ມ.2_ຄະນິດສາດ_Math

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[qfmu $# ,kf8kl;j o !D dyf9tde dyf9tdemu ! l;o8vo\\bj’gxaoI6[lujc9lkdF ];’d;hk’cmdwfh &W m c]t ];’pk;cmdwfh !&W m. gryoj wfh8afmk’zjkol;o8vofa’j dkj ;8k,];’pk; d;hk’ $m c]t 16sj jk’9kdIQ;g0Qk.ol;o #W m. gryjowfhc8h,czo;kfl;o8vooAo.ljg9Ap.[\\bj’ g-yj’.shI6h;jk I6[l;oohvpd;jk0t\\kfl;o8q;9y’ !WWW gmnvj D dD 9q’j -vdsk0t|kf0t|fI6[c8h,0v’l;o c]h;0Po8nj,.l8j k8t]k’]5j,oUG ];’pk; ];’d;hk’ ];’d;hk’0v’mk’ w]ptc8jmk’ skIQ; 0t\\kf8q;9y’ X m) !&W &W $ #W 0t\\kf.oI6[c8h, DDDDDDDDDD DDDDDDDDDD DDDDDDDDDD DDDDDDDDDD ສະຫງສ.ວວນ.ລິສຂະສິດ X cm) 0D gIqkwfh0t|kf0v’l;o.oI6[c8h, 3fpdko7yfw]9j kd0t|kf9y’0v’l;o fh;p;ymu.fL 7D 0t|kf.oI6[c8h, gxaovaf8kl;oj rq;raoda[0t|kf9y’0v’l;omudj q’da[,ao s^n [+L 4hkgxao 9qj’0Povaf8kl;j orq;raooUgxaoI6[g]dlj;oc]h;7af9hvoD Xg]dl;j omujwfh,koUgvUo;jk G ,kf8kl;j o 0t|kf.oI6[c8h,0v’l;oda[0t|kf8q; 9y’0v’l;oD phvo;kj I6[c8h,ohvpd;kj l;o8q;9y’ gryoj gvUo,kf8klj;ooU;kj G ,kf8kl;j osp=h?D dyf9tdemu @ I6[]5j,oUltcf’g4y’I6[lujc9lkd mu0j tspkp .spdj ;kj 8q;9y’ @ gmnvj D dD 9q’j cmd];’pk; c]t ];’d;hk’0v’ I6[lujc9lkd c]h;7yfw]]j ;’pk;9y’ c]t ];’ d;hk’9y’0v’I6[lucj 9lkdoUD 197

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ສະຫງສ.ວວນ.ລິສຂະສິດd=]toumu @G .shI6hw]ptsjk’mk’vkdkfc8;j P’9aoskg,nv’3I9u,yo ,u 980 kmF c8j16j.oczomujvyof69uomvj o-njmuj0uf8+c8;j P’9aosk3I9u,yocmdwfh 49 cm 4k,;kj ,kf8kl;j o0v’czomuoj Uc,oj gmqjk.fL [qfcdhG lq,,5f.sh x c,oj ,kf8kl;j o0v’czomufj a’j dkj ; c]t 980 km = 98 000 000 cm x  49  1 98000 000 2 000 000 d=]toumu #G .oczomuwj ]ptsjk’]ts;jk’lv’95fcmdwfh 5 cmD ,kf8klj;osp=h0v’czomucj ,oj 1 D 12 000 4k,;kj w]ptsjk’]ts;jk’lv’95foAo8q;9y’c,jogmqjk.fL [qfcdhG lq,,5f.sh L c,oj w]ptsjk’8q;9y’0v’lv’95ffaj’dkj ;D 5 1 L 12 000 9twfh L  512 000  60 000 cm  600 m D d=]toumu $G dhv’95]tmafvao\\b’j ,u0uf\\kp[vd0tspkp.spj 250 gmnvj wfhlvj ’gsaog-Nv 95]yo-u-toyf|b’j pk;xt,ko 1,25 cm. 9qj’-vdsk];’pk;8q;9y’0v’g-Nv95]yo-u faj’dkj ;D [qfcdhG lq,,5f.sh L c,jo];’pk;8q;9y’0v’g-Nv95]yo-ufa’j djk;D 0Pogxaovaf8klj;owfhG 1, 25  250 L 9twfh L  1, 25  0, 005cm  5103 cm. 250 199

