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2 I 01 1,, 4, 4_4_4=64() 2 (1) 4 (1, 3), (2, 2), (3, 1) 3 8 (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) 5 3+5=8() (2) 3 3, 6 2 2, 4, 6 3, 3 3 2_3=6() 3 X X,, 3 3_2_1=6(), X 2_1=2() 6_2=12() 4 2_1=2(), 3 3_2_1=6() 2_6=12() _6_ =35() m=35 6, 2 7_ =21() n=21 2 m+n=35+21= _5_4=120() 6 (1) 58 (2) dcbea ,, 3. G G, G 2,,

3 Middle School SUPRM Mathematics G 2 3_2=6() (1) a 4_3_2_1=24() b 4_3_2_1=24() c a 3_2_1=6() c b a 2_1=2() cbdae, cbdea cbdea =58() (2) a, b, c (4_3_2_1)_3=72() da, db (3_2_1)_2=12() dc =85. dc dcabe, dcaeb, dcbae, dcbea, y 88 dcbea. 0 (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) 6 1 (1, 2), (2, 3), (3, 4), (4, 5), (5, 6) (2, 1), (3, 2), (4, 3), (5, 4), (6, 5) 10 2 (1, 3), (2, 4), (3, 5), (4, 6) (3, 1), (4, 2), (5, 3), (6, 4) =24() ;3!;{;b;{2, ;3;{a{2b b=1, a=1, 2 2 b=2, a=1, 2, 3, 4 4 b=3, a=1, 2, 3, 4, 5, 6 6 b=4, a=2, 3, 4, 5, 6 5 fi b=5, a=2, 3, 4, 5, 6 5 fl b=6, a=2, 3, 4, 5, 6 5 ~fl =27() (-1, -3) (0, -3) (1, -3) (2, -3) -1 (-1, -1) (0, -1) (1, -1) (2, -1) 1 (-1, 1) (0, 1) (1, 1) (2, 1) 3 (-1, 3) (0, 3) (1, 3) (2, 3) m=4_4=16 4 x>0, y<0, (1, -3), (1, -1), (2, -3), (2, -1) 4 n=4 m-n=16-4=12,,, (, ) 3_2_1=6() 4. (4_3_2_1)_(2_1)=48()

4 4 2 4_ =6 2 1, 2, 3, 4.,,,, 4 2 4_ =6() , 2, _3_2=24() 2 3_3_2=18() 4 3_3_2=18(),, =60() 1 4_3=12() 2, 3, 4, 5 12 ( )_12= _ _10+180= =3() 3-1=2() 3_4_3_2=72() , 3, 3, 3 5_4_3_3_3=540() 03 y=3x-a, y=-2x+b x 1 3x-a=-2x+b 5x=a+b a+b=

5 Middle School SUPRM Mathematics a+b=5 (a, b)=(1, 4), (2, 3), (3, 2), (4, 1) =7() , 4, _11=44() _6_ =35() 6,,, =31() 2 4_ _3=18() 2, 3_4=12() =31() , , , 5_14-1=69() 08 (4, 6, 8), (4, 8, 10), (6, 8, 10), (6, 10, 14), (8, 10, 14)., a+1 2b x-b=0 x= a+1 (a, b) (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 3), (2, 6), (3, 2), (3, 4), (3, 6), (4, 5) (5, 3), (5, 6) 14.

6 10 2 4_3_2_1=24() 3 0 3_2_1=6() 3 1 3_2_1=6() 3 2 3_2_1=6() =42() 11 y=2ax-b y=8x 2a8, a=4 (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) =30() 12,,,,, 6,,,, 6 3 6_5_4=120()., 3,,, _1=120() 13 3_3_3_3=81() =42() 14 n (n-1) n n(n-1)., 2. n(n-1) n(n-1) =105, n(n-1)= =15_14 n=15, _ =6() _ =15() 2 6_15=90(), 5_3+4_2+3_1=26() 90-26=64()

7 Middle School SUPRM Mathematics k 9_9=81() 1 (9) 1 (9) 1 (, ) 1 k 9_9=81() 1 (9) 1 (9) 1 (, 001 1, ) 1 k 9_9=81() 1 (9) 1 (9),, =243() ,,,,, 6 (), 2 6+2=8() , 2, 4. 0, 1, 2, 3, _4=16() 2, 1, 3, _3=9() 4, 3_3=9(),, =34() _ =10() ,

8 10_2=20() H1357H9 0.H1359H7 0.H1537H9 0.H1975H _3_2_1=24() 3, 5, 7, 9 24 ( )_24= x+5y+z=29 5y y=1, x+z=24 (x, z)=(1, 23), (2, 22), y, (23, 1) 23 y=2, x+z=19 (x, z)=(1, 18), (2, 17), y, (18, 1) 18 y=3, x+z=14 (x, z)=(1, 13), (2, 12), y, (13, 1) 13 y=4, x+z=9 (x, z)=(1, 8), (2, 7), y, (8, 1) 8 fi y=5, x+z=4 (x, z)=(1, 3), (2, 2), (3, 1) 3 ~fi (x, y, z) =65() , 1, , 1, 2, 3, =36() 23 1, 2 x, y x+2y=5 y=0 (1 ) (1, 1, 1, 1, 1) 1 y=1 (2, 1 ) (2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2) 4 y=2 (2, 1 ) (2, 2, 1), (2, 1, 2), (1, 2, 2) 3,, 1+4+3=8() 24,, 1,,, _3=9(),, 2,, -, - 2

9 Middle School SUPRM Mathematics,, 2 (, ), (, ), (, ) 3, 2 3_2=6(),, 3 1,, 9+6+1=16() _3=12() 3_2=6() 2_1=2() 3_2=6() 4(),, =30() k 2_3=6() k 3() k 1(),, 3_2_1=6() 10_6=60() 26 (a-b)(b-c)(c-d)+0, a+b, b+c, c+d 6_5_5_5=750() a, b, c, d 6_6_6_6=1296() =546() 28 H 8 H O G,, G, H. 6, G, H 6. 6_4=24()

10 29 3 () 1 () (), (), () 3 (), (), () (), (), (), (), (), () 6 (), (), (), (), (), () 6 12,, =20() , 2, 4, 5, 10, 20 6 ;2 0;=;1 0; 2, 6_6=36() 2 (1, 1) 1 3 (1, 2), (2, 1) 2 5 (1, 4), (2, 3), (3, 2), (4, 1) 4 7 (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) 6 fi 11 (5, 6), (6, 5) 2 ~fi =;3!6%;=;1 2; 36 3 ( ) =1-( ) =1-{( ) +( )} =1-{;5#;+;4!;}=1-;2!0&;=;2 0;

11 Middle School SUPRM Mathematics 4 ;1 0;=;5$; ( ) =1-( ) =1-;5$;_;5$;_;5$;=1-;1 2 5;=;1 2 5; 5 ;1 0; ;1 0;., ;1 0;_;1 0;_;1 0; ;1 0;_;1 0;_;1 0; ;1 0;_;1 0;_;1 0; 3_{;1 0;_;1 0;_;1 0;}=; ; 5 5_4_3_2_1(),, 3_2_1(),, 3_2_1() (3_2_1)_(3_2_1) =;1 0; 5_4_3_2_1 3 ;5!0^;, 5 ;5!0);, 15 ;5 0; 3 5 ;5!0^;+;5!0);-;5 0;=;5@0#; 1-;5@0#;=;5@0&; 6, ;9%;_;8$;=;1 8;, ;9$;_;8%;=;1 8; ;1 8;+;1 8;=;1!8);=;9%; ;1 0; ;5@0&; ;1 8; ;5$4(; ;6 2 5; ;3 6; ;5@0!; ;6@4!; ;2!0!; (1) ;10(0; (2) ;1 5; 8 ;1 6; ;1 0; 6_6=36() x=;a; x a=1, b=1, 2, 3, 4, 5, 6 6 a=2, b=2, 4, 6 3 a=3, b=3, 6 2 a=4,b=4 1 a=5,b=5 1 a=6,b= ;3!6$;=;1 8; ab.,

