ມ.2_ຄະນິດສາດ_Math

A { 0ufmjvo-nj9kdc8]j t9v,skg,afg7y’j dk’mu1j 6j0hk’g-yj’|hkda[,aoD B C @D glAo9v,lkd dD oypk,G glAo-nj0uf9kd9v, c]t 8A’lkd A da[0hk’dq’|hk gvUo;jkG glAo9v,lkdD { AD ⊥ BC F AD c,joglAo9v,lkdmu0j uf BD 9kd9v, A D A C H BA 0D lk,glAo9v,lkd0v’I6[lk,c98afdao 16gj ,af|bj’ c]t g,affP; gmqjkoAog-y’j gvUo;kj G vvd38-a’ XOrthocentre)D C { H c,jovvd38-a’0v’I6[lk,c9 ABC ສະຫງສ.ວວນ.ລິສຂະສິດ B C { 4hkI6[lk,c9,u|bj’,5,gxao,5,s;kF vvd38-a’9t16oj vdI6[lk,c9oAoD H dD oypk,G glAo-njmujzjkog,afg7yj’dk’0v’0hk’|b’j #D glAodk’lkd c]t 8A’lkdda[0hk’oAo16jg,affa’j djk; gvUo;kj G glAodk’lkdD AL C { Fgxaog,afg7yj’dk’0v’ BC F FL ⊥ BC S FB = FC S FL c,joglAodk’lkd0v’ BC D BF { lk,glAodk’lkd0v’I6[lk,c98afdao A 16jg,af|b’j D g,afoahoskj ’ltg\\u9kd9v,0v’ C I6[lk,c9D OA = OB = OC O { g,af O gxaog,af.9dk’0v’;q’,qoco[I6[lk,c9D B 145

AO { 4hkI6[lk,c9,u,5,|bj’gxao,5,s;kF g,af8afdao 0v’lk,glahodk’lkd9t16oj vdI6[lk,c9D BC 0D ;ymulhk’glAodk’lkd 8q;1jk’G lhk’glAodk’lkd0v’0hk’ BC { 0uflv’mvj odq’j muj,ug,af.9dk’ A c]t B M ]aflt\\ugmqjkda[ BC A { lv’mvj odq’j 8afdao16jlv’g,af M c]t N BC M NA { 0uf8+g,af M sk N 9twfhglAodk’ສະຫງສ.ວວນ.ລິສຂະສິດ B C lkd0v’ BC D N A { lj;oglAodk’lkd0v’0hk’ AB c]t AC BO c,oj xt8y[afg-ajofP;daoda[dkolhk’glAo C dk’lkd0v’0hk’ BC D { lk,glAodk’lkd8afdao16 O Xfaj’I6[?D 146

$D glAoc[j’g7y’j ,5, dD oypk,G glAo-nmj ucj [’j ,5,.o0v’I6[lk,ສະຫງສ.ວວນ.ລິສຂະສິດ !D glAoc[’j g7y’j ,5,.o c9vvdgxaolv’,5,mu,j u7kj ;afcmdgmqjkdao gvUo;kj G glAoc[’j ,5,.oD A C { BnAD = DnAC F AD c,joglAoc[’j g7yj’ ,5, Al D BD { lk,glAoc[’j g7y’j ,5,.o0v’I6[lk,c9 A 8afdao16gj ,af|bj’F g,afoAo skj ’ltg\\u9kd lk,0hk’0v’I6[lk,c9D IM = IN = IP PN C { I c,jog,af.9dk’0v’;q’,qo8yf.oI6[ I lk,c9 ABC D BM 0D ;ymulhk’glAoc[’j g7y’j ,5, 8q;1jk’G lhk’glAoc[’j g7y’j ,5, Bl A { 0ufmvj odqj’mu,j ug,af.9dk’16j B 8af C 0hk’ BA c]t BC 16j M c]t N B 8k,]efa[D A { 0uflv’mjvodq’j mu,ug,af.9dk’16j MP C M c]t N S lv’mvj odq’j 8afdao16j P BN { 0uf8+ B sk P 9twfhglAoc[’j g7y’j ,5, Bl D A C MP BN { l;j oglAoc[j’g7yj’,5, Al c]t Cl c,oj xt8y[afg-aoj fP;daodaodkolhk’glAoc[j’ A g7yj’,5, Bl C { lk,glAoc[j’g7yj’,5,8afdao16j I Xfaj’I6[?D I B 147

@D glAoc[’j g7yj’,5,ovd dD oypk,G glAo-njmujc[’j ,5,ovd0v’I6[lk, c9vvdgxaolv’,5,mu,j u7kj ;afcmdgmqkj x daogvUo;kj glAoc[’j g7yj’,5,ovdD A { CnAJ = nJAx c]t (AJ) c,joglAo c[’j g7yj’,5,ovdD BC y x J A { glAoc[’j g7y’j ,5,.o c]t glAoc[’j g7yj’ ,5,ovdmuj0uf9kd9v,fP;dao 8A’lkddaoD B LC (AL) c,oj glAoc,joglAoc[’j g7yj’,5,.o { (AL) ⊥ (AJ) x ສະຫງສ.ວວນ.ລິສຂະສິດ Jy A { lk,glAoc[j’g7yj’,5,ovd8afdao16gj ,af|bj’g,afoAo O c,jog,af.9dk’;q’,qo8yfovdI6[lk,c9 { Oc,oj g,af.9dk’;q’,qo8yfovdI6[lk, B I c9 ABC D LC y &D I6[lk,c9ltg\\u J A .oI6[lk,c9ltg\\uF glAoc[’j g7yj’,5,.o0v’ m5dM,5,gxaoma’glAo9v,dk’F gxaoma’glAo9v, NM lkdF gxaoma’glAodk’lkd c]t 8afdao16gj ,af |bj’ c]t g,affP;gmqjkoAoD g,affaj’dkj ; gxaoma’ O C g,af4j;’|ad0v’I6[lk,c9F gxaoma’vvd38-a’F BL gxaoma’g,af.9dk’0v’;q’,qo8yf.o c]t ;q’,qo co[I6[lk,c9oAoD 148

#D [qfg/ydsaf !D 9q’j c8h,I6[lk,c9]5j,oU7noF 9kdoAodeoqfg,af4j;’|ad le]a[I6[ (a) F vvd38-a’ le]a[I[6 (b) F g,af.9dk’0v’;q’,qoco[I6[lk,c9le]a[I6[ (c) c]t g,af.9 dk’0v’;q’,qo8yf.oI6[lk,c9le]a[I6[ (d) HK AG B (a) CF (b) E I (c) J L (d) M @D 9q’j c8h,glAodk’lkd S X 0v’I6[ (1) c]t I6[ (2) D9kdoAo -Uc9’;kj G lv’glAodk’lkd0v’ (1) TW ສະຫງສ.ວວນ.ລິສຂະສິດ(2)Y 0hk’,5,lkdxtdv[da[ R lv’0hk’,5,lkd gxaoI6[lucj 9lkd c]t I6[9t85]afle]a[I6[ (1) c]t I6[ (2) 8k,]efa[D #D .shI6[lk,c9 ABC lkd16j A D 9qj’deoqfg,af8afdao0v’lk, glAodk’ lkd c]t g,af8afdao0v’lk,glAo9v,lkd0v’,ao c]h;.sh0=hla’gdfD $D .shI6[lujc90hk’0t|ko ABCD F I c,jog,afg7y’j dk’0v’ AB c]t J c,oj g,af g7y’j dk’ 0v’ BC D 9q’j -Uc9’;jkG [AJ) c]t [CI) 8afdao16jglAogoa’j 9v, BD 0v’ I6[lujc90hk’0t|koD &D .oI6[lk,c9mjP’ ABC (AB = AC) D lv’glAodk’lkdmuj0uf9kd9v, B c]t C 8afdao16jg,af I D 9q’j -Uc9’;jkG glAoc[’j g7yj’,5, Al zjko I. *D .shI6[lk,c9 ABC F M c,jog,afg7y’j dk’0v’ BC D glAo-njzjko M c]t 0t|koda[ AC 8af AB 16j K S 0t|ko AB 8af AC 16j N. 9q’j -Uc9’;jkG AM, KC c]t BN 8afdao16gj ,af|bj’ I. 149

ສະຫງສ.ວວນ.ລິສຂະສິດ(D .oI6[lk,c9mjP’ ABC (AB = AC) F glAo9v,lkdmuj0uf9kd9v, B c]t C 8afdao16gj ,af I. 9qj’-Uc9’;jkG AI ⊥ BC D )D .shlugj ,af A,B,C c]t D 16jglAo-nfj P;daoF I6h;kj AB = CD. 9qj’-Uc9’;kj G M c,jog,afg7y’j dk’0v’ [BC] d=c,oj g,afg7y’j dk’0v’ [BD]D _D .oI6[lucj 90hk’0t|ko ABCD mu,j ul6odk’ OF glAodk’lkd0v’ AB c]t AD 8afdao16j K. 9qj’-Uc9’;jkG OK ⊥ BD D !WD .shI6[lk,c9ltg\\u ABC F glAo-njzjko B 0t|ko AC c]t glAo-nzj kj o C 0t|ko AB 8afdao16jg,af E D 9qj’-Uc9’;jkG I6[lk,c9 EBC gxaoI6[lk,c9ltg\\uD !!D .shlugj ,af A,B,C c]t D 16;j q’,qo|b’j D 9q’j 0ufglAo9v,lkd CI c]t DJ 0v’I6[ lk,c9 ABD c]t ABD 8k,]efa[D glAo-nj (DJ) c]t (CI) ,udkorq;raodao co;.fL !@D .shI6[lujc90hk’0t|ko ABCD S M gxaog,af|bj’mu[j +16gj lAo-nj (AC) s^n (BD) D 9q’j -vdskg,af4j;’|ad0v’I6[lk,c9 AMC c]t BMD. la’gdfgsaoco;.fL !#D .shI6[lk,c9 ABC lkd16j A. H c,jo8uo0v’glAo9v,lkdmu0j u9kd9v, A. M gxaog,af|bj’8k,.916jmvj o-nj BC D ;q’,qo C muj,uglAozkj o.9dk’ AM 8af AB 16j I c]t 8af AC 16j J. 9qj’-Uc9’;kj G dD H 16j;q’,qo C D 0D AIMJ gxaoI6[lujc9lkdD 7D 4hk ABC c,oj I6[lk,c9lkdmPj ’F 9t;k’ M 16j[jvo.f0v’mjvo [BC] grnjv.sh [IJ] lAomujl5fL 150