#D [qfg/ydsaf !D .oczomujw]ptsjk’]ts;kj ’lv’95f A c]t B cmdwfh 4 cmD ,kf8kl;j o0v’czomujc,jo 1 D 120 000 4k,;kj w]ptskj ’8q;9y’0v’lv’95f A c]t B ,ugmqkj .fc,afL @D czomujltcf’glAomk’8kj ’Mvao\\b’j ,u,kf8klj;o 1 D 200 000 dD w]ptskj ’9kd[hko[ko sk [hkook7epk; 75 kmD 4k,;jk.oczomujw]ptsjk’oUpk;9ad-a’8uc,afL 0D .oczomuwj ]ptsjk’c8[j hkook sk [hko3ro cmdwfh 150 cmD 4k,;jkw]ptskj ’8q;9y’ c8j[hkooksk [hko3ro ,u9addy3]c,afL #D .shI6[lujc9lkd ABCD F I6[8Qoc[[,u];’d;hk’ 3cm c]t ];’pk; 5cm D 4hk8hv’dkosp=hI6[ 75% c]t 0tspkpI6[ 120% 9twfhI6[lucj 9lkd EFGH c]t PQRS mugj xaoI6[sp=h 75% c]t I6[0tspkp 120% 0v’I6[lucj 9lkd ABCD 8k,]efa[ fa’j I6[]5j,oU ສະຫງສ.ວວນ.ລິສຂະສິດ A 5 cm B E 3,75 cm F I6[lujc9lkd9y’ 3 cm I6[sp=h 75% 2,25 cm HG DC P 6 cm Q I6[0tspkp 120% 3,6 cm SR 9kd8q;1jk’faj’djk;0hk’gmy’F 9q’j 8v[7e4k,8+wxoUG 200

dD 4hk0tspkpI6[8Qoc[[ AB  6cm gxaoI6[0tspkp 150% c]h; AB 9t,u 7;k,pk;s^kpxko.f L 0D 4hksp=hI6[8Qoc[[ AB  20cm gxaoI6[sp=h 75% c]h; AB 9t,u7;k,pk; s^kpxko.f L 7D 4hk0tspkpI6[lujc9lkdmu,j u];’pk; 9cm c]t ];’d;hk’ 5cm gxaoI6[0t spkp 120% c]h; I6[0tspkp,u];’d;hk’ c]t ];’pk;gmqjk.fL ’D 4hksp=hI6[lujc9lkdmuj,u];’pk; 15cm c]t ];’d;hk’ 12cm gxaoI6[sp=h 50% c]h; I6[sp=h,u];’d;hk’ c]t ];’pk;gmqkj .fL 9D 4hk0tspkpI6[lk,c9mjP’muj,u0hk’pk; 8cm c]t rNopk; 5cm gxaoI6[0t spkp 120% c]h; I6[0tspkp9t,u0hk’ c]t rNopk;gmqjk.fL $D oad;ylt;tdvovvdc[[z6h|b’j 8hv’dkoc8h,czoza’gInvos^a’|bj’ 3fp.sh 13 m .o8q;9y’dq’da[ 5,2 cm .oczoza’D 4k,;jk ]k;9t8hv’ gvqk,kf8klj;ogmqjk.fL &D 9qj’0Po8nj,.l8j k8t]k’]5j,oU.sh7q[4h;o c]t g\\ktlq, w]pt.o8q;9y’ w]pt.oczomuj ,kf8klj;o ສະຫງສ.ວວນ.ລິສຂະສິດ 7,6 km 76 mm 1 156 m 4 000 65 cm 10,5 km 4,2 cm 1 87 km 3,5 mm 80 000 3,5 mm 1 1 500 000 125 *D l;o8vo|b’j gxaoI6[lujc9lkdF ];’pk;cmdwfh 245m c]t ];’d;hk’cmdwfh 125m D gryjoc8h,czomu0j v’l;ofaj’dkj ;3fp.sh];’pk;0v’I6[l;ocmdwfh 9,8cm D dD 3fp[+-vdsk,kf8klj;o0v’I6[c8h,F 9qj’7yfw]]j ;’d;hk’0v’I6[l;ofaj’dkj ;D 0D 9q’j -vdsk,kf8kl;j oI6[c8h,0v’l;ofaj’dkj ; c]h;7yfw]gj rnvj d;fg[yj’ 7e8v[.o0=h Xd?D 201