12 1-(ab ) =1-{1-;6%;}_{1-;9$;} =1-;6!;_;9%;=;5$4(; () {1-;5@;}_{1-;5@;}_{1-;5@;}_{1-;5@;} =;5#;_;5#;_;5#;_;5#;=;6 2 5; 2 (, ), (, ), (, ), (, ) 4, (, ), (, ) 2 ;4@;=;2!; 2 6_6=36(), 1 (1, 2), (2, 1), (1, 3), (3, 1), (1, 4), (4, 1), (1, 5), (5, 1), (1, 6), (6, 1) 10 ;3!6);=;1 8; ;2!;_;1 8;=;3 6;,,. ;1 0; 1-;1 0;=;1 0; {;1 0;_;1 0;}+{;1 0;_;1 0;}=;1 0 0;=;5@0!;,. 2, 3, 5 ;2!; ;2!;_;2!;=;4!; 4 ;2!;_;2!;_;2!;_;2!;=;1 6; 6 ;2!;_;2!;_;2!;_;2!;_;2!;_;2!;=;6 4;,, ;4!;+;1 6;+;6 4;=;6@4!;, ;2!; ;2!;_;5#;=;1 0; ;2!;_;6#;=;4!; ;1 0;+;4!;=;2!0!; (1) ;1 0; ;1 0; ;1 0;_;1 0;=;10(0;

13 Middle School SUPRM Mathematics (2) ;1 0; ;1 5; a ;1Å0;. ( ) =1-( ) ;2ª5;=1-;1Å0;_;1Å0; a 133 =;2!5^; 100 a =64 a=8 8. 4_4_p=16p 2 3_3_p-2_2_p=5p 2 5p 133 =;1 6; 16p 6 3 6_5_ =20() 6,, 2 ;2 0;=;1 0; 1 ;2!0!; 2 ;9%; 3 ;6!; 4 ;4!; 5 ;1 1; 6 ;7@2(; 7 8 ;3!; 9 ;1!6#; 10 ;1 8; 11 ;2 0ª0; 12 ;2 5 6; 13 ;3!2%; 14 ;9$; 15 ;4@8#; 16 ;2!; 17 ;1 8; 18 ;1 8ª0; 19 ;7@; 20 ;6%; 21 ;4!; 22 ;4!; 23 ;2 1 6; 01 1, 2, 3, 4, 5 5 5_4_3=60() , 2, 4, 5 3, 3 3_3=9() 4, _3=12() 5, _3=12(),, ;6#0#;=;2!0!; 02, ;9%;_;1 1;=;9#9%;

14 , ;9$;_; a =2_5, 14=2_7, 16=2, 20=2 _5 4 ;1 1;. 03 a +b }50 b }50-a a=1, b }49 a=2, b }46 a=3, b }41 a=4, b }34 b=6 a=5, b }25 b=5, 6 a=6, b }14 b=4, 5, 6 a +b }50 (4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6) 6 ;3 6;=;6!; 06 {;2!;} {;2!;} {;3@;} {;3!;} {;2!;} {;2!;} y y {;3!;} y y {;2!;} y y {;3!;} y y {;2!;}y _2_2_2=16() 4 x (4-x) P -2, x+(-1)_(4-x)=-2 2x=2 x=1 1, 3 (,,, ), (,,, ), (,,, ), (,,, ) 4 ;1 6;=;4!; 05 ={10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} a _ = _a 2 _a ;2!;_;3@;_;3!;=;9!; ;2!;_;3!;_;2!;=;1 2; ;2!;_;2!;_;3!;=;1 2; ;2!;_;2!;_;2!;=;8!; ;9!;+;1 2;+;1 2;+;8!;= =;7@2(; , ;3!;_0+;3!;_1+;3!;_1=;3@;.. (). 4, 5, 6. ;3@;.. ()

15 Middle School SUPRM Mathematics. 2 3 ;3!;., 9 1. ;3!;_;3!;+;3!;_;3!;+;3!;_1=;9%;.. (),,. 08 a, b,4 a}b+4 b}a+4 4}a+b. a}b+4 ;2!;_;2!;_;2!;_;2!;=;1 6; ;2!;_;2!;_;2!;_;2!;=;1 6; ;2!;_;2!;_;2!;_;2!;=;1 6; ;2!;_;2!;_;2!;_;2!;_;2!;=;3 2; ;2!;_;2!;_;2!;_;2!;_;2!;=;3 2; ;2!;_;2!;_;2!;_;2!;_;2!;=;3 2; ;2!;_;2!;_;2!;_;2!;_;2!;=;3 2; ~ ;4!;+;8@;+;1 6;+;3 2;=;1!6#; (6, 1), (6, 2), (5, 1) 3 ;3 6;=;1 2; b}a+4 (1, 6), (1, 5), (2, 6) 3 ;3 6;=;1 2; 4}a+b (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1) 6 ;3 6;=;6!; ;1 2;+;1 2;+;6!;=;3!; , ;2!;_;2!;=;4!; ;2!;_;2!;_;2!;=;8!; ;2!;_;2!;_;2!;=;8!; 10 6_6=36() 3x+by+2=0, (a+1)x+2y+2= =;2;+1 a+1 (a+1)b=6, a+2, b+2yy (a, b) (1, 3), (5, 1) 2 ;3 6;=;1 8; a, b 2 (a, b) 2 2 6

16 11 ( ) =1-( ) {1-;3!;}_{1-;4#;}_{1-;5@;} =;3@;_;4!;_;5#;=;1 0; m=1-;1 0;=;1ª0;, Y. Y ;3!;_;4#;_{1-;5@;}=;2 0; Y ;3!;_{1-;4#;}_;5@;=;3 0; Y {1-;3!;}_;4#;_;5@;=;5!; n=;2 0;+;3 0;+;5!;=;6@0#;, mn=;1ª0;_;6@0#;=;2 0ª0; m 12 4 ;4!;, 1-;4!;=;4#; n 1 mn _{;4#;_;4!;_;4!;_;4!;}= ;4!;_;4!;_;4!;_;4!;= =;2 5 6; {1-;2!;}_{1-;2!;}_{1-;2!;}_{1-;2!;}=;1 6; ;1 6;=;1!6%;, ;2!;. ;1!6%;_;2!;=;3!2%; 14 R 3. k k ;3!;_;3@;_;3@;=;2 7; k k ;3@;_;3!;_;3@;=;2 7; k k ;3@;_;3@;_;3!;=;2 7; ;2 7;_3=;9$;

17 Middle School SUPRM Mathematics ;6!; y=;a;x y (,, ), (,, ), (,, ) 3 1 O 4 5 x ;8#; ;a; ;6!;_;8#;=;1 6; 1 ;6%; ;4@;=;2!; ;6%;_;2!;=;1 2; ;1 6;+;1 2;=;4@8#; _19_ =1140() 6 ()+()+() ()+()+() ()+()+() 1, 3, 5, 7, y, _9_ =120() 6 ()+()+() 10 2, _ _10=450() 2, =;2!;,. ;4!; ;5!; ;5!;{;a;{;4!;yy (a, b) (4, 1), (5, 1) 2 ;3 6;=;1 8; 18. -;3!;_;4#;_;3!;=;1 2; -;3!;_{1-;4#;}_{1-;5!;}=;1 5; ;1 2;+;1 5;=;2 0; ;3!;_;3!;_{1-;5!;}=;4 5; {1-;5!;}_;3!;_;3!;=;4 5; ;4 5;+;4 5;=;4 5;, ;2 0;+;4 5;=;1 8ª0;

