 = 94 = 8.9 5 = 52 39 - 6 = 8 - 2 =4 (6) 9 C :E a // E` C = C E 031 C = 93 33 ` C = 33 * 01 = 011 3 - 95 84. : = / / = / / = / E ( ) 7 77 9 // E C // E = = 2 ` - 3 = 3 + 1 2 - 3 - 4 = 0 ( - 4) ( + 1) = 0 = 4 = - 1 2 - 1 = 31 9 C : C C a // ` = / C /C C / : = / / /C/ = / 9 E: / / E ` = E/ E/ / = / E C // // E C = E ` = =4 4 - 1 = 2 + 7 = 4 * 4 - 1 = 51 - 4 = 51 = 41 3 2( ) (2)/ (1) (2) : C E = = /C/ / / / E / / ` C : : /C = E/ : / / : E / . : C C E ECE C a // // E // C = ` E E 02 82 = E 51 = = 33// C E /C/ /C C82 C : C // // E // C = 82 = 02 E = 51 = 33. : E 02 / 15 E 33 ` 21 = E = 44. . ( ) 3 21 51+53 + 2 2 + 467=7( ) (1)95+1 (9) C // E = ` E = EC =4 2 + 3 = 3 - 1 =3 2 + 7 = 3 * 4 + 1 /60124 - 13 - 2 - ( ) . . : 8 87 . (01) 01 C // // // E 77: . ( ) E C q . q E C . = E C = 21 ` 2 5 C ` = 84 3 = E 3 = E 5 . 081 5 C ` 108 = E 06 - 85. 1- / C C C / / : // : C = C / C C : : = / C C : / // ///(5 - 2)2 + (5 - 3) C 2 : : = (11 - 5)2 + (9 - 5) = 31 = 1 31 2C /2 //CC2- . / 1 // 2 // 3 // 4 / / : C = = E: /C/ = / / = E 1//E/E 2 23: C = 1 = E C E 2 4 . C a // // E E = E ` C = : 2 - 3 = + 2` = 5 a E = E = 5 ` + 3 = 5 + 12 - 3C2//C `=3+2+3E: C = = 1 2 +1(11 9) (5 5)E)3 2( C . . ( ) E 2 C 2 + 7 3 -13 + 1 C2 - 33 + 1 2 - 1-41 - 2 E 4 C - 21 08 E` // E C // E a C C E // E C a // C E ` C = E : C = 21 + 80 21 80 ` C = 21 * 821 = 291 ` C - 91 80 : E (11 9)8 6 (5 5)C C = 081 = 2 C : : 3 : 4 : 5 = E E4(2 3) C 0184622C . CE 4.2 : C 14 C . 09. 42 .87 6C E :2 - 5 : : . E C C = 21 E = 08 = 21 .1 042 014 = 004 = 14 * 6 = 642 3 77004 3 C. . .C+3C301E518-124+ 3+8 : C { = E} 6 = 6 C 5 15 = C 1 5 2 = 52 E 10 = E10 1 +3EC3 2+7 : C = 014 = 09 C = 04 (9 C ) : C // C (9 )E = 9 = 4 = 5 = 4 (9 C ) 3 2 9 (9 25 = ) 5 ( = )E - - 16 86. C E C E // C C C : =5 = }{ // = = = . . 6 : : = E C = E = C = . C = = = E C - E = . C = = = = E C. = E C = + E = = .C : a // C C (1) = ` // E C a = ` E EC(2) C E C8 = = = C = . // E E7 // C9 C C E C C = . = E C E = . E // . . 01 C E . C = = = = E. C E .11 . 21 C C E E = E C C = C C . C = C = C . E . = E C a ( ) E 31 C E = E C = E C C E C E // C . C = . = (3) ` E (1) (3) C = ` . E41 C E . C . E C E . : // C . : : 83 04 . M51 : C = C = E ................................C=EC = C E = ................ ................ ................ C C= =C................E E C E C = ................................61 ) ( +++ +71 : E C } = E{ C C E // // E: C 81 C } = E{ C E // C // E : C * = E * 91 : C ++ + + E+ 02 C E C // E // C E C E C . : = C. 26 - C = E 87. 