(D 9qj’-vd,kf8klj;o0v’czoza’ c]h;-vdgoNvmuj0v’shv’8jk’Mrkp.ogInvo X9eo;omuj.sh16]j 5,j c,oj 0t|kf8q;9y’0v’gInvogxao-a’8uc,af?D shv’ovo shv’ovo shv’vk[ shv’l5 gInvo7q; shv’Ia[ shv’rad X!? X@? oE 0k c0d zvj o goNvmuj m2 15 372 15shv’ovo shv’ovo shv’Ia[ X!? X@? c0d ສະຫງສ.ວວນ.ລິສຂະສິດ shv’radzjvo gInvo7q; l50k shv’vk[oE 15 1425 10 )D czoza’gInvos^a’|b’j gxaoI6[lucj 9lkdF ];’pk;cmdwfh 16m c]t ];’d;hk’cmd wfh 10m g-y’j c[’j ’vvdgxaoshv’faj’oUG shv’ovo shv’ovo gInvo7q; shv’radzjvo shv’vk[oE shv’l50k X!? X@? goNvmuj 20 20 16 40 10 5 m2 dD 9q’j g]nvdgvqk,kf8kl;j oc]h;c8h,czoza’gInvoD 0D 9qj’[vd0t|kf8q;9y’0v’shv’ovolv’shv’F 0v’gInvo7q;F shv’radzvj oF shv’vk[oE c]t shv’l50kD 202

rkdmu ( s^adgdo8kc]af [qfmu $$ dkolkp8k,];’mu.j sh,k !D dyf9tde dyf9tdemu ! .shIa’oqddk’cd c]t g’qk0v’,aomujwfhIa[9kdcl’cff.oc8]j td=]touS gsao;jkg’qk0v’,aogxaomjvo-njmu8j kj ’dao8k,];’0v’cl’cffD ];’0v’cl’cffwfhltcf’fh;pglAo-nj ສະຫງສ.ວວນ.ລິສຂະສິດ c]t 4hkdtmq[.lj;af45|b’j gIqklk,kf c8h,g’qk0v’,ao.lj|hkfyowfh !D .olk,I6[]5j,oU gryjoltcf’|hkfyofh;pglAo-nj|b’j c]t ;af45fh;pmjvo-nj|b’j [AB] D 9q’j [vd;kj I6[.fgxaoI6[muj4nd8hv’L phvospa’L dD 0D B A 7D B B A A @D .olk,I6[]5j,oUcl’cffwfhltcf’fh;pglAo-nj d’F |hkfyowfhltcf’fh;pglAo-nj d   203