18 _7_ =56() 6 4 8_4=32() 8, =;5!6^;=;7@; M-m{1 M-m=0,, 6 M-m=1 =1, =2, ( =1, =2 { =2, =2 3 9 =2, =1 ( =1, =1 \ =2, =1 \ =1, =2 { 6 =3, =2 \ =2, =3 9 =3, =3 =3, 4, 5, =2 6 ( =5, =5 =6, { =6, =5 3 9 =5, =6, M-m{ _4+3=36() 21 1-;2 1 6;=1-;6!;=;6%; P 0 {x{180, 0 {y{180 POQ>90 x-y>90 y-x>90 yy y<x-90 y>x+90 yy, (x, y). O, ;4!;. y O 90 x y _6=36(). Q x

19 Middle School SUPRM Mathematics 2 (1, 1) 1 ;3 6; 6 (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) 5 ;3 6; 10 (4, 6), (5, 5), (6, 4) 3 3_;2 4;=;8!;, ;2 7;+;8!;=;2 1 6; , ;3 6;+;3 6;+;3 6;=;3ª6;=;4!; P ;3!;_;3!;_;3!;=;2 7; 1 0, 2 3 1, 4 ;6!;_;2!;_;2!;=;2 4; ;2!;_;6!;_;2!;=;2 4; ;2!;_;2!;_;6!;=;2 4; ;6!; 5 ;1 0; ;1 0 8; 9 336,,., 6 1 6, , 4 4_ =6() 2,. 6_5_6=180()

20 , a, b. P a b b b b a a a Q, P Q. P P a- -a-b a a -a-b-a k 4 a-b-a-a b -a-a-a P a -b-b-b a-a-b-b b k 4 - -a-b b a -a-b-a 4_4=16() Q Q 1, Q 1 1_1=1(), 16+1=17() y, x y O x,. x 8 2 8_ =28() 2 y 5 2 5_ =10() 2 28_10=280(), O ' ' ', ' O μ''. (, ', ') '' 'O'=60 (μ'' ) =;6!; ( ) 6 3 6_5_ =20() 6, , _4=4() 500 1, _1=2() 500 1, 100 1, _2_2=8(),, =14() ;2!0$;=;1 0;

21 Middle School SUPRM Mathematics (), () f(1)=1, f(3)=3, f(5)=5. 1. () 1, 2, 3, 4, 5, 6 3 2, 4, 6. 6_5_ =20() 6, f 1_20=20() a+b+c=9 (1, 2, 6), (1, 3, 5), (1, 4, 4), (1, 5, 3), (1, 6, 2), (2, 1, 6), (2, 2, 5), (2, 3, 4), (2, 4, 3), (2, 5, 2), (2, 6, 1), (3, 1, 5), (3, 2, 4), (3, 3, 3), (3, 4, 2), (3, 5, 1), (4, 1, 4), (4, 2, 3), (4, 3, 2), (4, 4, 1), (5, 1, 3), (5, 2, 2), (5, 3, 1), (6, 1, 2), (6, 2, 1) 25 a+b+c=1 (6, 6, 6) 1, =;2 1 6;=;1 0 8; 6 6 (k=1, 2, 3, y, n) _ =78() 2 2, 3, 4, y,14 78 a +a +a +y+a«=(1+2+3+y+14)_78 =8190 a, b, c 100a+10b+c, 9 100a+10b+c=9(11a+b)+(a+b+c) a+b+c 9. a, b, c 1 6 3{a+b+c{18 a+b+c= n 6 x n 1 x. k n, 1., n 2008 n =6_ n., k 1_334+2=336

22 II 01 1 (1) 3+2=5 (2) x=-2 x=0,.. (3) 6 3 (4) 2 (1) a, b. ab. () (2) ~ ~... () () 1 2. (3) ~ ~ {1, 2, 4},{1, 2, 4, 8} () 3 (1) ;c;>;c; ac>bc. () () ;c;>;c; c () ;c;_c >;c;_c ac>bc (2) 3 6. () () (3) ab=0 a=0 b=0() () a=1, b=0 ab=0 a+0. (4). () 4 (1)... (2) (3) (4).. 2. x-1=0 x=1 x -1=1-1=0 (). 2. (). 60. (). ''' =''' (), 2.

23 Middle School SUPRM Mathematics a=-2, b=1 a<b a >b () a=0, b=-1, c=3 ab=ac b+c () a=3, b=2, c=-1 ac<bc a>b () a=3, b=2, c=-1 ;c;<;c; a>b () (1) a >1 a>1. () () a=-2 a =4>1 a<1. (2) x+y=5 x=2, y=3. () () x=1, y=4 x+y=5 x+2, y+3. (3) 60 () ( ) = yy = yy ==60 =60 +=yy,, (SS ) = = yy = + = + = yy yy,, (SS ) =. (). () a=1, b=-1 a+b=0 a+0, b+0.. a=0 b=0 a+b=0. (). () a=1, b=0 a+b= =u, '= (). '= -=u(). ( '=, -=u.). n n. (). n n. (). ( n n n n.), (1) (2) p, q. () p q.() p q. () p, q.() p, q.()

24 02 (1). () (2) 2 4. () () x=1 3x-2>0.. ( x=1 3x-2=3 1-2>0) ().. (). (.). c-a>c-b a>b. (). () c-a>c-b c. -a>-b a<b. a+b a, b. (). () a=3, b=3 a+b=6 a, b.. >90 <. (). (... >90 <.,, () x=-1 x+1 x =1.() () x=y=1 xy+0 (x-y) =0. () () () () x=0.3, y=0.7 x+y=1 x, y. () xy>0 x>0, y>0 x<0, y<0 x+y>0 x>0, y>0 () 05.,. ( ) a=1, b=-1 a =b a+b..,.,. ( ) x=2, y=-1 x+y>0 x>0, y<0..,. ( ) ( ) ( ) m=3, n=2 m+n=5 06 =[x ;2!;{x-1{;2#;, x<] ;2!;{x-1{;2#; ;2#;{x{;2%;. ={2},. ={2}... =[x ;2#;{x{;2%;, x<]., (1) G =, G =, G= G (SS ) G =

25 Middle School SUPRM Mathematics (2) G GH=G, GH=G () H=GH+GH ( GH ) H=G+G H=90 (=90 ) =+a=180 a=180 - yy ++=+b=180 b=180 - yy, a=b. 09 =+ =- =;2!;-;2!; =;2!;(-) =;2!; a 2 b 2 a=b 1 5 ( =+) 10 2a-1{x{a+2 P, 0<x<5 Q P,Q. 2a-1>0, a+2<5 ;2!;<a<3 a a=1, a< ;3!;a<. 3. a=0 ={0}. a+0 =[a, ;3;, ;9;, ;2Å7;, y]. 0< a+0 =+ =;2!; 02a-1 =[0, a, ;3;, ;9;, ;2Å7;, y]. a=0 ={0} u, 2. () () a=3 =[3, 1, ;3!;, ;9!;, y] 3<, 1< 3+1 () P Q () a=6 =[6, 2, ;3@;, ;9@;, y] 6<, 2< 6 2=12 () a+2 5 x

26 0. () a=2 =[2, ;2 7;, y]. () 12 ==x ==y z x ==z z x y y 180 =180 -x-zyy =180 -x-y yy =180 -y-z yy, ++= =540-2(x+y+z) x+y+z=180 yy +x+z=180 +x+z=x+y+z =y ==y 13 a, b, a=2m, b=2n(m, n ) a=a+b=2m+2n=2(m+n) a, b, a=2m-1, b=2n-1(m, n ) a=a+b=(2m-1)+(2n-1) a=a+b=2(m+n-1), a+b a+b. a+b a, b. 14 n n{x<n+1, [x]=n. n+5{x+5<n+6. [x+5]=n+5=[x]+5 (). () x=0.5 [0.5]=0, [-0.5]=-1. [0.5]+[-0.5]=0+(-1)=-1 (). () x=1.5, y=1.7. [ ]=[3.2]=3, [1.5]+[1.7]=1+1=2. [x+y]+[x]+[y] (). 15 = = + = + = + = =., =60 ==60, =, =yy = (SS ) 1 = 02 = ==180 - = =65