3 3Angle Bisectors and Proportional PartsAngle Biesectors and Proportional Parts 1- C E C .E .C .2- E C E C . .E 3- C . E C 4- . . EBisector of an Angle of a Triangle 3 ()C 2 1E ()3 Exterior Bisector Perpendicular34 : E . . .C : // E C C . . - . 1: C E C c C ( C ). C E : = E Interior Bisector2 C4 Bisector97 . . . . - - - . - ( ) - - - . - - - . . 97 58 . 93 04.- (). - 36 88. E C C c // E C( 2c = 3c) C / CC E E = C . 3.C E E = C C C = 8 C = 6 = 7 E C c C .E E E C E E C a c C ` E = C ` 4 = 8 = E C a = 8 C = 6 E 6 3 a = E + E = 7 ` 4 = E 7-3 E ( ) 3 4 - 28 = E EC()E7 ( ) C C 51q (3) q 18 .E68+47+154 E 9 C EC a E c` = C E ` C = 41 = 7 81 9 a 9 C = 08 C = 41 + 81 = 23 14 C E81 ` C + = 08 - 23 = 8408 - ` C = 3 C = 5 ! 0 ` = 4 () : 9 C E C aC c +1 EC ` E = C ` 5+195EC C . E c C E 14 = EC E = 81. 9 C = 08 : C .q (1) (2) q (1) (2) . 6 8 7 ` 28 = E 4 = E 3 = E = 51 = 9 9 9 C a E c2c / 1c ( 4c / 1c)4c / 3c E = ` C CE +1 +4 ` 5 = 8 = 4 = 46 = 61 4 ` 9 C = 3 + 4 + 4 = 21 = 21 * 61 = 291 q q 9 C E C a C c ` = E ` C + 4 = 7 = 01 : 5 E C E C a C cC E ` C = C 3 42 = 5 E ` E = 04 q C E . q q C = C E q C // .C42 q q C C E E E = 46 C a = 3 C 546 - q (3) 18 q (3) 28 . 89. C a = 7 9 84 = 61 9` C + = 7 + 9 9 . ( ) .( )` = 72 C = 12 C . E C Cc .E 24 = E C : C = 3 : 5 9 C . 1- C C ! C:C E C c C C .C C EC : E = C = C E E = E E C E C . ()2- C C Cc E EE C . C E C = C E C C C E E . 3 C C = 6 C = 4 = 5. E C Cc E C Cc . E . C E C a C Cc Cc E C : E = = C ` = E = 6 = 3 E 4 2 a = E + E = 5 - = = 54 ` E .6E5 : E C a C c C C c 6 ` E 18 - E + E = 3 + 2 2 E 5 = 1 - = 3 - 2 2 ` = 01 2 E = E + E = 2 + 01 = 21 5E 3 C C = 3 = 7 6 = C. E C C c E C C c . C C . E C C . . E C C c 9 C E: C = E C * C - E * E: C E C c C E C = {}E C : ( C = 2)E C * C - E * E : C E E C C E C : 9 C C9 + E () C = C ` C * E C = C * C E + E C( * E C) = C * C ( C = 2)E C * C - E * E C E * E C = E * E ( C = 2)E C * C - E * E : C = E C * C - E * E C C = 72 C = 51. E C Cc .E 18 = E . E CE C = E C a * C - E * E28` 15 = 225 = 10 * 18 - 15 * 27 = E C81 51 E C a c C 81 = 72 51C E ` E = C ` E = 0172CEE3 : = E = C = 6 = 2 E C 3 1 a : = 2 : 1 ` = C C . = 5 ` E = 2 2C a 2 = E = E + E = 7 1E 7 E + E = 3 !E= 7 2+1 3 ` E = 1 ! E 1 = E + 28 = 7 + 7 = E 3 3 = 41 E a E C C . E C9( W) E ` C9( W ) = = 82 41 = 2 3 3 . q 28 q 4 . - - 56 90. q q 4 . ( ) E C = 61 7 2 = E C 3 3 = 51 3 3 = E C 2 2 = 45 7 3 2 = E C 74E6C ( ) C: E C C E C 9 C . : (C)2 = * - C * C : E C a C C 9 C . C E ` C = E C C = E = * E = * E (1) 9 E C C 2 2 2 ( C) = ( )E C( - ) E 2 = ( - ( )E + E) - ()E C = * + * E - * E E * E - ( )2( 2) E C (1) (2) : 2 ( C)2 = * - E * E ()E C 2 ( C = )E C * C - E * E ( ) ( C)2 = * - C * C q q (5) .CC2194219E6( + 1)C : E C 9 C E E Cc. C . : // . 9 E C: E a E Cc 9 E C: E a E Cc 9 C : E C a (1) (2) (3) E E Cc C .EC EC ` E = C EC ` E = ` E = EC C = (1) (2) (3) // . - 38q (5) 38 q (6) 48 . 9 C : c ` = = 6 = 1 2 12 C C 9 C = 4 = 1 2 8 C a = = 1 2 C C ` // 9 C :E c C ` = C EE 666E9CC - . : C = * - C * Cq q 38 : = 7 C = 2 12 =8 C = 3 01 C 5 : : C c C (E+ 345)4 C7 (1) 9 E C E E C c = E C C E(2) 91. : : // C8 6 4 9 C a :E cC C ` = C EE 9 C : ECE C E EC = C EE C = C : C Cc C . C C = C E E E ` = C C E C2- :: .E 9 C : C = 81 = 3 C 21 2 = E - 6 = 9 - 15 = E ` 3 = 9 = E E 6 2 EC E C c C E a = CCE C a = E C E = E = 21 = 2 (1) 3 18 C 9 C EE 21(3) : 9 C :E - ` EC = C21` E CC c . q 6 48 7 58 q 7 8 58 . C C = 81 = 51 C = 21 E 9 = E C = E C . E C c C . 81 (2) (1) (2) E . E C E C ` = C E EC1- 9 C : EC E E = C : E C c C48 a c : E . E(1)C C (1) (2) (3) ` = E a // E E C C ` = 9 E C a = E C C ` E = 2 C 2 a = 3 E 3 C ` // E ` = 2 (2) 3 C ` E ` // (1) (2) = C C : a = 9 C C C 9 : . a // ` = 48. : C81 ` c = - C E (1)76 92. a ` C = C = C C (1) (2) : = C(2) C a = E C C ` C Cc 9 C = C + a = + ` 51 = 81 = 03 21 C E C = 81 = 21. E C 2 C = 3 E // E C . Cc . ` C c C C . E C .E : 9 C : C a = c( X) = 09c ` c( X C) = )1( 45c c( X c 90 = )E C ` CEc( X) = 54)2( c (1) (2) c( X C) = CEc( X) = 54c E E C = : C CE E = C E . E E a = (1)` c 9 E . C a ` Cc( X ) = 09c = C E a c 9 C ` C E CE C = (1) (2)E CE : C = ( ) (2) C CE ` C = ( ) ( ) .C E . . C c . : C E C E = {}. C = = 24 56 = E C. C C` C Cc 9 C E : = 24 = 3 4 56 E 3 = 3 + 4 ` +E ` = 3 7 E C9( W ) = 3 C9( W 7 )EC - CE 58 : 4 E 3 = E C = 81 C . ( ) = 01 ` c ` (9C ) = 3 * 1 * 24 * 65 = 405 . 7 2 ` C = C `=5 ( ) C = 42 01 8+ -3E168CC0121EC E E : E , E C c c .86 : C CC = E C C Cc : = E C . - : 14 24 . 93. 3 3Applications of Proportionality in the Circle . . . .Applications of Proportionality in the C = C = = E C EE CCEC C E C 9 a + 9 C ( E) ` C = E C = 2 = * Power of a point Circle Chord Tangent Secant Diameter 3 51 42 . : Power of a pointConcentric Circles Common External Tangent Common Internal Tangent C H 2 X( )C: X( C( = )C)2 - H 68H 1 C :X ( 0) C C . X ( 0 = ) C C . X ( 0) C C . - CCircle ( ) . : . . . . - - - - - - - - . - ( ) - - - . - - - - . . 