[qfmu #$ ];’Iv[ c]t goNvmuj0v’I6[lucj 9lkdF I6[9t85]af c]t I6[fvd9ao !D dyf9tde dyf9tdemu ! de,tdvoz6h|b’j x6dk3ishv’ovomujgxaoI6[lucj 9lkdF I6h;jk dk3i c8j]tczoj ,u0t|kf 25cm× 25cm Xfa’j I6[?D A B dD 9qj’0Po8nj,.lj[jvo9Eg,afD AB = DC = ……… cm AD = BC = ………. cm 0D 9qj’7yfw]j c]h;xP[mP[zqowfhIa[]ts;jk’G ]A;B’+IvB[C0+v’CID6[+lucjD9Alkcd]gtmqjk2d×a[(ADDBDDDD+DDBDDDDCDDD)DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD C 7D de,tdvo.-hdk3ima’\\qf DDDDDDDDDDDDDDDDDDDDDDDDDczjo X!? ສະຫງສ.ວວນ.ລິສຂະສິດ { 0hk’ AB ,u DDDDDDDDDDDDDDDDDD czoj F 0hk’ AD ,u DDDDDDDDDDDDDDDDDDDDD czoj { 9eo;odk3i8k,0hk’ AB 76oda[9eo;odk3i8k,0hk’ AD ,u DDDDDDDDDDDDDDD czoj X@? { 9q’j xP[mP[zqowfhIa[9kd X!? c]t X@?D fa’j oAo goNvmuj0v’I6[lujc9lkdgmqjkda[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD dyf9tdemu @ 3fpvy’.ldj yf9tdemu !F gIqk4vowfhG { ];’Iv[0v’I6[9t85]afmuj,u0hk’gmqkj a c,oj DDDDDDDDDDDDDDDDDDDDDDD { goNvmu0j v’I6[9t85]afmuj,u0hk’gmqkj a c,jo DDDDDDDDDDDDDDDDDDDDDDDDDDDD dy9tdemu # a la’gdfI6[fvd9ao X!? mu,j uglAogoa’j 9v, D c]t d c]t I6[lucj 9lkd X@? muj,u];’pk; c]t ];’d;hk’c,oj D c]t d 8k,]efa[D 2 dD 9q’j xP[mP[goNvmu0j v’I6[fvd9ao X!? c]t goNvmu0j v’I6[lujc9lkd X@?D 0D 9q’j oe.-hl6f7yfw]jgoNvmuj0v’I6[lujc9lkdgrnvj 7yfw]jgoNvmuj0v’I6[lucj 9lkd X@?D 151

7D 9kd0=h Xd? c]t X0? gIqk4vowfh goNvmu0j v’I6[fvd9aomuj,uglAogoa’j 9v, D c]t d c,jo DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ’D I6h;kj I6[fvd9ao c8]j t0hk’,u];’pk;gmqjkdao faj’oAogIqkwfh];’Iv[0v’I6[ fvd9aomu,j u0hk’gmqkj a c,jo DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD D d d I II 2 IV I IV III d II III (1) 2 d 2 (2) II IV I III D @D .97;k, b ];’pk; ສະຫງສ.ວວນ.ລິສຂ];ະ’ສິdhດ;k’ !D I6[lujc9lkd a .sh a c,oj ];’pk;F b c,oj ];’d;hk’ P c,oj ];’Iv[F A c,oj goNvmuj gIqkwfhG P = 2× (a + b) c]t A = a ×b 8q;1kj ’G dtfkoczjo|b’j ];’d;hk’ c]t ];’pk;0v’,aocmdwfh 1,5m c]t 2m 8k,]efa[D 9q’j -vdsk];’Iv[ c]t goNvmuj0v’,aoD [qfcdhG 0=hlq,,5f a = 2, 40 m c]t b = 1,5m 1,5m 0=hlts^5[ -vdsk P c]t A vu’8k,l6f gIqkwfh 2,40m A = a×b = 2, 40×1,50 = 3, 6 m2 P = 2× (a + b) = 2× (2, 40 +1,50 ) = 7,8 m @D I6[9t85]af 0hk’ .sh a c,oj ];’pk;0v’0hk’ a P c,jo];’Iv[F A c,oj goNvmuj gIqkwfhG P = 4× a = 4a c]t A = a × a = a2 152

8q;1jk’G l;o8vo|b’j gxaoI6[9t85]af,u];’Iv[gmqkj da[100m D 9q’j -vdsk0hk’ c]t goNvmuj0v’l;o8vooAoD [qfcdhG a 0=hlq,,5f P =100 m 0=hlts^5[ -vdsk a c]t A vu’8k,l6f gIqk,u P = 4a 4vowfh A = a2 = 252 = 625 m2 a = P = 100 = 25 m 4 4 #D I6[fvd9ao a D .sh a c,oj ];’pk;0v’0hk’F d D c]t d c,oj lv’glAogoaj’9v, P c,oj ];’Iv[F A c,oj goNvmuj gIqkwfhG P = 4× a = 4a c]t A= D×d ສະຫງສ.ວວນ.ລິສຂະສິດ 2 8q;1jk’G 0hk’0v’I6[fvd9ao|b’j cmdwfh 10cm F lv’glAogoa’j 9v,cmd wfh 12cm c]t 18cm. 9qj’-vdsk ];’Iv[ c]t goNvmu0j v’,aoD [qfcdhG 0=hlq,,5f a = 10cm, D = 18cm 10 cm 12 cm c]t d = 12cm. 0=hlts^5[ -vdsk P c]t A 18 cm A= D×d = 18 ×12 = 108cm2 2 2 vu’8k,l6f gIqkwfhG P = 4a = 4×10 = 40 cm #D [qfg/ydsaf !D .shI6[lujc9lkd|bj’F ,u];’pk;]njo];’d;hk’ 5cm D 9qj’-vdskgoNvmuj0v’,aoD I6h;jk];’Iv[gmqjkda[ 50cm. 153

@D 9q’j 7yfw]j c]h;8nj,.l8j k8t]k’]5,j oU 3fp;kj G dD I6[lucj 9lkd 0D fvd9ao a bPA a D d P A 7m 4m 13 cm 24 cm 10 cm 58 cm 25 dm 6 m2 15 dm 6 cm 28cm 40 dm 16 cm 2, 4dm2 1, 2 m2 7D I6[9t85]af a P A 800 mm 80 cm 81dm2 #D gryjogIafmk’d;hk’ 2,5m vhv,,l;o8vo|b’j mujgxaoI6[lujc9lkd g-yj’];’pk; c]t ];’d;hk’0v’l;ocmdwfh 50m c]t 30 m. 9qj’-vdskgoNvmu0j v’ mk’ c]t ສະຫງສ.ວວນ.ລິສຂະສິດ goNvmu0j v’l;omupj a’gs^nvD $D gryjomkludecr’ l6’ 2,5m c]t pk; 40m ma’lv’fhkoD dD goNvmuj muj9t8hv’mkluma’\\qf,ugmqjk.fL 0D 4hk;jk lu4a’|bj’mkwfh 50m2 gryjo9t8hv’.-hluma’\\qf9ad4a’L &D 3I’ry,csj’|bj’8hv’dkozt]yf[afv;prvogxaoI6[lucj 9lkd .sh,ugoNvmuj 66cm2 c]t pk; 12cm. 4k,;kj ];’d;hk’0v’[af,ugmqkj .fL *D 9qj’7yfw]jgoNvmu0j v’rkdl;j omumj klu0v’I6[]5j,oUD 2 12 m 9m 2,5 2,5 10 2m 2 18 m 15 40m (D l;o8vo|bj’gxaoI6[lucj 9lkdF ,u 27m 2,5 m mk’lv’glAozjko Xfaj’I6[c8h,?D 9q’j 7yfw]jgoNvmu0j v’mk’ c]t goNvmuj0v’l;omujpa’gs^nvD 3m 154

[qfmu #& ];’Iv[ c]t goNvmu0j v’I6[lujc90hk’0t|koF I6[lk,c9 c]t I6[7k’\\6 !D dyf9tde dyf9tdemu ! 9qj’la’gdfI6[]5,oj U c]h;lts^5[gvqkl6f7yfw]jgoNvmu0j v’I6[lucj 90hk’0t|koD XI6[lucj 90hk’0t|ko? X8afI6[lucj 90hk’0t|ko? XI6[lujlkd? h II h II I II I h I b bb fa’j oAo goNvmuj0v’I6[lucj 90hk’0t|kogmqjk DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD dyf9tdemu @ 9q’j la’gdfI6[]5,j oU c]h;4vf4vogvqkl6f7yfw]jgoNvmu0j v’I6[lk,c9D ສະຫງສ.ວວນ.ລິສຂະສິດ b = h (1) + b h h (1) (2) h (2) bb h (1) = b h c8;j kj goNvmu0j v’I6[(1) gmqkj goNvmu0j v’(2) (2) 9kddyf9tdemu ! gIqkwfhG b =bh hI b ..... fa’j oAo goNvmuj0v’I6[lk,c9gmqkj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 155

dyf9tdemu # la’gdfI6[]5,j oU 9kdoAo4vf4vogvqkl6f7yfw]gj oNvmu0j v’I6[7k’\\6D h (I) b2 h= b2 (II) (II) + h (I) b1 b1 + =b1h b2h ......... + ........ 22 2 faj’oAo goNvmu0j v’I6[7k’\\6gmqkj DDDDDDDDDDDDDDDDD=DDDDDDDDDDDDDDDDDDDD(DD.D.D.DD.D.DD+DDD.D2.D.D.DD.D)DD×DDD.D.D.DD. @D .97;k, !D I6[lucj 90hk’0t|ko 120 m ສະຫງສ.ວວນ.ລິສຂະສິດ .sh b c,oj rNoF a c,oj 0hk’ c]t h c,oj ];’l6’ P c,jo];’Iv[ c]t A c,jogoNvmuj ha gIqkwfhG P = 2× (a + b) c]t A = b× h b 8q;1jk’G mk’glAo|b’j zkj omqj’okmugj xaoI6[lucj 9lkd Xfaj’I6[?D 9qj’7yfw]gj oNvmu0j v’glAomk’ c]t goNvmu0j v’okmugj s^nv9kddkogIafmk’D [qfcdhG 0=hlq,,5f I6[lujc9lkd,u a = 150 m c]t b1 = 120 m. I6[lu0j hk’0t|ko,u h = 120 m c]t b2 = 6 m. 6m 0=hlts^5[ 7yfw]jG A1 c]t A2 A1 A2 • djvovnjo7yfw]gj oNvmu0j v’I[6lucj 9lkd c]t A2 D 9kd0=hlq,,5fgIqk,uG 150 m A = a × b1 A2 = b2 × h = 150 m ×120 m = 6 m ×120 m = 18000 m2 = 720 m2 A1 = A − A2 = 18000 m2 − 720 m2 = 17280 m2 156

@D I6[lk,c9 A I6[lk,c9 ABC ,u rNc,joob c]t h c,oj ];’l6’D h gIqkwfhG P = AB + BC + CA c]t A= b×h BD C 2 b 8q;1jk’G 9q’j 7yfw]jgoNvmu0j v’I6[lk,c9 CAT D I6h;kj 0hk’rNo CT = 25cm c]t ];’l6’ h = 10cm. [qfcdhG 0=hlq,,5f b = CT = 25cm c]t h = 10cm. 0=hlts^5[ 7yfw]Gj A A h 9kd0=hlq,,5f gIqk,u A = b× h 2 C bT = 25cm ×10 cm 2 = 125cm2 ສະຫງສ.ວວນ.ລິສຂະສິດ #D I6[7k’\\6 A b2 B I6[7k’\\6 ABCD ,u b1 c,jorNo.spjF b2 c,jorNoohvp c]t h c,oj ];’l6’D h gIqkwfhG P = AB + BC + CD + DA D b1 C c]t A (b1 + b2 ) × h = 2 8q;1jk’G 7yfw]jgoNvmuj0v’I6[7k’\\6 g,nvj I6h];’pk;0v’lv’rNo c]t ];’l6’c,jo 12cm, 8cm c]t 7 cm 8k,]efa[D [qfcdhG 0=hlq,,5f b2 = 8cm,b1 = 12cm c]t h = 7 cm. 0=hlts^5[ 7yfw]Gj A 8 cm 7 cm 12 cm 9kd0=hlq,,5f gIqk,uG A = (b1 + b2 ) × h = (12 +8) × 7 2 2 = 70 cm2 157