27 Middle School SUPRM Mathematics =180 -(+) (2) I =180 -( ) =50 = ==65 I=;2!; I=;2!; I =- =65-50 =15 I =180 -(I+I) 2 M M =180 -;2!;(+) =180 -;2!;(180 -) M =M, M =M M=M=90 M M(RHS ) =90 +;2!;= =110 = =;2!;( )=70 5 Q Q Q =12 3 I r =I+I+I QQ =12 Q=;3@; Q P, P P =21 ;2!;_5_12=;2!;_13_r +;2!;_5_r+;2!;_12_r 13 r 12 QPPQ=21 PQ=;3!;Q 60=13r+5r+12r 30r=60 r=2 I 5 PQ=;3!;_;3@; PQ=;9@;_36=8 4 (1) O O =O =O (1) O, O, O O O=O, O=O, O=O O+O+O =;2!;_180 = p ;2#; , 61 2 ==x

28 , = ==x x+2(x+42 )=180 3x=96 x=32 ==;2!;( )=67.5 O O =O O=O=67.5 O=180-2_67.5 =45 O O =O O=O=67.5 O=180-2_67.5 =45 O=180 -(O+O) O= =90 O 90 p_6 _ =9p 360 =, = ==45 = =180 2=135 =67.5 ==90, = (RH ) = = - =5-3=2 =x =+ ;2!;_4_3=;2!;_5_x+;2!;_x_3 ( = ) 4x=6 x=;2#; =;2!;_ _ =, ==90, = (RHS ) = yy +=90 yy, +=90 G G=90 G=90 () =,, ==90 (RHS ) =;2!;_2_;2#;=;2#; I I =I =I II I,I =I I I=I=90 I I(RHS ) = I I = = =16- =16- =16-10=6

29 Middle School SUPRM Mathematics // I=I() yy I I=I yy, I=I I =I I =I ( ) = + + = +(I +I )+ = = + =10+8=18 = = =;2!;=;2!;_68 =34 I I=;2!;=;2!;_50 =25 I' I'=;2!;=;2!;_34 =17 O IOI'=180 -(I+I') II'=180 -( )=138 O, O=x, O=y O=O=x x O=O=y y x O y 2x 2y O O O=2x, O=2y OOO=657 O=;1 8;_360 =100, 2x+2y=100 =x+y=50 O, O, O O O, x z O x O z O=O=x y y O=O=y O=O=z. 2x+2y+2z=180 x+y+z=90 =x+z=68 y=22 O O O,O =O, O=O=90 O O x=z=;2!;_68 =34 =y+z= =56 =180 -(+) =180 -( ) =80 I I=;2!;=40 O O=2=2_65 =130 O=;2!;(180 -O) O=;2!;( )=25

30 OI=I-O IO=40-25 IO=15 // ;1 6; = = yy, // =, yy G : 2 : == 01 Q 15 = =;2!;, =12 =12 =;3!; Q P 45 P =;3!;_;2!; =P yy =;6!; =61 Q=P=90 yy P P=60, P P=90-75 =15 P P =32 PP=32 Q=90 -( )=15, Q=P=15 yy,, Q P(RH ) P=;3@;P =Q = =5, Q Q =21 PQPQ=21 P=3PQ P=;3@;P=;3@;_3PQ P=2PQ P =2 PQ 02 =I+I+I I I =(6+4+3)4 =134

31 Middle School SUPRM Mathematics 03, = O=90 ==;2!;( )=50 I I=;2!;=;2!;_50 =25 =180 -(+) =180 -( )=65 04 I I=90 +;2!;=90 +;2!;_80 =130 I=I=130 () I I+I=360 I+I=150 yy + =(180 -)+(180 -) =360 -(+) = ( ) = =, =, 50 y ==x x ==y x y =180-2x, =180-2y I O (180-2x)+(180-2y)=180 2x+2y=230 x+y=115 =180 -(x+y) = =65 06 O O =O, O 30 x O=O=x. O O =O O=O=30 +x O O =O O=O=40 +x =O+O =(30 +x)+(40 +x) =70 +2x, =180 -( )=110, 70 +2x=110 x=20 yy yy O O=180-2x=180-2_20 = = =x = =y O' = =1 =x+y=6 ( )=2(x+y)+2 ( )=2_6+2=14 y 6 y O 40 x x O' 1 x

32 S S=;2!;_1_( + + ) S=;2!;_1_14=7 08 =, ==90 =90 -= (RH ) = =14, = =30 = - =30-14=16 09, =G =G=30 yy G =G G=G=30 =21 =2 =;2!; yy =;2!;(+G+G) =;2!;( )(, ) =45 10 I I=I=30, I=I=30 // I I==60 () I=I=30 () =I yy // I I==60 () I=I=30 () I = yy I=I=I=60 I. I = =I yy,, = = =;3!; =;3!;_6=2 11 I I=90 +;2!;=110 =40 I=I=;2!;=20 yy, =60, =40 =180 -( )=80 O O=2=2_80 =160 O =O O=;2!;( )=10 yy, OI=I-O=20-10 =10 12 H G H=G=90 H // H=() yy yy yy = ==G() yy, H=G yy,, H G(RH ) H =G =4 H = =6 =H +H =4+6=10

33 Middle School SUPRM Mathematics 13, =;4#;, =;4#;_;4!;=;1 6; 4 O O O=O, O O=O=90 O O(RH ) 7 5 O =;1 6;, =;1 6; =-(++) =-{;1 6;+;1 6; +;1 6;} O =O yy =;1 6; O O(RH ) O =O yy, ;1 6;., O =O O O O O=O=90 O,O =O O O(RHS ) = =x yy =x, = =5-x =4+x =7+(5-x)=12-x 4+x=12-x, 2x=8 x= =180 ++=180 =, ==90 =, =() 14 =31 =31 =. =;4#; yy, =13 =13 =;4!;yy 2 = 2 2 6

34 16 // ==40 () ==40 = =100 I I=;2!;=;2!;_100 =50 ==;2!;( )=70 I' I'=;2!;=;2!;_70 =35 O O =180 O=55 I I' 17 P x, P y yyyyyyyyyy PP =PP x(y+15)=1020=12 y+15=2x 2x-y=15 yy O 2 7 y x P 20 PP=PP (x+10)y=2015=43 4y=3(x+10) 3x-4y=-30 yy, x=18, y=21 = =84 P=x, P=y 1 PP=PP 2 PP=PP 2 18 =x x, y = ==x =2x = ==2x ==;2!;=x, = ==x 180 x+2x+2x=180, 5x=180 =x=36 19 O O( ) =2=2_80 = O x x 2x 2 1 8

35 Middle School SUPRM Mathematics O( )= =200 y, O O( )=2 yy, 2=200 = P Q P =Q ==90 = P 20 P Q(RHS ) Q=P=20 PQ=90 P =Q PQ. QP=45 Q Q=90-20 =70 PQ=Q-QP=70-45 =25 21 // '' ='() yy ='() yy, '' ='' yy,, =, '=', ' =, ' = = + = +' =' = =4 4 5 ' ' 20 Q 22 P Q =180 -( )=90 =90 P, Q., Q =Q =Q, P =P =P Q =;2!; =;2!;_20=10 P =;2!; =;2!;_3=;2#; PQ =Q -P PQ =10-;2#; PQ =: 2 : 23 M 30 M =M =M M =M M=M=30 M M=2_30 =60, =60 M. =M =M yy =2 M =M M = yy, = = =150 =;2!;( )=15 =+ = =75

36 24 ' x y y ' P Q P =Q ( ), = ( ) P=Q=90 P Q(RHS ) P =Q = -P =5-4=1 ( ), Q =P =2 '' '==45 '=x =90 -='=x ' ==45 +x, = = (SS ) ==y =x+2y=90 yy = y+2(45 +x)=180 2x+y=90, x=y=30 ==45 +x = =75 25 P Q M M M,M =M M=M=90 yy P=Q P=Q= P Q P M Q (RH ) 2 P =Q =2, P =Q =4 yy M M(SS ) = yy Q=;2!:_Q _Q =;2!;_1_2=1 26,,, G. = =, G G. G = 3 2 G P, Q, R () P P R Q 122 = P P P = P P

37 Middle School SUPRM Mathematics () P = P-P = P P-P y Q 122 = Q yy R 122 = R yy,, P Q R =1 P Q R P = P = P P P P 2 P = P 2 Q =, R = Q R 4 P Q R =1 P Q R =54 =5x, =4x +=180 5x+4x=180 x=20 ==4 20=80 2 y,,,, (2, 2),, (4, 0) x. (a, b) a+4 b =;2@;, =;2@; 2 2 a=-2, b=2 (-2, 2) (6, 2), (2, -2) 3 = yy // =()yy = yy, =, =, = 4 yy (1) H H // // G // // HG G GH. =90 GH.