9q’j c8h,g’qk0v’mjvo-nj [AB] 16j d 8k,];’0v’ d’ B  d'   A  d'   d  A  d'   A  B  B d  d  dyf9tdemu @ A .oI6[lk,c9 ABC F gryoj .shglAo-nj (MN) 0t|ko da[glAo-nj (BC) M N 9qj’8n,j xt3spd7eg;Qk]j5,oUG dD g,af N c,jog’qk0v’g,af M .lgj lAo-njDDDDDDDDDD B C 8k,];’0v’DDDDDDDDDD 0D g,af B c,oj g’qk0v’g,af DDDDDDDD .lgj lAo-njDDDDDDDDDD 8k,];’ ສະຫງສ.ວວນ.ລິສຂະສິດ 0v’DDDDDDDDDD dyf9tdemu # .oc8]j td=]tou]5j,oUF I gxaog,afg7y’j dk’0v’mvj o-nj [AB] D !D 9qj’c8h,mjvo [A'B'] g’qk0v’mvj o-nj [AB] c]t g,af I' g’qk0v’g,af I .lgj lAo-nj (d) 8k,];’0v’ (d ') . @D 9q’j ;afcmd];’pk; A'I' c]t B'I' , c]h;.sh0=hla’gdfD d’  B d’  d' B A B d  d I I A I d  A @D .97;k, !D dkolkp8k,];’muj.sh,k dD oypk,G .shlv’glAo-nj d1 c]t d2 8afdao1j6g,af|b’j S   204

d1 A  { g,af A 8k,.9 X A [+16j d1 c]t A [+1j6 d2 ?S { zjko A , gryoj 0ufglAo-n0j t|koda[ d1 F 8af d2 1j6 A ' ; A’ d2 { g,af A' c,jog’qklkp0v’g,af A .ljglAo-nj d2 8k,];’0v’ d1 @D g’qklkp0v’g,afg7y’j dk’0v’mvj o-nj|bj’ d1 { g,nvj gIqklkpmvj o-nj [AB] gIqkd=wfhmvj o-nj ⎡⎣A'B' ⎤ ⎦ B I { g,njvgIqklkpg,af I mugj xaog,afg7yj’dk’0v’mvj o-nj A [AB] gIqkd=wfhg,af I' mujgxaog,afg7yj’dk’0v’mvj o-nj ⎣⎡A'B' ⎤ ⎦ d2 A' I' B' { g,njvgIqklkpmjvo-nj|b’j mu[j +0t|koda[];’lkp .ljglAo-nj|b’j 8k,];’muj.sh,kF gIqkd=9twfhmjvo-njvudmjvo -n|j b’j g-y’j ];’pk;0v’,aogxaovaf8klj;orq;raoda[];’pk;0v’mjvo-njmu.j sh,kD { .shglAo-nj d .f|bj’gxao];’lkpF glAo-nj d1 c]t d2 8afdao16jg,af O; g,af A ,ສະຫງສ.ວວນ.ລິສຂະສິດ B , C , D 1j6glAo-nj d1 g,af A', B', C', D' gxaog’qklkp0v’g,af A, B, C, D .lgj lAo-nj d2 8k,];’ 0v’ d gIqkwfGh OA = OB = OC = AB = BD = L OA' OB' OC' A'B' B'D' D' C' d OA B C A' B' d2 d1 D #D [qfg/ydsaf !D gryjo.shI6[lujc90hk’0t|ko ABCD F M c,jog,af|b’j 0v’glAogoa’j 9v, [AC] , zjko M, gryjo0ufglAo-n0j t|koda[0hk’0v’I6[lujc90hk’0t|koD 9qj’8n,j xt3spd7eg;Qk]j5,oUG   205