#D [qfg/ydsaf 0D I6[lk,c9 !D 9qj’7yfw]cj ]h;8nj,.lj8k8t]k’]5,j oUD dD I6[lucj 90hk’0t|ko a b h AP b h A 13 cm 1, 2 cm 5 cm 4,8 m2 1000 cm 50 cm 2,5 dm 1,8 m2 30 dm 50 cm 6 dm2 24 cm 3cm 12 cm2 16 cm 7 3x 3 5x 7D I6[7k’\\6 (m c,oj rNoltg]pj ? 9 3 36 12x b1 12 6,8 15 45 27 12 b2 8 3, 2 15 9 h 7 6,1 A 112 300 90 m 14 @D .oI6[lk,c9 ABC, M c,oj g,afg7y’j dk’0v’0hk’ BC. ສະຫງສ.ວວນ.ລິສຂະສິດ 9q’j ryl6f;jk goNvmuj0v’I6[lk,c9 AMB gmqjkgoNvmu0j v’I6[lk,c9 AMC c]t gmqkj g7y’j |b’j 0v’goNvmu0j v’I6[lk,c9 ABC. #D 9kdI6[ 9qj’7yfw]jG 0D A D E dD P Q 12 O 5 20 R 8 S BC { APQRS = ?, APSR = ? { AABC = ?, ABCE = ? AOSR = ?, APSO = ? { -Uc9’;kj ADBC = 1 ABCED 2 $D 9q’j 7yfw]gj oNvmu0j v’I6[7k’\\6 GROS I6h;jkgoNvmuj0v’I6[lucj 9lkd GRAS G 23cm R gmqjk 828cm2 Xg[y’j 8k,I6[? S A O 31cm 158

[qfmu #* ,5,.o c]t ,5,ovd0v’I6[lk,c9 !D dyf9tde dyf9tdemu ! 2 2 1 13 3 .oI6[c8h,F ,5, 1; 2 c]t 3 c,oj 31 2 ,5,.o0v’I6[lk,c9F 3fpdkola’ 31 2 gdf8k,I6[ gIqkwfhG ສະຫງສ.ວວນ.ລິສຂະສິດ 1 + 2 + 3 c,oj ,5, DDDDDDDDDDDDDDDDDDDDDDD 2 ltoAo Š zqo[;d,5,.o0v’I6[lk,c9 |b’j 9bj’gmqkj da[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 2 3 31 1 Š ,5,ovd0v’,5, 3 gmqkj da[ 1 + 2 = 180D −........ fh;p;ymufP;dao gIqkwfh ,5,ovd0v’,5, 2 gmqkj da[ 1 + 3 = 180D −........ c]t ,5,ovd0v’,5, 1 gmqkj da[ 2 + 3 = 180D −........ \\kp7;k,;kj 7jk;afcmd,5,ovd0v’I6[lk,c9gmqjkzqo[;d0v’DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD dyf9tdemu @ B .oI6[lk,c9lkd ABC F lkd16j A D 9kddyf9tdemu !F gIqkI6h;kj G Al + Bl + Cl = 180D c8j;jk Al = DDDDDDDDDDDDDDDDDD gIqkwfhG Bl + Cl = 180D − DDDDDDDDDDDDD = DDDDDDDDDDDDD A C 159

fa’j oAo zqo[;d0v’lv’,5,cs^,0v’I6[lk,c9lkdgmqkj da[DDDDDDDDDDDDDDDDDDDDDDDDDDD #D .97;k, A !D,5,.o0v’I6[lk,c9 dD zqo[;d,5,.o0v’I6[lk,c9|b’j gmqkj 180D \\kp7k,;jk B C Al + Bl + Cl = 180D 8q;1kj ’ !D .shI6[lk,c9 ABC g-yj’,u,5, Al = 36D c]t Bl = 55D.9q’j 7yfw]j,5, Cl. [qfcdhG A 0=hlq,,5f I6[lk,c9 ABC F Al = 36D c]t Bl = 55D 0=hlts^5[ -vdsk Cl 36D 55D C 9kd0=hlq,,5f gIqk,u Al + Bl + Cl = 180D gIqkwfh Cl = 180D − (Al + Cl) B = 180D − (36D + 55D ) = 89D ສະຫງສ.ວວນ.ລິສຂະສິດ 0D zqo[;d0v’lv’,5,cs^,0v’I6[lk,c9lkd|n’j gmqkj da[ 90D D \\kp7;k,;jk .oI6[lk,c9lkd ABC F lkd16j A F gIqkwfhG Bl + Cl = 90D { 75o]adltotxUo7noG I6[lk,c9muj,uzqo[;d0v’lv’,5,cs^,gmqkj da[ 90D c,oj I6[lk,c9lkdD 8q;1jk’ @D .oI6[lk,c9lkd SON, lkd16j O. S = x +10D c]t Nl = 2x + 5D D 9qj’-vdsk7jk0v’ x c]t 7jk0v’lv’,5,cs^,oAoD [qfcdhG I6[lk,c9lkd SON, lkd16j O. S 0=hlq,,5f S = x +10D c]t Nl = 2x + 5D 0=hlts^5[ -vdskG x; S c]t Nl 9kd0=hlq,,5f gIqk,u ON S + Nl = 90D gIqkwfh (x +10D ) + (2x + 5D ) = 90D x +10D + 2x + 5D = 90D 3x = 75D ⇒ x = 75D = 25D 3 160

faj’oAo S = 25D +10D = 35D c]t Nl = 2× 25D + 5D = 55D A 7D .oI6[lk,c9ltg\\uF 7kj ;afcmd0v’c8j]t,5,.o gmqkj dao c]t gmqkj da[ 60D D 60D \\kp7;k,;kj .oI6[lk,c9ltg\\u ABC gIqk,u 60D 60D BC Al = Bl = Cl = 60D A @D ,5,ovd0v’I6[lk,c9 ,5,ovd0v’I6[lk,c9|b’j gmqkj zqo[;d0v’ lv’,5,.omu[j +I;j ,9v,fP;da[,aoD \\kp7;k,;kj .oI6[lk,c9 ABC F AnCX c,jo,5,ovd|bj’0v’,ao B X gIqkwfhG AnCX = Al + Bl C 8q;1kj ’G .oI6[lk,c9 HOT F lj;ocdjpk;0v’glAo-nj (HT) xtdv[da[0hk’ OT gxa,5, 106D D 9qj’7yfw]j 7kj 0v’,5,.o Hl c]t Tl 0v’I6[lk,c9 HOT D I6h;jk Ol = 38D ສະຫງສ.ວວນ.ລິສຂະສິດ [qfcdhG O 0=hlq,,5f OnTY = 106D c]t Ol = 38D 38D 0=hlts^5[ 7yfw]Gj Hl = ? c]t Tl = ? 9kd0=hlq,,5f gIqk,u H 106D Hl + Ol = OnTY gIqkwfh Hl = OnTY − Ol TY = 106D − 38D = 68D c]t Tl + OnTY = 180D gIqkwfh Tl = 180D − OnTY = 180D −106D = 74D #D [qfg/ydsaf !D gryoj lk,kfc8h,I6[lk,c9muj,u,5,.ogmqkj da[G dD 25D, 75D c]t 35D D 0D 55D, 60D c]t 85D D 7D 55D, 60D c]t 65D. wfh[+L phvospa’L 161

@D 9q’j 0Po7kj ,5,.o0v’I6[lk,c9 ABC 8n,j .lj8k8t]k’]5j,oUF Al 28D 65D 115D 24D 26D 50D Bl 102D 48D 32D 60D 106D 74D 50D 45D Cl 25D 36D 118D 25D 60D 90D #D ,5,9v,0v’I6[lk,c9mjP’|b’j gmqkj 48D D 9q’j -vdsk7jk;afcmd0v’,5,rNoDສະຫງສ.ວວນ.ລິສຂະສິດ $D ,5,rNo0v’I6[lkc9mPj ’|b’j gmqkj 45D D 9q’j -vdsk7jk;afcmd0v’,5,9v,0v’,aoD &D ,5,cs^,|bj’0v’I6[lk,c9lkdgmqkj 35D D 9qj’-vdsk7jk;afcmd0v’,5,cs^,mujlv’D *D 9q’j -vdsk7kj ;afcmd0v’,5,.o0v’I6[lk,c9 ABC D I6h;jkr;d,aogxao vaf8kl;j o rq;raoda[G dD 1, 2 c]t 3 rhv,[vdxtgrf0v’0v’I6[faj’djk;D 0D 3, 4 c]t 5. 7D 5, 6 c]t 7. (D .oI6[lk,c9lkd|b’j lv’,5,cs^,0v’,aos^5f]njodao 36D D 9q’j 7yfw]7j jk0v’ c8]j t ,5,cs^,fa’j djk;D )D .oI6[lk,c9lkd ABC lkd16j A c]t Bl = 34D F gryoj 0ufglAo9v,lkd AH c]t glAoc[’j g7yj’,5, AD 0v’,5, HnAB D 9q’j 7yfw]j7kj 0v’,5, CnAD c]t CnDA. _D ,5,ovd0v’I6[lk,c9|b’j gmqjk 120D F |b’j ,5,.omu[+I;j ,9v,fP;da[,aogmqkj 40D D 9q’j 7yfw]7j kj c8j]t,5,.o0v’I6[lk,c9fa’j dkj ;D C !WD .oI6[lk,c9lkdmPj ’ ABC F lkd16j A, glAoc[’j D g7yj’,5, Bl 8af AB 16jg,af E D 9kd E 0ufglAo-nj E (BC) 8A’lkd16gj ,af D. 9qj’7yfw]j7jk;afcmd0v’,5, BnED. !!D .oI6[lk,c9 ABC F glAoc[j’g7yj’,5, Bl c]t AA B Cl 8afdao16gj ,af I. 4hk;kj Al = 80D D 80D C 9qj’-vdsk,5, BnIC I B ? 162

[qfmu #( ,5,.o0v’I6[lujc9l;f !D dyf9tde dyf9tdemu ! 9qj’0ufglAogoa’j 9v,.ljc8j]tI6[lucj 9muj16]j 5j,oU 3fp[+.shglAogoa’j 9v,8afdaoF c8j]tI6[wfh9adglAoL { glAogoaj’9v,c[j’I6[lucj 9vvdgxao9adI6[lk,c9L 12 5 34 dyf9tdemu @D ສະຫງສ.ວວນ.ລິສຂະສິດ .oI6[lucj 9 ABCD F glAogoa’j 9v, AC A c[j’I6[lujc9 ABCD vvdgxao 12 B lv’I6[lk,c97n ABC c]t ACD D { .oI6[lk,c9 ABC gIqk,uG D A2  B + C2  DDDDDDDDDDDDDDDDDDDDDDDD { .oI6[lk,c9 ACD gIqk,uG A1 + C1 +D  DDDDDDDDDDDDDDDDDDDD 1 2 c8j A = A1 + A2 c]t C  C1  C 2 C { .oI6[lujc9 ABCD gIqk,uG A +B + C +D = (A2  B + C1)  (A1 + C1 +B)  DDDDDDDDDDDDDDDDDDD  DDDDDDDDDDDDDDDDDDDD ltoAo zqo[;d,5,.o0v’I6[lujc9l;f|bj’9bj’gmqjkda[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD @D .97;k, A !D zqo[;d,5,.o0v’I6[lujc9l;f|b’j gmqjk 360 D \\kp7;k,;jk .oI6[lujc9l;f ABCD D B gIqkwfh A + B + C  D  360 C 163