38 (2) H H =G =;2!; =HG =;2!; = H =G = =HG G GH. 5 P, Q, R P=R P=R P+P =R+R 4 6 R P Q =;2!;QR+;2!;RQ =;2!;(QR+RQ) =;2!; =;2!;_4_6=12 =75-30 =45 x=180 -( )=70 // ==65 () y=65-35 =30 x+y= =100 x x x, y y y G. =2x+y G G=x+2y +G =(2x+y)+(x+2y) =3(x+y), +=180 2(x+y)=180 x+y=90 +G=3_90 = = =180 -( )=75, ==75 ' ' ' = = yy P // =() =' P P =P yy P = -P P=' -P (, ) P=P' P P' =11

39 Middle School SUPRM Mathematics = 3a-2=2a+7 a=9 = = = =25 O=90 = = == =, =;4!;_16=4 O O O =O O=O=90 O=O() O O(S ) O =O. +=180 +=90 =90 +=90 =90 H G H=G=90 GH. O O O =O O=O=45 O=90 -O=O O O(S ) = = + =5+3=8, 8 =8_8=64 =, ==45 (SS ) = =+= =65 =65 // = // G =G ( ) =++ =++G =G=15

40 // = =;2!; =;2!; =50 =- =50-;2!; 10 8=50-40=10 M // M=M =M+M =M+M=M, M =M M=;2!; M =;2!;=;2!; 24= x S // = O=-O O=-O O=O=6, OO=618=13 O O =13, OO=O O =13 O=;3!;O=;3!;_6=2 =O+O+O+O = =32 02 U, P, Q P;Q. n(p'q)=n(u)-n((p'q)ç ) U P n(p'q)=35-10=25 n(p'q)=n(p)+n(q)-n(p;q) 25=21+19-n(P;Q) n(p;q)=15, O 18 Q

41 Middle School SUPRM Mathematics () =(). = = //, //,. = =, //, =. // ==40 () =180 -( )=64 // ==64 () 05 = =2x, =2y 2x+2y=180 x+y= x x 40 y y G // G=G=G=x G x+y+g=180 G=180 -(x+y)= =90, =90 06 O O O M O =O 16 O ( ) N O=O 18 () O=O() O O(S ) 07 M=OM+O M=OM+O M=M=;2!; M=;2!;_{;2!;_16_18}=72 G G = x =G() =G() G(S ) G = = G G. = G

42 , ==x =90 - =90 -x 08 =, = = =x ;2!;_4_x=;2!;_8_(12-x) 2x=48-4x, 6x=48 x=8 09 O=G=90 G =O, G =O G G O 10 O=;2!; G =4O=2G =2_20=40 50 G = ( ) x 12-x G 4 8 =G ( G ) =90 -=90-25 =65 =G G(SS ) =G=40 G G=180 -( )=75 11 P, P+PQ =PQyy =23 =2k, =3k P =k, Q =2k, Q =k Q PQ =Q, Q =P Q=PQ=90 Q PQ(SS ) Q =QP yy, PQ=Q PQ+Q=Q+Q PQ+Q= =90 QP=90 yy, QP PQ=45 P+PQ=45 12 =G G G (SS ) G 45 Q 64 P

43 Middle School SUPRM Mathematics G=, G = G=G- G=90-45 =45 =G =180 -(G+) =180 -( ) =71 13 = yy, // = yy = yy, =yy =31, =31, =21 =;2#;, =;2!; =+ =+ =;2#;S+;2!;S=2S 14 P=R P=2R P=2_;6!; P=;3!; P=yy M P N S Q R P M P Q N N =NQ =QR QR=;3!;R=;3!;_;6!; QR=;1 8;yy =PQRS+2P+2QR =PQRS+2_;3!; PQRS=;9@; +2_;1 8; (, ) PQRS=29 15 P P=-P=90-60 =30,P = P=P P=;2!;( )=75 =45 P=P-=75-45 =30 16 M yyyyyyyy P P M 2 2 P 2 6

44 M M M =M M=M() M=M() M M M=M, M =M M(M+M) =M(M+M) =MM =M M =11 M 1 M M 2 2 MM 2 7 ' M. ' M+'M' =;2!; '' = M=;2!; M+M =-M =-;2!; =;2!; M(M+M)=11 17 =, = =60 -= (SS ) = = = yy (SS ) = = = yy,. 18 P, h, h' P=;2!;_9_h P=;2(;h P=;2!;_12_h'=6h' PP=32 ;2(;h6h'=32, 18h'=9h h=2h' =;2!;_(12+9)_3h'=: 2 :h' h 9 P h' 12 3 = 2 = 2 1 8

45 Middle School SUPRM Mathematics P+P =-(P+P) =: 2 :h'-{;2!;_9_2h'+;2!;_12_h'} =: 2 :h' (P+P) =: 2 :h': 2 :h' = P P =12 P=;3!;=;3!;_;2!; 21 = =25 =;5@;=;5@; 20=8yy, =+ =+ = =. // ==8 ( ) P=;6!;_40=: 3º: O OP=;2!;P=;2!;_: 3º:=: 3º: Q QP =32 OQ=;5#;OP=;5#;_: 3º:=2 20 OP OQ O =O PO=QO=45 OP=90 -PO OP=OQ OP OQ (S ) P =Q =8 =8+2=10 OP=OQ OQP=O=;4!; OQP=;4!;_10 =25 ( ) =-OQP =100-25=75 Q O P 2 G 22 N M N (SS ) M=N M=N M=3M=3_4=12 =4M=4_12=48 23 G G(S ) G =G H H(S ) H =H G, H, =G =H, GH. P=90 N M

46 24 = = = = =-60 =-60 (=) = (SS ) = =+ =+ =G(G ) =() =60. =60 25 =120 =180 -=60, =. = = =60 =- = =60 P Q(SS ) P =Q, PQ PQ=P+Q PQ=P+P PQ= PQ=60 PQ. P Q =H H H=90, G=90 HG HG=G-H HG=H-H HG= =G HG(SS ) HG = =8, HG==90 G=;2!;_ _HG =;2!;_5_8=20 27 =() ==90 = yy // O=O() yy O =O, O O=O yy,, =O G G=G yy, // G G=G()yy 5 O 8 G H G

47 Middle School SUPRM Mathematics, G=G G =G. =O 4 G=G 1 G=G 2 =G 2 9 G=G 2 ==O 2 O 2 2 O= G O Y X G. //O = G=G yy, //O ==O yy ( O ) O=O+ O=O+(G-G) =+G-G =O+G-G (, ) O= ;2#; ;1 5; I I=I I= O 2 I= I 6 yy, I I=Iyy I I+I=I I=I+, I=I, I =I =O -O I =6-2=4 =45 =x =y ' ' y x ' x+y=45 y x. ' y

48 '' ==x ='=y ' =' '='=x+y ='=y =90 x+(x+y)+y=90 x+y=45, += O=O=x O=x+x=2x =21 O = O=O=2x 180 x+2x+141 =180 3x=39 x=13 =2x=2_13 = G H. = =16 =20-16=4 // G, // G G G H =, H =;2!;H =;2!; =;2!;_4=2 2x 2x O x x 141 // ==90 O O. O =O =O O=x, 6. 6 a, b, c, d, e, f 1334= =;d; yy M =M M=M, a+b+c=d+e+f, c=f a+b=d+eyy 16 a d e 12 =x =2 =2x =12-x, =16-2x 16-2x ;a;= 1334= x 12-x ;d;= 1334= x 16-2x a+b=a a=;[*;a 2x 12-x 12 d+e=d d= 13 d x x 12 ;[*;a= 13 d 2a=3d x f b c M