68 29 34 44 () : - 96 94. . . C 5 : X ( X11 = ) C () = X () = -61 . ` C X a ( 011 = ) C X a ( C( = ) C)2 - C( = 11 ` 2H)2 - 52 ` X a () = ` X a () = -61 X a () = ( )2 - ( = 16- ` 2H )2 - 52 X( )C X( C( = )C)2 - (1) 78 : X( 0) C C C X( 0) C C X( 0) C q : q (1) 78 . C 3 : X ( ) = X ( ) = -4 X ( 15 = ) C 2 C : X( C( = ) C)2 - H2HCH 3 C : X( C( = ) C)2 - H2 = ( C - C()H + )H = - ( C - H)( C + )H = - C * C C /C//CC-HH C E C /2X( C - = ) C * C = - C/ * C/78 - 13. C 32 C C = 3 C : . : ` C 31 = H a C = 32 C X( C( = ) C)2 - C - = 2H * C (32)2 - (13)2 = -3 C * C ` C = 21 ` = 4 C = 21 * 4 = 84 EC = E = E ` E 24 = E a = E ` 196 - 385 = E ` ( 385 = 2)24( - 2)31( = 2)E X ( = )C. 2 8. 21 E = E E . 3 C. C C E E 9 = E E = 7 . ( X) = ( X). C = 01. C . 9 C a = 01 ` X() = ( C ( = )10 + C )2 = 441 ` 8 = C ( a 10 + 2)C 144 = C ` = 21 ( a )2 = 441 - E C7 a . ` X() = * E = * C (1) a . 2 (2) ` X() = * C = ( ) ` X() = X() = 9 * 61 = 441 (1) (2)88 - X(C( = )C) - 51 = (C)2 - 9 2 ( C) = 42 C=2 6 ` 2 6 . 207E C = ( C - C( )H + )H 2 = C * C = ()E C ` C = X()CX( C = ) C * C = C/ * C/ = ()E C X (4- = )C` = 3 2 : X ( C `015 = )C X () = 0 ` X () = -4 ` C = 2 6 = 3 = 5 C 3 : X (15 = )C` C = 6 ` = 5 95. . X ( X = ) C( ) C C . : X() = X() X ( X = ) C( = ) C X () = X() = ` C . 3 C C E . : C ( X) = 63 = 4 = 9. C E . : E : 1- . : C { = E}C E C : . E C C011c 06c cE E54c74cCC cC65c - 021cC c ( )2 = 08 24 8 = 2 01 = E` = 01 = E ( )2 = (8)2 - ( 01)2 = 45 = 3 6 .561c051c511cEc07cc: C c( X ) = 1 [ C ( X ) + E( X)] 2 2- . : C { = E} : C c( X ) = 1 [ C ( X ) - E( X)] 2 ` 2 2 () = ( ) - = * E 2 2 2 (21) - (8) = 2 ( ) E05cE 061ccE98q q ( = )C ( )C () = () C X (C( = )C)2 - ` 3 09 . . X ( C( = )C)2 - 9 = - 4 ( C)2 = 5! C = 5 a () = ( C)2 = * E () = ( C)2 = * 53 ` () = () (1) 4 ( = )C( = )C (2) . C E q (1) (2) q (2) 88 (2). ` C . 2 9 a () = 63 ` * ( = E = 2)C * = 63 : 4 * 36 = E ` 9 = E 5 = E 2 ` 6 = C ( 36 = )C ` = 4. * 9 = 63 - 17 96. q q : : . : . q q . ( ) . ( ) . : . : . E CC E c a 9 C ` c(X = )Cc(X c(X - )E )C = 1 (X (X 1 - ) E ) 2 2 E c a 9 C ` c(X = )Cc(X c(X - )E )C = 1 (X ) - 1 (X ) 2 2 = 1 [ (X (X - ) E )] 2 E = 1 [ (X ) - (X )] 2 . E (4) 09 (5) 19 . : = 58 = 07 : 1106 . 