8q;1jk’ !D .oI6[luc9l;f EFGH muj,u E  90, F  118 c]t H  77 D [qfcdhG 0=hlq,,5f E c,oj ,5,lkdF F  118 c]t H  77 E 0=hlts^5[ -vdskG G H 77 118 F 9kd0=hlq,,5f gIqk,u E + F + G + H  360 H  360  (E + F + G) G  360  (90 118  77 )  75 8q;1kj ’ @D .oI6[lujc9 ABCD F A  86 F C  110 F D  40 c]t BE c,joglAoc[’j g7y’j ,5, B D 9q’j -vdsk AEB D [qfcdhG A B 0=hlq,,5f A  86 S C  110 S ສະຫງສ.ວວນ.ລິສຂະສິດ 80 D  40 c]t ABE = EBC E _? D 40 0=hlts^5[ -vdskG AEB 110 9kdI6[ gIqk,uG C A + ABE + AEB  180 AEB  180  (A + ABE) c8j;jk ABE  1 B D 2 vudfhko|bj’ A + B + C  D  360 B  360  (A + C + D)  360  (86 110  40 )  124 Xgrkt;jk ABE  1 B  124  62 ? fa’j oAo AEB  180  (68  62)  50 22 @D ,5,.o0v’I6[lujc90hk’0t|koD A B .oI6[lujc90hk’0t|ko zqo[;d0v’lv’,5,.o4afdao gxao,5,7q[rP’D D C \\kp7;k,;jk .oI6[lucj 90hk’0t|ko ABCD gIqkwfhG 164

A + B = B + C = C + A  D + A  180 A = C c]t B = D A + B + C + D  360 8q;1kj ’G .oI6[lujc90hk’0t|ko ABCD F A 115 D 9qj’-vdsk,5,mujpa’gs^nvD [qfcdhG A B 0=hlq,,5f ABCD c,joI6[lujc9 115 0hk’0t|kF A  115 D C 0=hlts^5[ -vdskG B, C c]t D 9kd0=hlq,5f gIqk,u A + D  180 D = B  65 A = C  115 D  180  A  180 115  65 #D [qfg/ydsaf ສະຫງສ.ວວນ.ລິສຂະສິດ !D A, B, C c]t D c,oj ,5,.o0v’I6[lucj 90hk’0t|ko ABCD D 9q’j 7yfw]j c]h;8n,j .l8j k8t]k’D A B C D 120 85 140 @D .sh x; 2x; 3x c]t 4x c,jo7kj ;afcmd,5,.o0v’I6[lujc9l;f ABCD |bj’D 9qj’-vdsk7jk;afcmdc8j]t,5,.o0v’I6[lucj 9fa’j djk;D #D .oI6[lujc90hk’0t|ko EFGH. lv’,5,mu1j 6j8yfcxtdao s^5f]njodao 30 D 9q’j -vdsk7jk;afcmd0v’c8j]t,5,.o0v’I6[lucj 90hk’0t|kofaj’djk;D $D I6[lujc90hk’0t|ko ROSE 3fp;jk RES = 46o c]t SRO  34 D 9q’j 7yfw]j RSO D &D dD ABZY c,joI6[lucj 9 _B A_ 0D 9kdI6[.o0=h Xd? 0hk’0t|koD AY = BX 4hk;jk ZY = BX c]t 9qj’-Uc9’;jk 1  2  3 1_ _2 _3 1  2 9q’j -Uc9’;kj _X Z_ _Y I6[ljcu 9 ABZY c,oj I6[fvd9aoD 165

*D 9q’j 7yfw]j,5,.o0v’I6[lucj 9D I6h;jkr;d,aogxaovaf8klj;orq;raoda[ 2; 3; 6 c]t 7. (D .oI6[lujc90hk’0t|ko EFGH D HG dD EFG  70 c]h; HEF  ? 0D 4hk HFG  32 F EHF  ? 7D 4hk EHG  2x  5, HGF  5x F x  ? E F )D 9q’j -vdsk7kj 0v’ x c]t y .oc8]j td=]tou]5j,oUD (7 y  6) (4x  6) 3y (4x 15) 48 9x (1) 3y (2) 80 25 y x x (4) 62 60 130 (3) ສະຫງສ.ວວນ.ລິສຂະສິດ _D .shlugj ,af X , P, Q, Y 16gj lAo-nj|bj’3fp;kj PX = QY Dc8h,I6[lujc90hk’0t|ko PQRS F PQR  130 c]t QR = QY D glAo-nj (XS) c]t (YR) 8afdao16j Z D 9q’j 7yfw]j7jk0v’,5, XZY D X !WD .shI6[fvd9ao ABCD c]t I6[lk,c9ltg\\u ABX B Xfaj’I6[?D 4hk;kj BCD  82 D A 82 C !!D 9qj’7yfw]j ADX c]t DBX D  60  D gryoj .oI6[lujc9 ABCD, DAB lj;o;jklk,,5,vnoj ,u7jk;afcmdgmqkj daoD 0ufglAo-nj (CE) 0t|ko (BA) 8af AD 16jg,af E D 9qj’7yfw]j ABC c]t ECD. !@D la’gdfI6[F 7jk;afcmd0v’,5, x c,jo0=hG a) a  b  c b) 360  (a  b  c) b c) a  b  c 180 d ) 360  a  b  c a c x 166

[qfmu #) ;q’,qo c]t czoj ,qo !D dyf9tde dyf9temu ! A .sh A xyoj vhv,g,af O wfhIv[|bj’3fpIadlkw]pt OA 7ngdqjkF g,af A g7njvomuj8k,glAo|bj’D { glAooAo gvUo;kj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD O { g,af O gvUo;jk DDDDDDDDDDDDDDDDDDDDDDDDDD 0v’;q’,qoD { w]pt O sk A gvUo;jk DDDDDDDDDDDDDDDDDDDDDDD 0v’;q’,qoD { m5dMg,af0v’;q’,qo]h;oc8jskj ’ltg|u9kdg,af DDDDDDDDDDDDDDDDDDD O 0v’;q’,qoD { |hkrP’muj16.j o;q’,qo oa[ma’glAo0v’;q’,qo gvUo;kj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD { 4hkgIqkra[czjo,qo8k,glAozkj o.9dk’0v’,aoF gIqk9twfh DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD g8a’daor=fuS glAomuwj fh 9kdIvpra[oAo gvUo;jk DDDDDDDDDDDDDDDDDDDDDDDDDD0v’czjo,qoD ສະຫງສ.ວວນ.ລິສຂະສິດ s^n 0v’ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD dyf9tdemu @ EF la’gdfI6[D A { 9qj’0ufglAo-njzkj og,af C c]t D D T glAo-nj (CD) gvUo;jk DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ;q’,qoD D O G { 9q’j 0uf8+lv’g,af F c]t G F mvj o-nj FG gvUo;kj DDDDDDDDDDDD B { 9qj’0ufglAo-nj d 8yfda[;q’,qo16gj ,af T F glAo-nj d gvUo;kj DDDDDDDDDDDDDD C { lv’g,af C c]t D c[’j ;q’,qovvdgxaolv’lj;oF c8j]trkdlj;ogvUo;kj DDDDDDD dyf9tdemu # oadIPoz6h|b’j ;afcmdglAozjko.9dk’0v’dq’]qf4u[wfh 70cm c]h;dyh’dq’]qf wx8k, mk’rP’wfh|b’j Iv[r=fuD ]k;;afcmdw]ptmk’mudj q’]qfwxwfhc,jo 2,2mD l;j ooadIPovudz6h|bj’ wfh;afcmdglAozjko.9dk’ c]t ];’Iv[0v’ xkd9vd cdh; |;j p|bj’’wfh 8,8cm c]t 27,6cm8k,]efa[D { 9qj’0Po8nj,.l8j k8t]k’]5,j oUD 167

oadIPo ];’Iv[ glAozjko.9dk’ ];’Iv[ ÷ glAozkj o.9dk’ ! @ ]; 'Iv[ gloA zjko.9dk' { 4hkdeoqf.sh π = g-y’j ;kj π 3,14 9twfh ];’Iv[0v’;q’,qo gmkjq DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD dyf9tdemu $D I6[;q’,qo I6[lujc90hk’0t|ko h=r b =πr { goNvmuj0v’czoj ,qogmqjkgoNvmuj0v’I6[ DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ສະຫງສ.ວວນ.ລິສຂະສິດ fa’j oAoF goNvmuj0v’czoj ,qogmqkj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD @D .97;k, !D ;q’,qo dD oypk,G 4hkg,af A xyjovhv,g,af O 3fpIadlkw]ptskj ’ltg\\u9kdg,af O F g,af A g7nvj omuj8k,glAo|b’j ’gvUo;kj ;q’,qoD { g,af O gvUo;jkg,af.9dk’D Or A { w]ptc8j O sk A gvUo;jk]aflt\\uF laopt]adfh;p8q;vadlvo r F r = OA D 0D 75o]adltotD m5dMg,af0v’;q’,qo]h;oc8sj kj ’lt\\udao 9kdg,af.9dk’0v’r;d,aoD 7D glAo8kj ’M0v’;q’,qoD { glAo-njmuzj jkog,af.9dk’0v’;q’,qo gvUo;kj G glAozjko.9dk’F laopt]adfh;p 8q;vadlvo d. d = 2R = ABD { mjvo-nmj u8j +]ts;kj ’lv’g,af0v’;q’,qo gvUo;kj G glAogoaj’mvj odqj’D [FG] c,joglAogoaj’mjvodq’j D 168