dD g,af Q c,jog’qk0v’g,af M AN B .ljglAo-nDj DDDDDDDDD8k,];’0t|koda[DDDDDDD R P 0D g,af N c,oj g’qk0v’g,af M .lgj lAo-njDDDDDDDDDD 8k,];’0t|koda[DDDDDDD M C 7DDDDDDDDD c,jog’qk0v’g,af M .ljglAo-nj (AD) DQ 8k,];’0t|koda[DDDDDDD @D .sh O gxaog,afg7y’j dk’0v’mvj o-nj [AB] A O B'  d A'   B d1 c]tglAo-nj d muzj jko O. gryjolkpg,af A c]t B 8k,];’muj[+0t|koda[ d .lj glAo-nj d’ dD 9qj’-Uc9’;kj O c,oj g,afg7yj’dk’0v’ mvj o-nj [A'B'] 0D I6[lujc9 AB’BA’ gxaoI6[lujc9xtgrf.fL #D gryjo.shI6[lucj 9lkd ABCD, I c,oj g,afg7yj’dk’ A J K B ສະຫງສ.ວວນ.ລິສຂະສິດ 0v’0hk’[AD] ,c]t glAo-nj (IJ) 0t|koda[glAo I C -nj (DK) . 9qj’-Uc9’;jk J c,jog,afg7yj’dk’0v’mvj o-nj [AK] D $D .shlv’glAo-nj d c]t d’ 8afdao1j6 OF M c,oj g,af|b’j 0v’ d, N c,oj g,af g7y’j dk’0v’mvj o-nj [OM] D zkj og,af M c]t N 0uflv’glAo-nj0t|kodao c]t 8afglAo-nj d’ 1j6 E c]t F 8k,]efa[. dD 9q’j c8h,I6[ 0D gryjog;Qkwfhco;.fdP;j da[g,af FL phvospa’L &D gryoj .shI6[lk,c9 ABC F I gxaog,afg7y’j dk’0v’ A 0hk’[AB] , J c,oj g,afg7yj’dk’0v’[AC] F I J M c,jog,af|bj’0v’0hk’[BC], K glAo-nj (AM) 8afglAo-nj (IJ) 16j K D B MC 9qj’-Uc9’;kj K gxaog,afg7y’j dk’0v’mvj o-nj [AM]   206

[qfmu $& glAo-nzj jkog,afg7y’j dk’0v’lv’0hk’0v’I6[lk,c9 !D dyf9tde dyf9tdemu ! A N .shI6[lk,c9 ABC , M gxaog,afg7y’j dk’0v’ [AB] M zjko M , gryjo0ufglAo-n0j t|koda[(BC) F 8af(AC) 16j N dD 9q’j -vdg’qklkp0v’g,af M , B c]t A 16gj lAo-nj B C (AC) 8k,];’mu0j t|koda[ (BC) D 0D 9q’j -Uc9’;jk N gxaog,afg7yj’dk’0v’mvj o-nj [AC] dyf9tdemu @ ສະຫງສ.ວວນ.ລິສຂະສິດ !D .shI6[lk,c9 ABC , M c]t N gxaog,afg7yj’dk’0v’ A [AB] c]t [AC] 8k,]efa[D M N dD 9q’j -vdg’qklkp0v’mjvo-nj [AB] c]t0v’g,af M 16gj lAo-nj (AC) 8k,];’muj0t|koda[ (BC) D B P C0D 9q’j -Uc9’;kj (MN) (BC) @D .sh P gxaog,afg7yj’dk’0v’mjvo-nj [BC] dD gryjolk,kfg;Qkwfhco;.fdPj ;da[glAo-nj (NP) c]t glAo (AB) L 0D 9q’j -Uc9’;jk I6[lujc9 MNPB gxaoI6[lujc90hk’0t|koD 7D 9q’j 0Po8n,j .l[j vj o9Eg,af BP = ...×BC MN = ... = ...× BC @D .97;k, !D I6[lk,c9c]tg,afg7yj’dk’ dD s^adgdo s^adgdo !G { .oI6[lk,c9|b’j glAo-nmj uzj kj og,afg7y’j dk’0v’lv’0hk’8hv’0t|koda[0hk’mulk,D { .oI6[lk,c9|bj’mjvo-nmj u0j uf8+g,afg7y’j dk’0v’lv’0hk’,u];’pk;gmqjkda[g7yj’|bj’ 0v’0hk’mulk,D .oI6[lk,c9 ABC , I gxaog,afg7y’j dk’0v’ [AB] F J gxaog,afg7y’j dk’0v’[AC] F 207