49 Middle School SUPRM Mathematics 1334=;d;=;2#; G=H-GH G=(4a+4x)-4x=4a =;4!; = // G I =O O O 1334= H=H // IH H=O H O 1334= H G = =O O 1334= H O O O = = =1 5a=;4!;_2_2 a=;5!; =;4!; 4x+2x+a=1 6x=1-a=;5$; x=;1 5; HG=a+2x=;5!;+;1 5;=;1 5; y 2 (2, 2) y, G x, 1 H 1 2 O() x y= x y=-2x+2 y=x y= x y=;2!;x+1 G=a 2 G=a a G H H = 4x 2 a HH=21 2x H a+x GH=2x GH=4x, H H =21 H=2a+2x H=a+x H=4a+4x ( H H =21) H{;3@;, ;3@;}, G{;5@;, ;5^;} HG=H-G HG=;2!;_2_;3@;-;2!;_1_;5@; HG=;1 5; G G H H, G, H 15 15

50 =G=GH=H = =60, G, GH, H. =G =GH =H H =H, H==150 H=;2!;( )=15 =15 = =60 P P P =P P P =P, P =P. = = Q R = =180 -(+) =180 -(+) ( =()) = (SS ) P =;2!; =;2!; =;2!;7=;2&; Q =;2!; =;2!;6=3 R =;2!; =;2!;5=;2%; P +Q +R =;2&;+3+;2%;=9

51 III 01 1 O O =48=12 2 =, ª( ) =, 63=(3+ )2 3(3+ )=12 =1 3 = =32 ª(SS ) = =32, 9=32 =: 2 : 4, ==90 ª =, 108=9 =: 5 : 5 ª = 12 =16 =9, ª = =9 25=225 =15 Middle School SUPRM Mathematics MÚN // =NM() ªNM( ) N = M 4(6+N )=6(4+12) 6(6+N )=4 16, 6N =28 N =: 3 : : 3 : ;5*; : 2 : : 4 : =, ª( ) = 108=8, 10 =64 =: 5 : =+= = =8 = - =8-: 5 :=;5*;

52 =, ==90 ª( ) =, 2 = 4 =8 = ` = =8 G GªG G =G `G =10 40=400 G =20 M M = M = M =;2!;_50=25, GH M GHªMG G = H M, 20 = H 25 H =16 =180 -(+) =180 -(+) = ª( = 3(15-x)=7x 7(15-x)=3x, 10x=105 x=: 2 : =() =() =, G =G =;2!; =5 Gª( ) G =G G 6=58 ( =, = ) 8G =30 G =: 4 : ª( ) = 35=15 =25 = 15 = 25 =9 = =25 (25+9)=850 1 ;2&; ;3&; : 3 : 11 ;2@5!; 12 : 2 : =x =7, =x 7 x ;5^; 18 y=;5@;x 19 ;1 2 5; 20 : 2 : ==60 ==60 7 x =16, =20 23 ;3$;

53 Middle School SUPRM Mathematics 01 = =( ) ª( ) = 68= 6, 8 =36 =;2(; = - =8-;2(;=;2&; 02 =+ =+ = =+ =+ = ª( ) = =68=34 03 =, = = yy yy = = yy, = yy, ª( ) = 126=6, 12 =36 =3 04 P =P=90 yy P+P=90 P+P=90, P=P() P=P =P yy ªP( ) = P ( +3)4=43 3( +3)=16 P =;3&; 05 P = = =21 =13 =a, =3a = =8a. = =180 -(+) =180 -(+) = ª( ) = 8a3a=a =;8#;a = - =8a-;8#;a=: 8 :a =: 8 :a;8#;a=613 8a a 3a

54 06 = ( // ) ª = yy, = ( // ) ª = yy, = = =x 12 =8 (12+x) 8x=48 x=6 07 M =M=90 +=90 +M=90 =M ªM( ) M =21 =21 =2 M =21 =2M, =4M M =;5!;M =;5!;_10=2 8 4 x yy yy 08 ª = yy = yy, ª(SS ) ==40 =80-40 =40 09 =, ª( ) = = =1520=34 =;3$; =;3$; =;3$; ;3$; =: 9 : = - =: 9 : - =;9&; =7 =9 10 ª = 8 =510 = , = // =() =() ª( ) = =12yy ( = =12) 70 30

55 Middle School SUPRM Mathematics =x = - =16-x (16-x)x=12 x=2(16-x), 3x=32 x=: 3 : 11 M= M M M= _ yy, ª = 3 = 5 =;5(; 13 ª = =10(10+15) =25yy // =()yy // =() =()yy, ª( ) =, 15=25 () =6 M =M - =;2%;-;5(;=;1 0; yy M= 132 _{;2!;_3_4}=;2@5!; 5 12 = =32 = ª(SS ) = 64= 4 =6 = =23 = ª(SS ) = 69=5 =: 2 : + =6+: 2 :=: 2 : 14 GI G = =10 GI==90 G GH G=HG=90 G=90 -G=HG GªGH G =G H =G GH 68=8H =10GH H =: 3 :, GH =: 3º:, GHªHI I HI =G H HI =GI -GH = -GH =16-: 3º:=;3*; I ;3*;=8: 3 : I =2 =I =2

56 15 ª = =12yy, ª = ( +2)=12 () +2=2 =2 ª ==90 =() ª( ) =, 1810=30 =: 3º: ª 2 2 7, ==90 ª( ) =, 40{18+: 3º:}= 26, :;!3); : =40_26 =30 =;2!;_ _ =;2!;_30_40= =+ =+60 yy =+ =+60 yy, =, ==60 ª( ) =, 53=2, 5 =6 =;5^; ª ª = ª

57 Middle School SUPRM Mathematics 18 G =G=90 =G ( // G ) ªG( ) G = yy = +2y=x+4 =x+4-2y (x+4-2y)4=(8+y)8 4(8+y)=8(x+4-2y), 5y=2x y=;5@;x 19 ª = 3 = 5 =;5(; M =M =;2%; M =M - =;2%;-;5(;=;1 0;yy = 3 4=5 =: 5 : M M =M =M =;2%; yy yy M =M H,, : 5 : ;1 0;=;2%; H H =;1 2 5; 20 O O==90 Oª( ) O = 1016= =200 =: 2 : O O O=O(), O =O O=O=90 O O(S ) = =: 2 : 21 = = = = PQQR=P P PQQR= PQQR= PQQR=3 1 PQQR=91 22 = =12 =+ =+ = ª(SS ) =12 =2 =2_10=20 = =21 = ª(SS ) =21 =2 =2_8=16

58 23 ==60 () // HªH( ) H H =, 69= 3 =2 HªHG H H =G yy = =2 H =H - =6-2= =G 2 G =;3$; G 6 H ==;9@;=4 =-(+ +) = =6 25 =a =2a = = ( // ) =yy = = yy, ª( ) = =, a =2a =;2; 2a a Q P, P, Q ={2a-;2;};2; =;2#;a;2; =31 QªP Q P = =23 Q =;3@; P, =;3!; =;2!; Q =;2!; ;3!; ;3@; P 26 P // P P =;9@; {;2!; P } =;9@;=;9@; 18=4 ªP( ) P = =21 P =2 yy

59 Middle School SUPRM Mathematics P=+ P= =67.5 P=90 -P P= =67.5, P=P P. =P yy, =2 =2_4=8 27 O O O=O O=O O () OªO ( ) 50 O O =O O yy, O O O=O() OªO(SS ) O=O=50, == =;2!;( ) =65 =180 -(+O) =180 -( ) =65 28 N =N M N N =N, N =NM M. // M OªO OªO O Mª( ) =M =12 =2 = N = ;2!; N=2 ;2!; N= =4N 9 N =N 2 // M 2 3 =4N 2 9