{ glAo-njmuj,ug,afIj;,da[;q’,qolv’g,af d T gvUo;kj G glAo8af;q’,qoD (DC) c,oj F B glAo8af;q’,qoD d C { glAo-njmuj,ug,afIj;,da[;q’,qorP’g,affP; EO gvUo;jkG glAo8yf;q’,qoDglAo-nj d c,joglAo 8yf;q’,qoD A D { lv’g,af C c]t D c[’j ;q’,qovvdgxaolv’lj;oF c8j]trkdlj;ogvUo;jkG mjvodqj’F laopt]adfh;p CDD vjko;jk < mjvodq’j CD<D • ly’j 7;ogvqk.9.lDj { glAozjko.9dk’c,joglAogoa’j mvj odq’ mupj k;l5f0v’;q’,qoD { glAo8yf8A’lkdda[]aflt\\u0v’;q’,qoD @D czjo,qo |hkrP’mu1j 6j.o;q’,qo oa[ma’glAo0v’;q’,qooAo gvUo;jkG czjo,qoD C { g,af A c]t D 16.j oczjo,qo phvo;kj OA = OD = r B { g,af B 16j.oczoj ,qo phvo;jk OB < r ສະຫງສ.ວວນ.ລິສຂະສິດ { g,af C 16oj vdczoj ,qo phvo;jk OC > r O rA r #D ];’Iv[0v’;q’,qo s^n czjo,qo D .sh C c,oj ];’Iv[F r c,oj ]aflt\\u c]t d c,oj glAozjko.9dk’ gIqkwfGh C = 2× r × π = d×π dr $D goNvmu0j v’czoj ,qo O .sh A c,oj goNvmujF gIqkwfh A = πr2 = πd2 4 8q;1jk’G 7yfw]]j ;’Iv[ c]t goNvmu0j v’;q’,qomu,j uglAozkj o.9dk’ 15cm. cdhG i C = 3,14×15cm i A = 3,14×15cm ×15cm 4 = 47,1cm. = 176, 625 cm2. &D cdog7y’j 7n c]t l6odk’g7y’j 7n0v’;q’,qo c]t czjo,qo { ;q’,qo c]t czoj ,qo,us^kpcdog7y’j 7n g-yj’c,oj glAo zjko.9dk’0v’,aogv’D { ,u|bj’l6odk’g7yj’7n g-yj’c,oj g,af.9dk’0v’,aogv’D 169

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{ oadIPo7yf;kj dkoxPj o[jvo 0v’8q;xtlko .o76j9eo;o9tgIaf.sh7;k,\\kpxPj o s^n [+L dyf9tdemu @ 4- A 9kdI6[F 4hkoa[9kdg,af O 8k,glAog7Qkovo -3 wxmk’0;k,n # sq;|;j p c]t oa[8k,glAo 2 -3 -4 g7Qk8A’0Bowx @ sq;\\j;pd=9twfh76j9eo;o 2 X#S@? g-yj’gxaog,affP;daoda[g,af AD ltoAo g,af A dq’da[76j9eo;o X#S@?D 1 -Œ4 - 3 - 2 Œ1 0 1 Œ1 -2 -3 { 9q’j -vdsk769j eo;omujdq’da[g,af Œ- 4 O; B; C; D c]t E. { 9q’j |kpg,af P; Q; R c]t S 4- B mujdq’da[769j eo;o X@S!?S X!S@?S A XWF!? c]t X!FW? 8k,]efa[D -3E { g,afdq’da[76j9eo;o X#S!? C c]t X!S#? 16gj ,affP;dao s^n [+L 2 D 1 2 3- 4- - 3 - 2 Œ1 0 1 ສະຫງສ.ວວນ.ລິສຂະສິດ { 769j eo;o (x;y) mudj q’da[g,af -1 M gvUo;kj 8q;xtlko0v’g,af -2 M laopt]adfh;p M(x;y). -Œ3 dyf9tdemu # .oczjorP’mu,j u]t[q[glAog7QklkdF .sh8q;xtlkolk,9v,0v’I6[lujc9lkd ABCD |b’j c,jo A(1;1); B(1;3) c]t C(6;1) D 9q’j -vdsk8q;xtlko0v’9v,muluj D D dyf9tdemu $ .shI6[lk,c9 ABC mu,j u8q;xtlko A(−5;2); B(−3;5) c]t C(−2;1) fa’j I6[]5j,oUG dD g,af A '; B' c]t C' c,jog’qk0v’g,af A; B c]t C mP[.lcj do y D faj’oAo 8q;xtlko0v’g,af A ' c,jo A '(....;....) 8q;xtlko0v’g,af B' c,oj B'(....;....) 8q;xtlko0v’g,af C' c,oj C'(....;....) 172

g’qk0v’I6[lk,c9 ABC mP[.ljcdo y c,oj I6[lk,c9 DDDDDDDDDDDDDDDDDD y 6 B 4 A C2 2 4 6x - -4 -2 0 -2 -4 -6 0D g,af A ''; B'' c]t C'' c,oj g’qk0v’g,af A; B c]t C mP[.ljg,af O D faj’oAoສະຫງສ.ວວນ.ລິສຂະສິດ 8q;xtlko0v’g,af A '' c,oj A ''(....;....) 8q;xtlko0v’g,af B'' c,jo B''(....;....) 8q;xtlko0v’g,af C '' c,oj C''(....;....) g’qk0v’I6[lk,c9 ABC mP[.lcj do y c,joI6[lk,c9 DDDDDDDDDDDDDDDDDD 7D g,af A '''; B''' c]t C''' c,oj g’qk0v’g,af A; B c]t C mP[.ljcdo x D fa’j oAo 8q;xtlko0v’g,af A ''' c,oj A '''(....;....) 8q;xtlko0v’g,af B''' c,jo B'''(....;....) 8q;xtlko0v’g,af C ''' c,oj C'''(....;....) g’qk0v’I6[lk,c9 ABC mP[.lcj do y c,oj I6[lk,c9 DDDDDDDDDDDDDDDDDD @D .97;k, !D ]t[q[glAog7Qksq;\\j;p8A’lkd 16|j hkrP’ gIqkc8h,lv’cdo (xx’) c]t (yy’) 8afdao c]t 8A’lkddao16jg,af O. { g,af O gvUo;jkG “g,afg7Qk” { cdo9eo;o (xx’) gvUo;kj G <cdog7Qkovo s^n cdova[-yf< { cdo9eo;o (yy’) gvUo;kj G <cdog7Qk8A’ s^n cdovvdfvogo<D 173

-y -3 2 -1 x x′ 1 2 -3 - 3 - 2 Œ1 0 Œ1 -2 -3 - y′ @D dkodeoqfg,af|b’j gmy’czjorP’mu,j u]t[q[glAog7Qk ສະຫງສ.ວວນ.ລິສຂະສິດ dD dko-vdsk8q;xtlko0v’g,af|b’j -y .oczoj rP’mu,j u]t[q[glAog7Qkmu.j shdvj o gIqklk,kf-vd8q;xtlko0v’g,af A -3 wfhfaj’oUG A2 1 9kdg,af A gIqk0ufglAo-nj8A’lkdda[lv’ - 3 - 2 Œ1 0 Œ1 1 2 3- - x glAog7QkF [vj oglAo-nj8afda[glAog7Qkma’lv’ .shI6h8q;xtlko0v’g,af A D - 2 8q;1jk’G 8k,I6[gIqkwfh8q;xtlko0v’g,af -3 P A c,jo X{@S@? laopt]ad A(-2;2) D 0D dko-vdskmuj8A’0v’g,af|bj’g,njvI6h8q;xtlko0v’,aoD gIqklk,kfI6hmu8j A’0v’g,af P(x;y) wfhfaj’oUG { 9kd x c]t y 0uflv’glAo-nj8A’lkdda[glAog7Qk r;d,ao8afdao16gj ,af|b’j g-y’j c,jo g,af P mu8j hv’dkoD 174

8q;1jk’G -vdskmu8j A’0v’’g,af P(2;-3)D ສະຫງສ.ວວນ.ລິສຂະສິດ { 0ufglAo-n8j A’lkdda[glAog7Qkovo16gj ,afmuj,uva[-yfgmqjk @D { 0ufglAo-nj8A’lkdda[glAog7Qk8A’16jg,afmuj,uvvdfvogogmqjk −3 D { lv’glAo-nfj aj’dkj ;8afdao1j6g,af P Xg[y’j I6[0hk’gmy’?D 7D dko-vdsk8q;xtlko0v’g,afg7y’j 7nmP[.lcj doD .shg,af P(x;y) .oczoj rP’mu,j u]t[q[glAog7QkD { gIqklk,kf-vdg,afg7y’j 7n M 0v’g,af P mP[.lcj do y wfhfa’j oUG 9kdg,af P 0ufglAo-n8j A’lkdda[cdo y F 95f8afdao0v’glAo-njfaj’dkj ;da[cdo y c,jog,afg7y’j dk’0v’mvj o-nj [PM]D faj’oAo 8q;xtlko0v’g,af M c,oj (−x; y) c]t 0Po M(−x; y). { gIqklk,kf-vdg,afg7y’j 7n N 0v’g,af P mP[.ljcdo x wfhfaj’oUG 9kdg,af P 0ufglAo-n8j A’lkdda[cdo x F 95f8afdao0v’glAo-nfj a’j djk;da[cdo x c,jog,afg7y’j dk’0v’mvj o-nj [PN] faj’oAo 8q;xtlko0v’g,af N c,oj (x;−y) c]t 0Po N(x;−y). { gIqklk,kf-vdg,afg7y’j 7n Q 0v’g,af P mP[.ljg,afg7Qk O wfhfaj’oUG 9kdg,af P 0ufglAo-njzjkog,af O F g,af O c,jog,afg7yj’dk’0v’mvj o-nj [PQ] fa’j oAo 8q;xtlko0v’g,af Q c,jo (−x;−y) c]t 0Po Q(−x;−y). #D [qfg/ydsaf !D 9kd8k8t]k’8+wxoU 9qj’0Po769j eo;o3fp.sh8q;xtlkomu|b’j ltcf’9eo;o.oc4; mu|bj’ c]t 8q;xtlkomulv’ ltcf’9eo;o.oc4;mulv’D 9kdoAo.shc8h,769j e o;o]q’.o]t[q[glAog7Qk8A’lkdD c4;mu|bj’ ! @ # $ & * ( ) c4;mulv’ @ $ * ) !W !@ !$ !* @D 8k8t]k’8+wxoU ltcf’9eo;o7qog0Qk-q,s=ryryfmtraocs’j |bj’ .og;]k !W okmuF 3fp9qf9eo;o7qom5dokmuD 9q’j 0Po769j eo;o c]t c8h,]q’.o]t[q[glAog7Qk8A’lkdD 175

okmumu ! @ # $ & * ( ) _ !W 9eo;o7qo !@ ) ( !W !& !( * ) !W !& #D 9qj’-vdsk8q;xtlko0v’g,af A; B; C; D; E; F; G c]t H 9kdI6[mujde oqf.shD y 10 C 8A 6 4B D2 -10 -8 -6 -4 -2 02 4 6 8 10 x -2 E -4 F -6 G -8 H ສະຫງສ.ວວນ.ລິສຂະສິດ -10 $D 9q’j -vdmu8j A’[aofkg,affa’j ]5,j oU .o]t[q[glAog7Qk M(4;0); N(-2;0); P(0;3); G(0;0); R(0;-4). &D 9q’j c8h,I6[lk,c9 ABC .o]t[q[glAog7Qklkd muj,u9v,c,oj A(4;-2); B(-2;2); C(3;4) (1) y 6 *D 9qj’-vdsk8q;xtlko0v’g,af 4 P; Q; R; S; M; N c]t T 16gj lAoltcf’mujgxaoglAo-n.j o I6[ X!? c]t X@? 0hk’oUG P2 -6 -4 -2 0 x 24 6 Q -2 -4 R -6 176