gIqkwfhG A IJ (IJ)//(BC) c]t IJ = 1 BC 2 BC s^adgdo @G .oI6[lk,c9|bj’glAo-nmj ujzjkog,afg7y’j dk’0v’0hk’|b’j c]t 0t|ko da[vud0hk’|b’j 8hv’8afg,afg7y’j dk’0v’0hk’mulk,D A .oI6[lk,c9 ABC , I gxaog,afg7y’j dk’0v’ [AB] F IJ J gxaog,af|bj’0v’ [AC] c]t (IJ)//(BC) gIqkwfhG B C J gxaog,afg7y’j dk’0v’ [AC] 8q;1jk’ !D ສະຫງສ.ວວນ.ລິສຂະສິດ gryjo.shI6[lk,c9 ABC F M c,jog,af|b’j 0v’0hk’ [BC], R c,jog,afg7y’j dk’0v’[AM] D glAo-nzj jkog,af R 0t|koda[ (BC) 8af [AB] c]t [AC] 16gj ,af S c]t T 8k,]efa[D 9qj’-Uc9’;jk S gxaog,afg7y’j dk’0v’mjvo-nj [AB] . [qfcdhG A I6[lk,c9 ABC 0=hlq,,5f M c,jog,af|bj’0v’0hk’[BC] S RT C R c,jog,afg7yj’dk’0v’[AM] BM glAo-nj (d) c]t glAo-nj (BC) 0t|kodaoD 0=hlts^6[ -Uc9’;kj S gxaog,afg7yj’dk’0v’[AB] { .oI6[lk,c9 ABM, gIqk,uglAo-nj (SR) zjkog,afg7y’j dk’ R 0v’0hk’mu|b’j [AM] c]t 0t|koda[0hk’mulv’[MB] D vu’8k,s^adgdo @F glAo-nj (SR) 8hv’zkj og,afg7yj’dk’0v’0hk’mulk, AB 0v’I6[ lk,c9D gIqk9bj’wfh S gxaog,afg7y’j dk’0v’mvj o-nj [AB] . 208

8q;1kj ’ @D gryoj .shI6[lk,c9 ABC F M c,jog,af|b’j 0v’0hk’[AB] , N c,oj g,af |b’j 0v’0hk’[AC] F glAo-nj (MN) 0t|koda[glAo-nj (BC) F I6h;kj AM = 2 cm, AN = 3cm c]t AB = 8 cm. 9qj’7yfw]j];’pk; NC. [qfcdhG I6[lk,c9 ABC A 0=hlq,,5f M c,oj g,af|bj’0v’0hk’[AB] MN N c,oj g,afg7y’j dk’0v’[AC] glAo-nj (MN) c]tglAo-nj (BC) 0t|kodao BC 0=hlts^6[ 9q’j 7yfw]j];’pk; NC {.oI6[lk,c9 ABC F M c,jog,af|b’j 0v’0hk’ [AB] , N c,oj g,af|b’j 0v’ 0hk’ [AC] F c]tglAo-nj (MN) 0t|koda[glAo-nj (BC) F ສະຫງສ.ວວນ.ລິສຂະສິດ fa’j oAo gIqklk,kf0PowfhG AM = AB AN AC \\kp7;k,;kj 2 = 8 s^n 2× AC = 3×8 3 AC 4vowfh AC = 24 =12cm F faj’oAo NC = AC − AN =12 − 3 = 9cm 2 #D [qfg/ydsaf !D gryoj .shI6[lk,c9 ABC, I c,jog,afg7y’j dk’0v’0hk’[AB] , J c,oj g,afg7yj’dk’0v’0hk’ [AC] c]t BC = 7cm D dD 9qj’-Uc9’;kj glAo-nj (IJ) c]t glAo-nj (BC) 0t|kodaoD 0D 9q’j 7yfw]j];’pk; IJ. @D .shI6[lujc90hk’0t|ko ABCD muj,u.9dk’ O, E c,oj 95fg7yj’7n0v’ D mP[.lj CD dD 9q’j -Uc9’;kj (OC) c]t 0t|kodao (EB) 0D I6[lucj 9 ABEC gxaoI6[lujc9-toyf.fL 7D 9qj’7yfw]]j ;’pk;0v’mvj o-nj [BE] , I6h;kj AC=2 cm 209