60 02 1 (1) ª = ( +2)=57 5( +2)=7 =5, =23, =;2#; =;2#;_5=: 2 : = 52=: 2 :x x=3 (2) 4(x+6)=510 5(x+6)=40 5x=10 x=2 4 M, N,,,,, M N MN =;2!; = = = =30 5 G G G G =21 G G =;3@; =;3@;_12=8 =;2!;_ _G =;2!;_8_8=32 =2=2_32=64 2 (1) ª = =64=32 // = =32 (2) // = =52 6 =52 =: 5 : 3 = =34 =;7$; =;7$;_14=8 4 : 5 : : 3 : 3 : 3 : ;1 2; // 12-x 1334= 1334 x = =x 12-x =;6{;, 72-6x=12x 12 18x=72 x=4 x 6

61 Middle School SUPRM Mathematics, = ª( ) = 2025= 20 =16 // = 25(25-16)=30 =: 5 : I. =, 12 =87 =: 2 : = 1215= {: 2 :- } 15 =12{: 2 :- } 27 =126 =: 3 : PQ // PI=I(), I PI=I PI=PI P I =P =6 Q I =Q P =PQ 1218=PQ 16 6 P 12 I 16 Q = 63=(6- ) 6 =3(6- ) =2 = 63=(6+ ) 6 =3(6+ ) =6 = + =2+6=8 PQ =: 3 : Q =Q I =PQ -P I Q=: 3 :-6=: 3 : = =;2!;( + ) =;2!;(6+18)=12 G =GH =h 6 h G h 18 H 106= =;2!;(6+12) h =;8#; =;8#;_8=3 (RH ) = =3 =9h=63 h=7 =;2!;(12+18) 7=105

62 HG =;2!; =;2!;20=10 =;2!; =10 H =;2!; M =G M M =;2!; =10 H =G =5 GH =30 = G =H =;2!; =;2!; = =GH, GH. H =;2!;( + )=;2!;(4+8)=6 GH=;2!;_4_6=12 G M M =M M=;2!; G =;3!;M G=;3!;M G=;2!;G G=;2!;;3!;M G=;2!; ;3!; ;2!; H 20 M G G M G =;1 2; O O=;3!; =;3!; ;2!; =;6!; 15=;2%; = O=+O O=: 2 :+;2%; O=10 P N O M GG',, H G M =23 G H G' H =;3@; =;3@;_18 M 18 H=12 M GH =;2!; H =;2!;_12=6 GG' =;2!; GH =;2!;_6=3 yy GM =;3!;M =;3!;_16=: 3 : yy M =;2!; =;2!;_20=10 MG' =;3@; M =;3@;_10=: 3º:yy,, GMG' GG' +GM +MG' =3+: 3 :+: 3º:=15

63 Middle School SUPRM Mathematics 1 : 2 : : 3 : ;1ª0; a+c b ;2#; 16 ;2(; 01 O, // O=O O=O() O=O O=O(), O O = + =9 // = 9=1510 =: 2 : 02 G G =, // G =G G 5 O 8 10, // G =12 G G =12 =G +G =3G =3 =13 03 M PM N Q M. Q QN =21 QM =2MNQ =20 M=M M=2MN M=2(MNQ+QM) yy M=60 yy, MQ=M+QM MQ=60+20 MQ=80 04, =2, // =, =,. 21 =;3!; =;3!;_11=: 3 : N M L Q P

64 05 = =., // = = =;7$;. =;7$;=;7$;_35=20 06 // G, =G G, // H, =H H. PQ // SR, PS // QR PQRS. SR =PQ, R, P, Q S =SR =PQ =Q =2R =S +SR +R =5R, R =51 R=;5!; R=;5!;_;2!; R=;1 0;_;2!; R=;2 0;_20=1 Q Q = =52 P P(S ) P =P yy PQ P(S ) QP =P, P P =(Q +QP )P PP=61 08 (RH ) = =6+4=10 ª( ) yy = 610= 5 =3 = 610=3 =5 = + =5+3=8 09 = =35 =;8#; =;8#;_{;2!;_4_3} =;4(; = _ 07 3 = _5 =;5(; ª( = =3;5(;=53 Q P Q =23 =;5@;=;5@;_;4(;=;1ª0;

65 Middle School SUPRM Mathematics 10 H, H MN S MN // P=S P Q M S P N R H ª 2, =;2!; SH ;2!; H =SH H RH QS // RH yy SH H =QR R yy, P=QR R b(a+b+c)=qr R b(a+b+c)=qr (Q +QR ) QR (a+b+c)=b(q +QR ) (a+c)qr =bq Q a+c 1334= QR b 11 ª( = 612= 6 =3, =12-3=9 = yy = =12 12= (9- ) 2 =9- =3 12 H O G O, O G. G GO =21, O H. H HO =21 O =O G =GH =H =3GH=3_8=24 ==24 =;2!;=12 =;2!;=6 G H 3 =3GH

66 13 = 12 =43 =9, = 712= =;1!9@; =;1!9@;_9=: 1º9 : = =12: 1º9 :=199 =199yy = =412=13 =13 =S =3S 3S=199 =;1@9&;S =S;1@9&;S= , = P P P = P Q QP Q PQ =Q P Q P P. Q =21 =;3@; Q =;3@;;2!; =;3@;;2!;12=4 Q P 15 G ªG yy G // G = =13 =3G =31 =;3!; =;3!;3G =G, G = yy 1334=1 yy, G =, G G =12 = =;2!; yy, =;2!;+1=;2#; 16 P P =P =P=90 P(S ) =P, =P P P M, M // P, M =;2!; P P G M

67 Middle School SUPRM Mathematics 17 M =;2!; P =;2!;(P + ) M=;2!;( + ) M=;2!;(5+4) M=;2(; G 5 = =23 =2S, =3S =25 =;2%;=;2%;_3S=: 2 :S G Gª G G = =32 G =3, G =2 Gª G G = =52 = =52 =;5@; =;5@;(+) =;5@;{3S+: 2 :S} =: 5 :S =+ =: 5 :S+: 2 :S =: 1 0 :S =2S: 1 0 :S = =a =2=2a 6 2a a 2a a // ==2a = = = ==ayy =+=2a =a yy, = =;2!; =;2!;_6=3 19 ªM M = M M =2821=43yy ªM = M =2821=43 yy, // M M = M 21=47 7 =4 21 =12

68 20 H G P, Q // R =21 P =x, PQ =y Q R = =31 Q x+y R = 133 = Q QR = =32 G G =;3@; yy P P P = =21 P =2 HªPH QR =;3@;Q =;3@; ( ) R PQ // R PQ R =Q R =Q (Q +QR ) x+y, y =3{3+;3@;} 3 x+y=: 3 :y x=;3*;y G G =P PQ =83 P =;2!; 2 P=14( = ) H HP =14 H =;5!; P =;5!; 2 =;5@; yy, HG =G -H =;3@; -;5@; =;1 5; H HG =;5@; ;1 5; = ,, L, M, N Q QM =R RN =21 QR // L P Q M N [ 1] R 21, P, Q, R. // Q G G =P PQ, G G P PQ. = Q =21 Q =;2!; =;2^;=3 Q = -Q =4-3=1yy 6 x y P G Q R 4 MN =M +N =;2!;( + )=;2!;, Q QM =21, LP P =12 2 2b P P QR 3a Q O R,. M b N =3a =2b,QR [ 2] M O QR =QO +OR =;3!; +;3@; MN 2b 2b QR= =;3$;b 3 3 =6ab=45 ab=: 2 :

69 Middle School SUPRM Mathematics PQR=;2!; QR a=;2!;;3$;ba PQR=;2!; ;3$; : 2 : 24 PQR=5 23 I I=I = =2012 =53 =;8%; =10 =;8#; =6 I=I I IÆ = II=2010=21yy G G G =21 yy 20 G I 12 16, G G = I IÆ G I // 20 16, I IÆ G G I G I // P Q R, PR //, PR =;2!; yy QR //, QR =;2!; yy = PQR PR =QR. +=120 =60 QR //, PR // PRQ==60 PQR. 3 PQR 2 2 PQR 2 9