(2) Y T 15 -15 -10 S N 10 M 5 -5 0 5 10 15 20 x -5 -10 ສະຫງສ.ວວນ.ລິສຂະສິດ (D 8q;xtlkolk,9v,0v’I6[9t85]af|bj’c,jo A(−2;3); B(−2;−1) c]t D(2;3) D 9qj’-vdsk8q;xtlko0v’9v,muluj C 0v’I6[9t85]affaj’djk;D )D 8q;xtlkolk,9v,0v’I6[lujc90hk’0t|ko ABCD c,oj A(0;0); B(−6;0) c]t C(−8; −3) D 9q’j -vdsk8q;xtlko0v’9v, D D _D 8q;xtlkolv’9v,0v’I6[9t85]af|bj’c,jo (0;a) c]t (0;−a) -bj’ a ≠ 0 D 9q’j -vdsk8q;xtlko0v’lv’9v,mujgs^nvD !WD 8q;xtlkolk,9v,0v’I6[9t85]af|bj’c,jo A(−2;3); B(−2;−1) c]t D(2;3) D 9qj’-vdsk8q;xtlko0v’9v,muluj C 0v’I6[9t85]affaj’djk;D !!D .oczoj rP’mu,j u]t[q[glAog7QkF .shI6[lk,c9 ABC ,u8q;xtlkoc,jo A(2; −5); B(5; −4) c]t C(1; −2) D dD 9qj’-vdsk8q;xtlko0v’I6[lk,c9 A 'B'C' mugj 7y’j 7nda[I6[lk,c9 ABC mP[.lgj ,afg7QkD 0D 9q’j -vdsk8q;xtlko0v’I6[lk,c9 A ''B''C'' mujg7y’j 7nda[I6[lk,c9 ABC mP[.lcj do x D 177

7D 9q’j -vdsk8q;xtlko0v’I6[lk,c9 A '''B'''C''' mujg7yj’7nda[I6[lk,c9 ABC mP[.lcj do y D ’D 9q’-vdsk8q;xtlko0v’g,af D grnjv.sh ABCD gxaoI6[lucj 90hk’0t|koD !@D ]qf7ao|bj’c]oj fh;p7;k,w; @ dy3]c,af8+okmuD g;]k Xokmu? 0 1 2345 6 61 71 2 2 8 w]ptmk’ X km ? 0 2 4 dD 9qj’8n,j 9eo;omugj \\ktlq,.l8j k8t]k’D 0D 9q’j 0Po76j9eo;omujdq’da[8k8t]k’D 7D 9qj’c8h,glAoltcf’dkog7njvomu0j v’]qffaj’djk;D !#D dko[vdv5ootr6,.os^kpxtgmfoypq,[vd3fp.-hsq;|j;pgxaovq’lkc-ol5f (DC) F c8j,u[k’xtgmf[vdv5ootr6,3fp.-hvq’lk/kgIaoIkp (DF) D 7;k,lerao0v’v5o otr6,ma’lv’sq;|j;pltcf’wfhfh;plq,zqo ສະຫງສ.ວວນ.ລິສຂະສິດ F = 9 C + 32 5 g,njv C cmov5ootr6,sq;|;j pgxaovq’lkc-ol5f F cmov5ootr6,sq;|j;pgxaovq’lk/kgIaoIkp C -30 -20 -10 0 10 20 30 F dD 9qj’8n,j 9eo;omujg\\ktlq,.l8j k8t]k’0hk’gmy’ 0D 9qj’0Po769j eo;omujdq’da[8k8t]k’ 7D 9q’j c8h,769j eo;omudj q’da[8k8t]k’.o]t[q[glAog7Qksq;|;j p8A’lkdD 178

[qfmu $W dkogxaovaf8klj;orq;raodq’ !D dyf9tde dyf9tdemu ! 38ho-y’mu8j kj ’dao 4ndshvpw;hlQoltxy’F dko1nfvvd0v’ltxy’,uvaoc8d8jk’dao c]t zqoIa[wfh[aombdw;h.o8k8t]k’0hk’]5j,oUG 38ho-y’ (kg) ! @ #$& * ( ) !@ !* @W @$ @) dko1bfvvd (mm) $ dD lq,,5fgIqkc0;o38ho-y’mu|j ad x (kg) gIaf.shltxy’1nfvvd y(mm) Dສະຫງສ.ວວນ.ລິສຂະສິດ 9q’j 8nj,.l[j vj omu9j eg,af c]t 8k8t]k’.sh7q[4h;o 4hk x = 1 c]h; y = 1× 4 = 4 4hk x = 2 c]h; y = 2×...= .... 4hk x = 3 c]h; y = 3×...= .... 4hk x = 5 c]h; y = 5×...= .... 4hk x = 6 c]h; y = 6×...= .... x ! @ #$& * ( y $ ) !@ !* @W @$ @) y 4 = 4 x 1 dkorq;rao]ts;kj ’ x c]t y lk,kfltcf’fh;plq,zqo y = ......... dkorq;rao]ts;kj ’ 38ho-y’ c]t dko1nfvvd0v’ltxy’c[[oUgvUo;jkG dkogxaovaf8kl;j orq;raodq’D 0D 9qj’c8h,glAoltcf’dkorq;rao]ts;kj ’ x c]t y .o]t[q[glAog7Qk8A’lkdD dyf9tdemu @ .shlq,zqo y = 5x ltcf’g4y’dkogxaovaf8klj;orq;rqodq’]ts;kj ’ y c]t x D dD 9q’j 8n,j .l8j k8t]k’]5,j oU.sh7q[4h;o 179

x …. -4 -3 -2 -1 0 1 3 4 …. y …. -20 -15 0 5 10 15 …. 0D 9qj’c8h,glAoltcf’dkorq;rao]ts;jk’ x c]t y .o]t[q[glAog7Qk8A’lkdD la’gdfgsaoco;.fdjP;da[glAoltcf’L dyf9tdemu # 4hk a gxaovaf8kl;j orq;raodq’da[ 1 c]t a =5 g,njv b = 3D 9qj’0Polq,zqoltcf’ b vaf8klj;orq;raodq’D @D .97;k, oypk,G .sh x c]t y cmoxt]y,ko.f|bj’F y gxaovaf8klj;orq;raodq’da[ x g,njv y = kx g-y’j ;kj k gxao9eo;o7q’7kj c]t k ≠ 0. lq,zqo y = kx g-yj’;kj k gxao9eo;o7q’7kj c]t k ≠ 0 gvUo;jkG lq,zqoltcf’g4y’dkogxaovaf8kl;j orq;raodq’F 9eo;o7q’7kj k gvUo;jk lextlyf0v’vaf8klj;orq;raodq’D ສະຫງສ.ວວນ.ລິສຂະສິດ 9kdlq,zqo y = kx, k ≠ 0 c]t x ≠ 0 lk,kf0Powfhfa’j oU y = k D x faj’oAo gIqklk,kfg;Qkwfhvudc[[|bj’;jkG y c]t x gxaovaf8klj;orq;raodq’ g,nvj y gxao9eo;o7q’7jkle]a[m5dM7jk0v’ x c]t y g-y’j x ≠ 0 c]t 9eo;o7q’7jk x oUgvUo;kj lextlyf0v’vaf8kl;j orq;raodq’D ltoAo .odkory9k]tok;kj G xt]y,ko y gxaovaf8klj;orq;raodq’da[xt]y,ko x s^n[+L gIqkvkfry9k]tok9kd7kj 0v’ y le]a[ m5dM7jk0v’ x c]t y g-yj’ x≠0 x ;kj gxao9eo;o7q’7jk8q;fP;dao s^n[+L 4hkgxao9eo;o7q’7jk8q;fP;daoc]h;dkorq; rao]ts;jk’lv’xt]y,kooAogxaovaf8klj;orq;raodq’F c8j4hk,u x c]t y [k’76gj -y’j 7kj 0v’ y 8kj ’9kd x c]t y 76jvnoj c]h; dkorq;rao]ts;kj ’lv’xt]y x ,kooAo[+gxaovaf8kl;j orq;raodq’D 8q;1kj ’ 1. 8k8t]k’]5j,oUltcf’g4y’dkorq;rao]ts;kj ’xt]y,kooE,aocvaf-a’ c]t 9eo;o 180

g’yomu9j kj pD xt]y,kooE,ao X]yf? 0 1 2 3 4 5 …. 9eo;og’yomu9j kj p Xraodu[? 0 7 14 21 28 35 …. dD 9q’j 0Pol6fdkorq;rao]ts;kj ’ 9eo;og’yomu9j jkp c]t xt]y,kooE,aomuj-ND 0D 9qj’c8h,glAoltcf’dkorq;rao]ts;kj ’ 9eo;og’yomu9j jkp c]t xt]y,ko oE,aomu-j nhD [qfcdhG dD .sh x cmoxt]y,kooE,ao X]yf?D .sh y cmo9eo;og’yomu9j kj p Xraodu[?D 9kd y = kx cmo x = 1 c]t y = 7 9twfh k = 7 faj’oAo 9b’j lk,kf0Podkorq;rao]ts;kj ’ x c]t y wfh7nG y = 7x D 9tgsao;kj g,nvj xt]y,kooE,ao x gruj,0No 9eo;og’yomu9j jkp y d=gruj,0No8k,vaf8k 9eo;o7q’7kj (WWW du[8+]yfF 0D 9kd8k8t]k’F gIqklk,kfc8h,glAoltcf’dkorq;rao]ts;jk’xt]y,kooE,ao da[ 9eo;og’yomu8j hv’9jkpfaj’]5j,oUG ສະຫງສ.ວວນ.ລິສຂະສິດ y 9eo;og’yomuj9jkp Xraodu[? 35 28 21 14 7 0 1 2 3 45 x xt]y,kooE,aomuj-nh Xgxao]yf? 181

8q;1jk’ 2. .odkomqf]v’xa[v5ostr6,0v’shv’7;[75,v5ostr6,cs’j |bj’ lk,kfxa[v5ootr6, 0v’shv’7;[75,wfh s^5f]q’g]NvpM.ovaf8k7q’7jk9kd 0 D C 8k,g;]kmuzj kj owx9kd 0 g4y’ 5 okmu wfhzqodkomqf]v’fa’j 8k8t]k’]5j,oUG g;]k Xokmu?? 0 12345 v5ootr6, (D C) 0 -2 -4 -6 -8 -10 dD 9q’j 0Pol6fdkorq;rao]ts;kj ’ v5ootr6, c]t g;]kzkj owxD 0D 9qj’c8h,glAoltcf’dkorq;rao]ts;kj ’ v5ootr6, c]t g;]kzjkowxD [qfcdhG dD .sh x cmo g;]kmujzkj owxgxaookmu .sh y cmo v5ootr6,0v’shv’7;[75,gxao D C 9kd y = kx cmo x = 1 c]t y = −2 9twfh k = −2 fa’j oAo lk,kf0Podkorq;rao]ts;jk’ x c]t y wfh 7nG y = −2x D 0D 9kd8k8t]k’F gIqklk,kfc8h,glAoltcf’dkorq;rao]ts;jk’v5ostr6,da[ g;]k faj’]5,j oUG ສະຫງສ.ວວນ.ລິສຂະສິດ v5ostr6, XDc ? x g;]kzjkowx Xokmu? y 0 12345 -2 -4 -6 -8 -10 182