45 .c: = 53 = 59 . . = 06 = 051 q q 09 ( ) 09 : . ( X[ 1 = )Cc( X - ( X )] 2 451c07c( + 7)cE541ccC07cc05ccC52ccC - (5) 09 . : = 53 = 011 = 001 = 56 = 04 04 ( X = 2 * c ) - 063 ( X + c )` ( X ) = 081c220 = c 40 + c` = 022 * 2 * 9 * r 063 a(144 = )C : . : (4) - 6, 7. 27 - = 111975 - r` = ()C = 21 9 C : C = 51 ` C * 51 = 441 ` C = 69 () 97. C : (X ) = 45 c = 1106. a = 063 c` (X E ) = 063c306 = c54 - c (X[ 1 = )Cc(X E ) - (X )] 2E = 1 (603c126 = )c54 - c 2 1106 = 45c 2 * c360 H * r ` - 8736. ` 637787 = H = .C 04 .c 9. : 9 C C . C E E C . X( 144 = ) C:CCE C C .: : C 6 : (18 = )C (12 - = )C ( = )C. (36 - = )C : EC(3 +071c5)c 07c : . E .C X( C 100 = ) C = 5 . C E (X .)EEC 021c( +01)c 65c75cC5 E19 - : : . . . . = 03 04 a(100 = )C ` C = 01 : C a = 5 ` 001 = 5 (5 + )E ` 15 = E : . ( X [ 1 = )Cc( X (X - ) E )] 2 (X = 2 * c57 a c56 - ) E` (X c170 = c56 + c 114 = ) E - 37 98. IMdG4HGdG IMdG d Trigonometry Trigonometry:.. ). : . : . : . : . : . : . : . . ) (. . . . . .C C...=C=:=. . .. .. Degree Measure Radian Measure) (i ! c.(i ! c) ) (i ! c.. :(..)(i ! cSecant Trigonometric FunctionCotangentSineCircular FunctionPositive Measure(Negative MeasureCosineRelated AnglesRadianEquivalent AngleTangentStandard PositionQuadrant AngleCosecantDirected Angle) 99. .) - (:.) - (: ) - (:.) - (:. .) - (: ) - (: . ---. . . . )( -) ( ) )) (i ! c ) (i ! c(i ! c (i ! c . () .. . . i! c ,i! c i!c i!c . . . . . . - - - . - - - - - . - - - - . . 100. 4-1 4-1Directed Angle Directed Angle . C: M , ,M M M:) (c-Positive measure Negative measure)1/( )1//( : =cEquivalent AngleQuadrantal Angle. Degree Measure System:-Degree Measure= (.C CStandard Position= )CcCjhGd dG SdGDirected angle.Directed Angle) C(CC) (.CM)C -----I(C) (. )( : C = C: Cc c .C - c c . : c + c = c = * + c = c c + + = c * = c M M SQdG GAGLEG dGSQdG VY : : . . 101. ,dG dGM ) C ( ! ) ( C C . .: C)?( C(=).LdG jhGd SdG VdGStandard position of the directed angle. Cc.?C.?(E )C( )(C ) C( )( ) C( )CE dG dG : E.C1.? M1 . :M 1 . 2 . : Positive and negative measures of a directed angleC.C. C 21 2 369 C3 21 2 96i1: cciicc iiM )-= i =ic cc- = (cc=cc =i )-= i=c c cc c- = (c 2) ( c: c2c:edG KGMEG idG a jhGdG beAngle's position in the orthogonal coordinate plane c. c cc c - c c -c I M , 102. dG dGCc)(i :3 #ccc . #ccc . #c . #ccc . #ccc . CiciCiiccc iCiciCcic,cc Quadrantal anglecccc.2c:cccc cccccc cc:OTQEGc. .cc. cc.. )( : = c | | + | | = c .3c:cccc:)(ci ) ci(c i) (ci ) + ci(c M: 4 -c -c -c -c3.c M) = (c = |c | + |c: |c =c c c = c +c c c c5 c c 6 ccc . :ccc4cc +c) = (c = |c | + |c: |=c =cc c 5c: c ) (i : : : c c c -c : 4: cc:6 c.adG jGhdGEquivalent angles)(iCci.i)( i) ( ) ( ) (:* +i i:) i CCi:iiC?. c. i = (i cc.:+i ic. 103. c )( . : i * !i:* !ic* !icc* !i.....Nc. 5: cc:= c +c = c -cc: : =c +c =c -cc: cdG dGcMcMcc c- c c cc:7 c- c c- c c- c 8 : c ! i N )( ..? 7: c c c:8 - -cccc -c :cc 1: ccccccc2ccdGh jQdG c:3c: - - cc -c1 2 3 c M5cc - cc6........................................c7c8 :: c........................................ )( : c ! i c ! i.c ........................................: -c........................................c........................................ -c........................................: - -c -cc -c: ? ) )(E )(( )(C ) C?E(E )(C ECdG :c9 c c c c:01)-(c =- =- ?+c -cc =c c -=c=-+c=--c=c -=cc -c cc c- c c- c c- c.M c - c -c c -c c c c c c c- c c- c c- c 104. 4-2 4-2Degree Measure and Radian MeasureDegree Measure and Radian Measure of an Angle of an Angle I=.= ? M M IRadian Measure- I , , M C.--Degree Measure MCC?Radian Measure Radian Angle.:CCC=CC==C.. = M EiH: = EiIH:)(Ei MIHiH: = H * Ei=HiE --I M ---) = = ( .SQdG VY iFGdG SdG :J fhJ Y )( E i )(. . SQdG GAGLEG E : = i H H EiE i = .c . 105. ) (E(.)i Radian angle M M I : .......HH: H I =i:.?4.r :I= H * Eir = Ei=H:r=` -* 1.rdG dGr1:jhGd iFGdG SdGh dG SdG H bdG Relation between degree measure and radian measure of an angle:.:Hrc I :)r:)r((cc.Ei MdG dG( = - cr)E c2 ) (c c c .: Ei= c r c M5dG dG .rc = EiGrad3Ec ci = rr*c ,, i= cr, c c 2,:)(=i rdGh jQdG r r(.ccrcccc. crcErcc c6c.1 c .rcrcrr=c* rc=cc=: MathD''77.71 ''54 86 ,,,c=r80121. 3: EE -EE:7. . c c c cE=r) 106. dG- ?r - C C .: =Ca` =CCC+=+a)= r(= r`:I= H * Ei, : r = Ei=Hr=*:8.c .:jQdGh fCG c jQJ Y Me. C91 = r C = r = * r C) c. *- r=cE:4c:1c. M: C91)= ( C cC C.:02?:12.:22 C: c) XCC(=C c=. C.:32c..r . 42?rr:.. .= (= = c = * c = r = * r *c c 42 = = r : = + .cCc`c = = c ca02 c = r * = r 12 = r = r = r * = r 22 = * = H )X ( = c )X ( = c )X ( = - * r * c c 32 c = * c = r = r c = * c = c(=) C c:C c) X(ccc 107. 3-4 3-4Trigonometric FunctionsTrigonometric Functions . C9: C = C===C =C=C M -C E . .? Trigonometric Function?Sine:?( M? ii=( - -- I ---I -ICotangent:-Ec) X : (CH=a-EISecant. -aaCosecant:E =i H Ec) XHCHiI -=C.i=TangentCE E =)Cosine: M M.( )SQdG GAGLEG dG bfh a . 