70 03 1 G G=;3!;=;3!;_30=10,, //, =;2!; GªG( ) 12 GG=1 2 GG=14 G=;4!;G G=;4!;_10 G=;2%; =64125= = =827 V 827=80pV V=270p 270p-80p=190p(cm ) r r, 2r ª r (12-2r)12=r5 5(12-2r)=12r, 22r=60 r=;1#1); = 2r 5 5 =xcm x(x+2)= (x+2)=6x, 1.5x=9 x=6 ( )=6_50000=300000(cm) ( )=3(km) ( ) = P=Q=90 PªQ( ) =1620=45 PQ=4 5 = p 36 28cm 84pcm = p 40p. ( )-( ) =40p-10p=30p

71 Middle School SUPRM Mathematics ªGª G =1 2 3 =149 GG =1(4-1)(9-4)=135 G=;5#;G=;5#;_60=36,,, +, (+)(++) =1 2 3 =1827 =m, +=8m, ++=27m(m ) =m, =7m, =19m m=108 m=4 =7m=7_4=28(cm ), P PªP P P = P P =36=12 P 3 4 6, P (P +4)=12 P = =18 ( )=7_( ) ( )=7_{;3!;_p3 _4} ( )=84p(cm ) S, T V, V, r, r V V =21 r r ST=r r V V =r r Sfl Tfl =r r =V V =2 1 = pcm ;2#; m , ''' h, h', G,, ', '' M, M' h' GM =;3H;, GM' = 14 3 MM' =;6H; GM =GM' +M'M h' h ;3H;= h=2h' 3 6 hh'=21 '''=h h' =41 =4''' h' G M' M, ''' 4. ' r ' h r '

72 02 ªPGQ, G M GM =31 PGQ=91 PGQ=;9!;=;9!;_54=6 03 V, V [ 1] ( )( ) =12 (V -V )V =1 2 =18 V V =8V -8V 133 =;8&; V ;8&; [ 2] V V =4 h =87 h =56 G P M Q , =41 05 P R 14 2., OP OR =12 OP =h, OR =2h PQ =QR =;2!;h O, O, O OP OQ OR =h;2#;h2h= =82764 O V 8V V 27V-8V=19V V 64V-27V=37V, V V =1937 V =;3!7(;V =;3!7(;_74p=38p(cm ) P Q R O V V V 04 ª 123=41 21., =21, = = = =2 1 =41 06 G, H, GH //, GH =;2!; GH=14 GH=;4!;yy, =48 -GH

73 Middle School SUPRM Mathematics ;3!;GH =;3!; ;4!; ( ) ;3!;H =;1 2; 48=4 07 P P (S ) P = P = =;2!; PªP, P =;4!; P P P =116 P=;1 6;P=;1 6;_64=4 =P-P =32-4=28 08 // G ªG =12 G=1 2 =14 G=;4#;Gyy G =G, G =G, G =G = G=;3@; G=;3@;_;2!; G=;3!;_{;2!;_3_4} G=2 G=;4#;G=;4#;_2=;2#; P =31 ª 31 =3 1 =91 =9=9 =31 // = =41 =41 =161 =16 =- =15 = t P Q xm, ym PQª PQ =Q 31.65=(x+y)y 90m x=90t P Q x 31.65=(90t+y)y 3y=1.65(90t+y) y=110t 6 y 110_;1 0;=11(m) 1.65 y

74 11 O G O H O. G, H G GO =H HO =21 O =O G =GH =H, ª 12 =14, =;4!;=12 =48 GH=;3!;=;3!; GH=;3!;_48=16 =;4!;=;4!;_16=4, // = =21 =21 =;2!;=;2!;_16=8 ==8 =++ =16+8+4= = G, H 2 G =GH =H 1 2 GH a b 9 c 12 ª = =41 1, 4, 9 a, b, c =a+b+c 149 a b c =149=1 2 3 abc=123

75 Middle School SUPRM Mathematics =a+2a+3a=6a 16. ( ) =1 6 =136 =36_( ) =36_1=36 14 R P 21., '''. 21 '''=41 '''=;4!;=;4!;_12=3 15, =, = G H G G =21, H H =21 GHª G =13 GH=19 =9GH=9_10= _4= Q =Q Q =21 Q =;3!; ' ' ' P, N =;2!; NQ =N -Q =;2!; -;3!; =;6!; MN // NPQªQ NQ Q =;6!;;3!;=12 NPQQ=1 2 =14 Q=4NPQ=4_1=4 Q Q =21 QQ=21 Q=2Q=2_4=8 =Q+Q =8+4 =12 17 ', ' P, P M, N. P M, N P, P MN =;2!;, MN // ', ' P, P ' 'M =' 'N =21 '' =;3@; MN =;3@; ;2!; =;3!; '' =;3!;, '' =;3!; ' M P N '

76 '''ª 13 '''=19 '''45=19 '''=5 P, P M, N 3 '' =;3!; 2 '' =;3!;, '' =;3!; 2 ''' 2 ''' 1 10, MQ N =12 OMQ OQ, h N 2h. N=;2!; N 2h=;2!;6OQ 2h() N=12;2!; OQ h=12omq =2N =212OMQ =24S O O OM =21 M P, PM // ONªOMQ( ) , ONOMQ=41 ON=4S ON=;6!; =6ON=6 4S=24S 1 24S 2 ;3@; (1) 14 (2) 12 5 ;7%;a ;3@; 8 ;8#; M N O., PM // Q QN =M M =11 OQ =O -Q OQ=;3@; N -;2!; N OQ=;6!; N yy P, Q, R, S, =2a =3a ªS S Q S = 2aS =21 S =a SªQ =Q S 34=Q 4a Q =3a OªOQ O O = Q =2a3a=23 O 1334=;3@; O O P R

77 Middle School SUPRM Mathematics =;2!; yy P =G =G G=;3!; Q G (2), =;2!; 2 =14 a 4a b a+b 4b G=;3!;;2!; G=;3!;;2!;70=: 3 : PªP =P P =;2!; ;3!; =32 ( = ) P =35 PG=;5#;G=;5#;: 3 :=7 a, b (1) =14 =4a, =4b, = ==a+b =2(a+b)4(a+b) =12 =12, QªGQ G =Q GQ =;2!; ;3@; =34 ' Q G =37 PQPG=37 PQ=;7#;PG=;7#;7=3 (1) ==90 =90 - =- = ª( ) =,12= =2 yy, = 12= a G ' '. // 'ª'G ' ' = G =21 =;6%;a G =;2!; =;1 2;a, Gª G =;1 2;a;6!;a=52 =52 =;7%; =;7%;a

78 9 M Q 6 6 P N 4, M, N, P M // P M =;2!; =;2!;_12=6 MªP M = =3 M P P =M =6, MN Q MQªPNQ MQ NQ =M NP =612=12 MN =6 MQ =2, QN =4 MN MQ=N=60 M =N =6, MQ =N =2 MQ N, MQ=N =MN=60,,, M ', ', ', ' 'ª' =' ' yy, M'ªM' M =M M' M' ' =' yy =t =2t ' // ' =' ' 3a2t=' ' 3a ' = 133 ' 2t ' // ' =' ' at=' ' ' =;t; ' 3a ' ' = 133 ' ;t; ' 2t ' ' =32 () 1334=;3@; P 2a Q H G ' a 3a ' ' M =a, =3a. ' 2a, G, H H(S ) =H H H

79 Middle School SUPRM Mathematics H = H a =2a H H =;2;, G(S ) G = =2a HG =G -H =2a-;2;=;2#;a H G Q HQ // P HQ P =GH G =;2#;a4a=38 P QH(S ) P =QH P P =QH P =38 HQ 1334=;8#; P

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