#D .shI6[lk,c9 ODEF I6h;kj P c,jog,afg7yj’7n0v’ O mP[.lj D, R c,jog,afg7yj’7n0v’ O mP[.lj E dD 9qj’c8h,I6[ 0D 9q’j [vddkorq;rao]ts;kj ’lv’glAo-nj (PR) c]t (DE) $D gryjo.shI6[lk,c9 ABC F I, J c]t K c,oj g,afg7yj’dk’0v’0hk’ [AB] , [AC] c]t [BC] 8k,]efa[ dD gryjo.sh AB = 4 cm; AC = 3 cm c]t BC = 6 cm. 9qj’-vdsk];’pk;0v’mvj o-nj [IJ] , [JK] c]t [KI] 0D 4hk ABC gxaoI6[lk,c9mPj’S IJK 9tgxaoI6[lk,c9xtgrf.fL 7D 4hk ABC gxaoI6[lk,c9ltg\\uS IJK 9tgxaoI6[lk,c9xtgrf.fL ’D 4hk ABC gxaoI6[lk,c9lkd1j6 A; IJK 9tgxaoI6[lk,c9xtgrf.fL &D 9qj’7yfw]]j ;’pk;0v’mvj o-nj [O'C'] ,[CD] XI6[ !? c]t [OC] ,[C'D'] XI6[ @?D A 6 O4C D D'   C'   ສະຫງສ.ວວນ.ລິສຂະສິດ A2 O 8 A' 4 C2D A’  3 O’ C’1 D’ I6[ @ I6[ ! *D gryjo.shI6[lk,c9 ODE muj,u0hk’ DE = 4 cm, P gxaog,afg7y’j 7n0v’ O mP[.lj D c]t Q gxaog,afg7yj’7n0v’ O mP[.lj E. dD 9q’j c8h,I6[ 0D 9q’j -Uc9’;kj (PQ)//(CE) 7D 9qj’7yfw]]j ;’pk; PQ. (D gryoj .shI6[lk,c9 SEC F A gxaog,afg7yj’dk’0v’0hk’[SC], zkj o A gryjo0ufglAo-n0j t|koda[(SE) S 8af (EC) 16j B. S zjko B, gryoj 0ufglAo-nj0t|koda[(SC) S 8af (SE) 16j D. D A dD 9qj’-Uc9’;jk D gxaog,afg7y’j dk’0v’mjvo-nj [SE] 0D gryjog;Qkwfhco;.fdP;j da[glAo-nj (AD) c]t (EC) E B C 210

[qfmu $* s^adgdo8kc]af (Thalès) !. dyf9tde dyf9tdemu ! .oc8j]tI6[]5j,oUF gryjo.sh (MN)//(BC) dD 9q’j ;afcmd];’pk;0v’ AB, AC, BC, AM, AN, MN gxao cm c]h;8n,j .l8j k 8t]k’G d d’ AM AN MN I6[lk,c9 AMN A I6[lk,c9 ABC MN AB AC BC BC d d’ AM AN MNສະຫງສ.ວວນ.ລິສຂະສິດ A I6[lk,c9 AMN BC I6[lk,c9 ABC MN AB AC BC AM AN MN N B d’ I6[lk,c9 AMN A I6[lk,c9 ABC M Cd AB AC BC 0D 9q’j 7yfw]jG AM ; AN ; MN c]h;.sh0=hla’gdfD AB AC BC 7D 9q’j 8nj,xt3spd7eg;Qk]5j,oUG 211