#D [qfg/ydsaf !D 9kd8k8t]k’]5,j oU 8k8t]k’.fltcf’;jk y gxaovaf8kl;j orq;raodq’da[ x F 4hkgxao .sh-vdsklextlyf0v’vaf8kl;j orq;raodq’ c]t 0Polq,zqoltcf’ g4y’vaf8klj;orq;raodq’oAoD a). b). x2 34 5 y 10 15 20 25 x 1 -1 2 -3 y -1 3 -3 7 c). 2 34 5 d). -4 -6 -8 -10 x x 1 -1 2 -3 y y -3 3 -6 9 @D 9q’j c8h,glAoltcf’vaf8kl;j orq;raodq’0v’ x c]t y 8+wxoUG dD y = −3x 3fpzjkog,afg7Qk c]t g,af (1;−3) ສະຫງສ.ວວນ.ລິສຂະສິດ 0D y = 4 x 3fpzjkog,afg7Qk c]t g,af (3; 4) 3 #D 4hk y gxaovaf8kl;j orq;raodq’da[ x c]t y =12 g,njv x = 3 D 9q’j -vdsklextlyfvao8kl;j orq;raodq’D $D 4hk y gxaovaf8kl;j orq;raodq’da[ 3x + 2 c]t y = 4 g,njv x = 2D 9q’j -vdsk y g,nvj x = 5 D &D 4hk s gxaovaf8klj;orq;raodq’da[ t2 c]t s = 15 g,nvj t = 5D 9q’j -vdsk s g,njv t =10 D *D lq,,5f7qoz6h|bj’pkj ’fh;p7;k,w; 4km 8+-q;j 3,’ c89j 5f A g4y’95f B g-y’j ,uw] ptmk’ 12 km D 9qj’0Pol6fmulj tcf’dkorq;rao]ts;jk’ x(h) mu7j qoz6hoAopjk’ c]t w]ptmk’ y(km) muwj xwfh rhv,da[c8h,glAoltcf’dkorq;raoD (D xjvpoE.ovaf8k 2 ]yf8+okmu]q’.l4j a’muj[ao95oEwfh 18 ]yfD 9qj’oe.-hl6f c]t glAolt cf’grnjvvtmy[kpdkorq;rao]ts;kj ’ x g-y’j gxaog;]k Xokmu? mujoExvj p]q’ c]t y g-y’j gxaoxt]y,ko0v’oE X]yf? g0Qk.o4a’D 183

)D ,uoadIPolv’7qoF z6h|b’j 0u]j qf4u[ c]t vudzh6|bj’0uj]qf9ad9kd3I’IPogrnvj da[[hkoDສະຫງສ.ວວນ.ລິສຂະສິດ z6h0u]j qf4u[g]u,j vvdgfuomk’g;]k !$GWW fh;p7;k,w;lt\\jeltg\\u !W km/hF lj;oz6hmuj0u]j qf9adg]uj,vvdg;]k !&GWW fh;p7;k,w; @W km/hF I6h;jk w]ptmk’c8[j hko sk3I’IPoc,oj $W km. 3fpvu’.ljglAoltcf’G dD g;]k !*G #W z6hmuj0uj]qf4u[wxwfh9addy3]g,afL 0D g;]k !*G #W z6hmu0j u]j qf9adwxwfh9addy3]g,afL 7D z6hmu0j uj]qf9adIvf[hkog;]k9ad3,’L ’D z6hmu0j u]j qf4u[Ivf[hkog;]k9ad3,’L 9D z6hmu0j u]j qf9adwxmaoz6hmu0j uj]qf4u[g;]k9ad3,’ c]t skj ’9kd3I’IPo9addy3]c,afL w]ptmk’’ (km) 40 30 20 10 14:00 15:00 16:00 17:00 18:00 g;]k 184

[qfmu $! vaf8kl;j orq;raoxUo !D dyf9tde dyf9tdemu ! lq,,5f;jkgryoj .-hg;]k y -qj;3,’ .odkogfuomk’ 240km fh;p7;k,w; x km / h fa’j ltcf’.o8k8t]k’]5,j oUG x 40 60 80 100 120 y 6 4 3 12/5 2 dD 9q’j 8n,j .lj[vj o9Eg,afc]t8k8t]k’.sh7q[4h;o ສະຫງສ.ວວນ.ລິສຂະສິດ 4hk x = 40 c]h; y = 240 / 40 = 6 4hk x = 60 c]h; y = .... 4hk x = 80 c]h; y = .... 4hk x = 100 c]h; y = .... 4hk x = 120 c]h; y = .... x 40 60 80 100 120 y 6 4 3 12/5 2 y/x x×y dkorq;rao]ts;jk’ x c]t y lk,kfltcf’fh;plq,zqo 240 = ...... s^n y = ....... dkorq;rao]ts;jk’ 9eo;o-qj;3,’ c]t 7;k,w;c[[oUgvUo;kj G dkogxaovaf8klj;orq;raoxUoD 0D9qj’c8h,glAoltcf’dkorq;rao]ts;jk’ x c]t y .o]t[q[glAog7Qksq;|;j p 8A’lkdD dyf9tdemu @ .shlq,zqo y = − 2 ltcf’g4y’dkogxaovaf8kl;j orq;rqoxUo]ts;jk’ y c]t xD x 9qj’c8h,glAoltcf’dkorq;rao]ts;kj ’ x c]t y .o]t[q[glAog7QklkdD 185

dyf9tdemu # 4hk y gxaovaf8kl;j orq;raoxUoda[ x c]t y = 30 g,njv x = 4D 9qj’0Polq,zqoltcf’vaf8kl;j orq;raoxUoD @D .97;k, !D oypk,G .sh x c]t y gxaoxt]y,ko.f|b’j mu[j +gmqkj l6oF y gxaovaf8kl;j orq;rao xUoda[ x g,nvj y = k × 1 = k g-yj’;kj k gxao9eo;o7q’7kj c]t k ≠ 0. x x lq,zqo y = k × 1 = k g-y’j ;jk k gxao9eo;o7q’7kj c]t k≠0 gvUo;jkG x x lq,zqoltcf’g4y’dkogxaovaf8kl;j orq;raoxUoS 9eo;o7q’7jk k gvUo;jk lextlyf0v’vaf8kl;j orq;raoxUoD 9kdlq,zqo y= k , k≠0 c]t x≠0 lk,kf0PowfhG xy = k c]t k ≠ 0. x 8q;1jk’ 1. 9qj’c8h,glAoltcf’0v’lq,zqo y= 6 g-yj’ltcf’g4y’vaf8klj;o x ສະຫງສ.ວວນ.ລິສຂະສິດ rq;raoxUo]ts;jk’ y c]t x. [qfcdGh 9kdlq,zqo y = 6 s^n xy = 6 gIqkwfhlextlyf0v’vaf8klj;o x rq;raoxUo7n 6 c]t lk,kflhk’8k8t]k’wfhfaj’oUG x … -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 … y … -1 -1.2 -1,5 -2 -3 -6 × 6 3 2 1,5 1,2 1 … 4hkgIqkc8h,m5dMg,afmujdq’da[ (x, y) y pqdg;Aog,af (0;0) 9twfhglAo37h’g7y’j 7nmP[.lgj ,afg7Qkfa’j I6[G 6 3 6 x 1 -6 -3 -1 01 2 3 --23 -6 186

9kdI6[gsao;jk glAoltcf’[+zkj og,afg7Qk c]t [+gxaoglAo-njD 8q;1kj ’ @D 9q’j c8h,glAoltcf’0v’lq,zqo y = − 6 g-y’j ltcf’g4y’vaf8kl;j orq;raoxUo]t x s;kj ’ y c]t x. [qfcdhG 9kdlq,zqo y = − 6 s^n xy = −6 gIqkwfhlextlyf0v’vaf8kl;j orq;raoxUo7n −6 x c]t lk,kflhk’8k8t]k’wfhfaj’oUG x … -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 … y … 1 1,2 1,5 2 3 6 × -6 -3 -2 -1,5 -1,2 -1 … 4hkgIqkc8h,m5dMg,afmudj q’da[ (x; y) pqdg;Aog,af (0;0) 9twfhglAo37h’g7y’j 7nmP[ .ljg,afg7Qkfaj’I6[]5j,oUG y 66 ສະຫງສ.ວວນ.ລິສຂະສິດ 44 22 x 65 −6 -5 −2 2 −2-2 −4-4 8q;1kj ’ #D 6-6 4hk v gxaovaf8kl;j orq;raoxUoda[ t c]t v = 2 g,nvj t = 3D 9q’j 0Polq,zqoltcf’ vaf8klj;orq;raoxUo c]t c8h,glAoltcf’0v’,aoD [qfcdhG deoqf.sh v = k g-yj’ k c,oj lextlyf0v’vaf8kl;j orq;raoxUo t cmo v=3 c]t t=4 .llj q,zqo v= k 9twfh 3 = k t 4 k = 3× 4 = 12 fa’j oAo lq,zqomulj tcf’g4y’vaf8kl;j orq;raoxUo 7n v = 12 . t 187

v 6 4 23 -6 -4-3 -2 O 234 6 t --32 -4 -6 @D dkocdh39fdPj ;da[vaf8klj;orq;raoxUo 8q;1jk’D !D -jk’w,h z6h|b’j gIaf86h|;j p|bj’.-hg;]k # -q;j 3,’S okp-kj ’g/yd’kogIaf86h|j;p oAo9t.-hg;]k & -qj;3,’D 4hkma’lv’7qo-j;pdaogIaf 9t.-hg;]kgmqkj .fL ສະຫງສ.ວວນ.ລິສຂະສິດ [qfcdhG ;k’ x c,oj g;]kmuj.-h.odkogIaf86h X-qj;3,’? { okp-jk’.-hg;]kgIaf86h x -q;j 3,’ 9tc]h; x 0v’86hD 3 { okp-jk’g/yd’ko.-hg;]kgIaf86h x -qj;3,’ 9tc]h; x 0v’86hD 5 9kd[qfg]dgIqkwfhG x + x = 1 3 5 8x = 15 x = 15 -q;j 3,’ 8 d;f7no7e8v[G 15 { okp-jk’.-hg;]kgIaf86h x -q;j 3,’ 9tc]h; x 8 5 0v’86hD 3= 3 =8 15 { okp-kj ’g/yd’ko.-hg;]kgIaf86h x -qj;3,’ 9tc]h; x = 8 3 0v’86hD 5 5 =8 188