108. SQdG VY : : . : i : . i i ...),(]-C ,[, E[.= The basic trigonometric functions of an angle)-(=i-, i= i:Ci i=i-= i:= i )j= i!:i i!i(`=ii)(ii=i=i The reciprocals of the basic trigonmetric functions)i(:-= i:i- ii: = i: : = i = i = i * = ) ( : = i = i =i !)-H: ]-+:GOTQEG i((( .( - ) E),,)C: iThe unit circle== ii= i=i:i-!i=!:i,Ci! M The signs of The Trigonometric FunctionsEcrc,,iciEcrErrErcEcrcE. .ErcEcriiccE = i ! = i ! Err,, EcErcr = i !.: : : i ?i . i i ]- [ i i - [- ] .,,,]+++r]r+--[r]r--+]r-+-r[ [[rr,+ ,,+ r1c: ccc + r`c)- (c+ 109. cc-c`c.` )- (c)- (c )- (c-=cc 1: cccCc2C ) ) - ( ( )-(=i+-=i-=i)(=j` +=`=`=i= = - = =-=i )- ( + ) ( = `()=i``====- -=i:=i%i3-= i-=i%iC c) Xi(=i)C(a=-=ia+=`= ` -j + i`- =i`=i=ii=)ii ,=i-=idG dG-=i(?+ + + +1 . 2 . 3 ..i : M: 1 . . 3 = i = i ic2=iciii 4 )-.i(. -= -=i=i= i=i-=i -= -=i,ii i3i: )() )C- (C::= i:=iCH,: ) =cr = c: ) =cC(r =c:i ,( =c=i =c =c=c -=ci,C. c = i =c : =c)-,( .( - ) Ci(The trigonometric functions of some special angles: ( -) Cc)-=c i=-=-ii( )C . ..+ + fhdG dGc- i i i i.` +c i iccc`=c :C)=: GOTQEG , ) ( : = i = i = i .(( =c 110. :: = = ) ( r r r : c c c = i ) ( c .c ) ( c ) (4c : i ) (C C . i C -ii : : + c c + c + c c c - c c c -c + c c : = c * c ) =c(c: 4 = * * - * + )-( ) ( = * * - - = - c5c:-c=ccr=c `= ra=c *=c r= ,=c- * = - =c==cj=c=`` =. 5c:-c+ccc- =ii:6c=i?c=i. : =c-cr=rr- M = - = . : i i r = r* =(i) X i dGh jQdG = - - = * - = c = - = c = r - r = r dG :dG dGC1 + c ] c : [ ) (cc5 cC : = = = ) .=c-( C=c =cc .r: ) - ( -=c2 -c c ] c : [ 111. 4-4 4-4Related Angles c,i ,ic,ii.)i(.!ib= ac .CcCc/?/b/) )/i/(, ,i!c/=-Cii, +-: )= (i - c)- =(i - c),,(Related Angles,i )(i - c i c/=- M?- b=a Iici/: =aCC!ic,i,!icRelated Anglesi- = (i - c:=c =c ) )i i= (i - c - = (i - c-c- = (c= (c - cii!ci - = (i - c)) ), i!ci-=c M= c 1c:)+i ic= (i - c i-c,ic c,i:,i.. i!c,i:.i ! c .i - c .i ! c .i ! c. -I -- M M )(.SQdG GAGLEG dG . 112. : cci- /)+i/(/,( /=-i +c=- i:i )- = (i + ci)- = (i + ci)c iii:) /i)(i + c- = (i + ci)- = (i + ci= (i + c)i= (i + c),i:+c= c = c = c- = c- = (c) - = (c + c ) ) = (c + c- =c= c : i i ! c i ! c . . : : c c r r cc- /) )(i - ci:/(/),( //==-Ci: )- = (i - c)= (i - c - = (i - c)i)i)i- = (i - c = (i - c )i ci,i- = (i - ci: = c = c) )-c- = (c- =c-c= (c=c i 3icc: :ccc)- ( c)- (c.c)- (c M c -r 2c.cSQdG VY dG dG, 1 = c ) - = (c+ c - = c = c ) = (c+ c = c = ) - ( = - - = c 2 = c )- = (c+ c - = c = c ) - = ( + c - =c (i-c) i C : : c3 = c )- = (c- c - = c - =c = c ) - = (c - c =c = c ) - = (c- c - =c :+i =c ) (c +c = c ) - = (+ c = - = c ) = (c + c = c ) - = (c - c - =c ) = (c - c =c ) - = (c - c = - =c ) = (c +c c =c ) - = (c -c - =c C = i ) C = ( i-c C : ) = ( i-c i ) = ( i-ci ) = ( i-ciCi 113. ) = ( i-ci ) =( i-ci .1 c+ (c)-cc c= (c)--c)(c= =c)-+c)=* -c== c= (c(c= c =c= c)+c(c=-=)+c(c=c=*+ )-=-)- ) = ( i- c = i -=c= cc==-+ (cdG dG(* ) = (i - c = i 4cc- = (c)-- i ) (i - c.,i .HC:/ /=H,: / ) = ( i + c = i H=ic iCi) (i -c) = (i - ci) = (i - c) = (i - ci) = (i - cii) = (i - ci) = (i - cii1) (i - c: )() (i - c ) = (i + c - = = i 114. i : ) = (c + i ) (c - i ) = (c - i ) ( i+c =ii ) = (c + i ) (c +i : c c c ca) = (i - ci`) = (i - ca) = (i - ci`) = (i - c ) (i - c5) (i - c M/i ) (i + c//=,/=-,: ) = (i + ci) i = (i + c6i) - = (i + ci2 = +i = - i : -= i = ii) - = (i + ci) - = (i + c : :i( + c ) i) - = (i + c i c c =c + c =(i c) X (i- c)i )( .ii) (i + c) ) (i + ca) - = (i + ca`i) i = (i + c) - = (i + c`) = (i + c :) (i + c) (i + c M=-( 115. M/i)dG dG(i - c/,(i - c) i:i)- = (i - c)- = (i - ci= (i - ci)7 ) = (i - c - = i ) = (i - c = i (i + )ci )( .ii - = (i - c)- = (i - c)i= (i - ci)i,i3)- (i)- (ia)- = (i - ci`)- = - = (i - ca)= (i - ci`)= (i - c(): =: dG dG 7)(i - c)(i - c :/i)8 ) - = ( i+c - = i (i + c/(i + c) i,: ) - ( i+ c - = i 9 r =i!i r r = i = i r = i r = i r : + r + r r ) = (i - r = i i)- = (i + c)= (i + ci- = (i + ci)ii - = (i + c)= (i + c) )- = (i + cii,ii4:(i + c) ) )((i + c : + r r + r r r + r = i ! i : + r = i r r + = i r + r = i r + r = i r 01 ) r - - = (ii ) = (i - ri 116. dGh jQdG - = (i + c(i + c=)`i(i + c-=) `ia)= (i + c ) a .(i + c(i + c)8)i i = i a { r } = i (b X =a X ,b b =a b ,b L =a L) :IQdG Y dG dG Od dG dG General solution of trigonometric equations as the form [tan(a) = cot(b), sec(a) = cscb(b), sin(a) = cos(b)] bc=a =i(cr = i + i ` i = i : ` r = i ` r =i` r =i+ r +i { r } ` = i` = i { r } ` = (i - r ) : r ( = i - r ) ` r = i r = i - r ` dG i : ) -( b a) b ac =b+ab=ab=ai? r r:b a(=b+ab- r =a=b- ab+r =a:: (N:)!br + r =b!abr + r =b!ab =ab- r =a:b ab - r =a:(b - r ) = a(=b+a(N r( +r,rNr + r =b+a)-)b =a (b - r ) = a:: r !b=a! ar =b+a rr) r=aN)(Nr ( +r)(b +r-)b =a (b - r ) = a:rb=a)!a Mi=i5: i(Nr + r=i:= ir + r =i!i)r + r = i +i::bdG d FGKEG f r + r=iM r +r =ir + r=i-i: r +rr +r : 9: i=i= (i - r )i=i( r - i)