{ ma’lv’7qo.-hg;]k 15 -q;j 3,’ 9tc]h;86h 5 + 3 =1 |j;pD 8 8 8 @D g,njvw]ptmk’7q’muFj gIqkwfh 7;k,w; v gxaovaf8klj;oxUoda[g;]k t. 8es^;f9t ]k9vowfhoe.-hdkorq;raofa’j dkj ;.odko9a[7;k,w;0v’]qf.oglAomk’D le]a[w] ptmk’7q’muj|bj’ 8es^;fI6h;jk ]qfmucj ]oj fh;p7;k,w; 50 dy3]c,af8+-qj;w,’9t.-h g;]kc]oj .oglAomk’oAo * ;yokmuD 4hk]qf7ao|bj’.-hg;]k.odkoc]oj .oglAomk’ faj’djk;rP’c8j & ;yokmuD 4k,;kj ]qf7aooAoc]jofh;p7;k,w;gmqjk.fL [qfcdhG ;k’ v c,jo7;k,w;0v’]qf Xdy3]c,af8+-q;j 3,’? t c,oj g;]kmu.j -h X;yokmu? 9kd[qfg]d gIqklk,kf0Povaf8klj;orq;raoxUowfhfa’j oUG v = k -vdsk k t cmoj v = 50 c]t t = 6 = 1 h 9twfhG 3600 600 50 = k ສະຫງສ.ວວນ.ລິສຂະສິດ 1 600 k = 50 = 1 600 12 -vdsk v g,nvj t = 5s = 5 = 1 h 9twfhG 3600 720 1 v= 12 = 720 = 60 km / h 1 12 720 #D [qfg/ydsaf !D dD 9kd8k8t]k’]5j,oU 8k8t]k’.fltcf’;jk y gxaovaf8kl;j orq;raoxUoda[ x F 4hkgxao .sh-vdsklextlyf0v’vaf8kl;j orq;raoxUo c]t 0Polq,zqoltcf’g4y’ vaf8kl;j orq;raoxUooAoD 189

a). b). x 1 23 4 x 3 6 9 12 y 12 6 4 3 y 6 9 12 15 c). -2 -1 1/2 6 d). 2 4 -8 -2/3 x x 3 2 1 1/2 y y 1/2 3/4 3/2 3 0D 9kd0=h dD .od=]toumuj y gxaovaf8kl;j orq;raoxUoda[ x c]h; x gxaovaf8klj;o rq;raoxUoda[ y s^n[+L 4hk x gxaovaf8klj;orq;raoxUoda[ y F 9qj’-vdsklextlyfvaf8klj;orq;raoxUo c]t lq,zqoltcf’vaf8klj;orq;raoxUoD 7D oadIPo7yf;kj d=]toumq;j wx 4hk y gxaovaf8kl;j orq;raoxUoda[ x c]h; x gxaovaf 8kl;j orq;raoxUoda[ y s^n [+L phvospa’L ’D oadIPo7yf;jk 4hk y gxaovaf8kl;j orq;raoxUoda[ x c]h; lk,kfg;Qkwfh;jk y ສະຫງສ.ວວນ.ລິສຂະສິດ gxaovaf8kl;j orq;raodq’da[ 1 s^n [+L phvospa’L x @D 9qj’c8h,glAoltcf’vaf8klj;orq;raoxUo0v’ x c]t y 8+wxoUG dD y = − 3 3fpzjkog,af (1;−3) c]t g,af (−1; 3) x 0D y = 2 3fpzkj o (2;1) g,afg7Qk c]t g,af (−2; −1) x #D 4hk y gxaovaf8kl;j orq;raoxUoda[ x c]t y = 8 g,nvj x = 3 D 9qj’-vdG dD lq,zqoltcf’dkorq;rao]ts;kj ’ y da[ x D 0D y 9tgmqjkda[9adgmnvj 0v’xt]y,kogfu,g,njv x gmqkj da[lv’gmnjv0v’xt]y,kogfu,L 7D y 9tgrU,0No s^n s^5f]q’9adgxug-ao g,vjn x grU,0No 20%. $D de,tdvo !@ 7qo gIafmk’c]h;rkp.o !& vkmyfD 4hk,ude,tdvo #W 7qogIafmk’ glAofP;dao9tc]h;rkp.o9advkmyfL &D mhk; lk]ugfuomk’ 40 dy3]c,af .-hg;]kgmqkj da[ mhk;7efumu0j u]j qf4u[ 15 dy3]c,afD I6h;kj G mhk;7efu0u]j qf4u[-hkd;jk7;k,w;0v’mhk;lk]u 20 dy3]c,af8+-qj;3,’D 9qj’-vdsk7;k,w;0v’]qfc8]j t7aoD 190

[qfmu $@ g]dlj;oIhvp !D dyf9tde dyf9tdemu ! dD 9qj’7yfw]zj qosko]5j,oUG a. 8 ÷100 = 0, 08 b. 35 ÷100 = ...... c. 125 ÷100 = ...... d. 0,5 ÷100 = ...... e. 0, 02 ÷100 = ...... f . 0, 003 ÷100 = ...... g. 250, 6 ÷100 = ...... h. 61,3 ÷100 = ...... i. 7, 42 ÷100 = ...... j. 3 ÷100 = ...... 5 k. 0 ÷100 = ...... 0D 9q’j 0Pozqosko0hk’gmy’gxaog]dl;j o.sh,ur6fgmqkj !WW a. 0, 08 = 8 b. ............ c. ............ 100 ສະຫງສ.ວວນ.ລິສຂະສິດ d. ............ e. ............ f . ............ g. ............ h. ............ i. ............ j. ............ k. ............ g]dl;j omuj,ur6fgmqjkda[ !WW g-jao 8 ovd9kd9tvjko;jk “ ) lj;oIhvp ” c]h; 100 gIqkpa’lk,kfvkj owfh;jkG “ ) gxug-ao ” 0Pocmofh;p ):D : c,jolaopt]ad0v’ gxug-ao D 7D 9qj’0Pog]dl;j oIhvp.o0=h X0? gxaogxug-aoD a. 8 = 8% b. ............ c. ............ 100 d. ............ e. ............ f . ............ g. ............ h. ............ i. ............ j. ............ k. ............ 191

dyf9tdemu @ 9qj’0Po8nj,.l8j k8t]k’]5j,oU.sh7q[4h;o c]t g\\ktlq, gxug-ao g]dlj;oIhvp g]dlj;o *W: 60 3 100 5 !&: #F&: WF!@&: WF@: 3 % 4 dyf9tdemu # dkocdh39f[aoskdPj ;da[g]dgxug-ao X s^n g]dl;j oIhvp ? gIqk9trq[[aosk16j # d=]tou fa’j 8q;1jk’ 8+wxoUG ສະຫງສ.ວວນ.ລິສຂະສິດ d=]toumu !G #W: 0v’ *W gmqkj da[gmqkj .fL [qfcdhG .sh x gxao9eo;o #W: 0v’ *WD ltcf’;kj *W gxao9eo;oma’\\qfF x gxao9eo;o8k,gxug-ao 0Pogxaovaf8kl;j orq;rao9twfh x = 30 ⇒ x = .......... 60 100 d=]toumu @G _ gxao9adgxug-ao0v’ $&L [qfcdhG .sh _ gmqkj da[ x% 0v’ $&D ltcf’;kj _ gxao9eo;o8k,gxug-aoF $& gxao9eo;oma’\\qfF x c,oj gxug-ao 0Pogxaovaf8klj;orq;rao9twfh 9 = x gIqkwfhG x = .....% 45 100 d=]toumu #G ) gmqkj da[ @&: 0v’9eo;o.fL [qfcdhG .sh ) gmqkj da[ @&: 0v’ xD ltcf’;jk ) gxao9eo;o8k,gxug-aoF x gxao9eo;oma’\\qf 0Pogxaovaf8kl;j orq;rao9twfh 8 = 25 gIqkwfhG x = ......... x 100 192

@D .97;k, lj;oIhvp X s^n gxug-ao ? c,jog]dl;j o|bj’ g-y’j r6f0v’,aogmqjkda[ !WW g]NvpMD gryoj cmo7e;kj “ gxug-ao s^n l;j oIhvp ” fh;p : g-ajoG _ gxug-ao0Pocmofh;p _: ,u7;k,\\kp;kj 9 D 100 d=]toumq;j wxG a gxao9eo;o8k,.9 a% = a 100 0Pogxug-ao gxaog]dl;j oG 76ogxug-aoda[ 1 c]h;7af9hvo X 4hk7af9hvowfh ? 100 8q;1jk’ 25% = 25 × 1 = 25 = 1 D 100 100 4 0Pogxug-aogxao9eo;omqfltoypq,G skogxug-ao.sh !WW 8q;1jk’G 5% = 5 = 0, 05 100 0Po9eo;omqfltoypq,gxaogxug-aoG 76o9eo;omqfltoypq,oAoda[ 100 100 c]h;0PozqowfhIa[gxaogxug-aoD ສະຫງສ.ວວນ.ລິສຂະສິດ 8q;1jk’G 0,156 = 0,156 × 100 = 15, 6 = 15, 6% 100 100 0Pog]dl;j o a gxaogxug-aoG 76og]dl;j ooAo.sh !WW c]h;7af9hvo b X 4hk7af9hvowfh ? 9kdoAod=0Pogxaogxug-aoD 8q;1kj ’G 1 0Pogxaogxug-aogmqkj 1 ×100 = 25% 4 4 0=h7;ogvqk.9.lGj • 4hkg]dl;j oskd,ur6f c]t 9eo;or6fc,jo9eo;ofP;daoc]h;g]dlj;ooAo gmqkj da[ !WW:D 8q;1kj ’G 16 = 0, 7 = 100% 16 0, 7 • 4hkg]dl;j oskd,u9eo;or6f s^kpd;jkr6fc]h;g]dl;j ooAo9ts^kpd;kj !WW:D • 4hkg]dl;j oskd,u9eo;or6f |hvpd;kj r6fc]h;g]dl;j ooAo9t|hvpd;kj !WW:D gIqklk,kfoe.-h dkoskrqfmulu0j v’vaf8klj;orq;rqo g0Qk-;j p.odkocdh39f[ao skdPj ;da[gxug-aofa’j oUG 193

a c,jo9eo;o8k,gxug-ao b c,jo9eo;oma’\\qf 0Pogxaovaf8kl;j orq;raowfh a = m b 100 m c,oj gxug-ao X:? • 8q;1jk’dkooe.-hl;j oIhvp Xs^n gxug-ao ? dD oe.-hg0Qk.odko7yfw]jfvdg[hp X fvdg[hpc,oj dew]mujwfh9kd dko.shz6hvboj 1n,g’yo s^n gvqkg’yo/kdmtok7koD 9eo;og’yomu.j sh1n,gvUo;jkG 8Qombo ?D • 9q’j xtdv[8k8t]k’]5j,oU.sh7q[4h;o c]t g\\ktlq,D 8Qombo vaf8kfvd w]ptg;]k fvdg[hp8hv’9jkp 9eo;og’yo8hv’9jkp7no (c) g[hp8+xu d5h1n, ( t ) (i) (M) (r) &WWWWW du[ @W: @ xu &WWWWW ×12000 &WWWWW + @WWWWW = (WWWWW × 2 = 200000 @WWWWW du[ @&: & xu ສະຫງສ.ວວນ.ລິສຂະສິດ $&WWW du[ !*: # gfnvo !WWWWW du[ &: * gfnvo !WWWWW du[ !@: ! xu s^addko7yfw]Gj fvdg[hp gmqkj da[8Qombo 76oda[gxug-ao 76oda[g;]k \\kp7;k,;jkG i = c×r× t c]t M = c + i 0D oe.-hg0Qk.odko7yfw]]j k7k -toyflyo7hk ]k7k8yf gxug-aos^5f 9eo;og’yomujs^5fma’\\qf ]k7k8q;9y’ X s^n w;h ( p ) ]k7k ( r ) 3lh’0kpk; (d ) ]k7k? muj8hv’9jkp glNvc0opk; (&WW du[ &: lYow\\]k; ( sp ) &WWW du[ #: !WWWW du[ @F&: (&WW × 5 = #(& du[ (&WW - #(& 100 = (!@& du[ 194