lv’I6[lk,c9 AMN c]t ABC g-yj’ M 1j60hk’ [AB] , N 1j60hk’ [AC] c]t (MN) 0t|koda[ (BC) , ];’pk;0v’0hk’0v’I6[lk,c9 AMN gxao DDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD0v’I6[lk,c9 ABC dyf9tdemu @ A .shI6[lk,c9 ABC \\bj’,u (MN)//(BC), M B vy’.lzj qowfhIa[.o8k8k]k’0v’dyf9tde ! N 9qj’8nj,leo;o]5,j oUG ... = ... = ... AB AC BC @D .97;k, C !D s^adgdo8kc]af lv’glAo-nj d c]t d’ 8afdao16jg,af A, ສະຫງສ.ວວນ.ລິສຂະສິດ d d’ g,af B c]t M 1j6glAo-nj d’ A g,af C c]t N 16jglAo-nj d MN 4hkglAo-nj (BC) c]t (MN) 0t|kodao BC gIqk9twfhG AM = AN = MN AB AC BC @D s^adgdoxUo0v’s^adgdo8kc]af lv’glAo-nj d c]t d’ 8afdao16jg,af A, N B d’ g,af B c]t M 16jglAo-nj d’ A g,af C c]t N 16jglAo-nj d M Cd 4hk AM = AN AB AC gIqkwfh glAo-nj (BC) c]t (MN) 0t|kodaoD 212

8q;1jk’ !G .shI6[lk,c9 ABC muj,u AB =12cm c]t BC = 6cm D grujo0ufglAo-nj 0t|koda[ (BC) F 8af (AB) 16j D c]t 8af (AC) 1j6 ED 9q’j 7yfw]Gj DE c]t EC F I6h;kj AD = 8cm c]t AE = 6cm D [qfcdhG A 0=hlq,,5f I6[lk,c9 ABC F AB = 12cm F BC = 6cm DE (BC) / / (DE) F AD = 8cm c]t AE = 6 cm B C 0=hlts^5[ 7yfw]j DE c]t EC { 7yfw]j DE .oI6[lk,c9 ABC, gIqk,u (BC) / / (DE) vu’8k,s^adgdo8kc]af gIqkwfhG AD = AE = DE AB AC BC \\kp7;k,;kj AD = DE sn^ 8 = DE ສະຫງສ.ວວນ.ລິສຂະສິດ AB BC 12 6 gIqkwfh 8×6 =12× DE gIqk4vowfh DE = 48 = 4 cm 12 { 7yfw]j EC 9kd AE = DE sn^ 6 = 4 AC BC AC 6 gIqkwfh 6×6 = 4× AC gIqk4vowfh AC = 36 = 9 cm 4 gIqk9b’j wfh EC = AC − AE = 9 − 6 = 3 cm 8q;1kj ’ @G .oI6[7k’\\6 ABCD, gryoj 0uf8+lv’0hk’goU’ [AD] c]t [BC] 9qor;d ,ao8afdao16jg,af FD Ih6;jk FD = 8 , FB = 2, 25 cm c]t AB = 3 cm FA 5 9q’j 7yfw]j];’pk;0v’ BC. 213

[qfcdhG F 0=hlq,,5f I6[7k’|6 ABCDF FB = 2,25 cm A 0=hlts^5[ AB = 3 cm D B ( AD) c]t (BC) 8afdao1j6 F, FD = 8 C FA 5 7yfw]j BC c]t DC { 7yfw]j BC FAB gxaoI6[lk,c9muj,u(DC) (AB) F vy’8k,s^adgdo8kc]afF gIqkwfh FD = FC = DC FA FB AB 9kd FD = FC s^n 8 = FC FA FB 5 2,25 gIqkwfh 8× 2, 25 = 5× FC ສະຫງສ.ວວນ.ລິສຂະສິດ gIqk4vowfh FC = 18 = 3, 6 cm 5 gIqk9bj’wfh BC = FC − FB = 3,6 − 2, 25 =1,35 cm { 7yfw]j DC 9kd FD = DC s^n 8 = DC FA AB 5 3 gIqkwfh 8×3 = 5× DC gIqk4vowfh DC = 24 = 4,8 cm 5 #. [qfg/ydsaf !D 9q’j 8v[7e1no1ao.o[qfg/ydsaf8+wxoU fh;p7e;kj G 4nd s^n zyf dD AB AC BD AH B CD = AE = BE 0D HB = HD = BD D FG HG BF FG C BH BD HD 7D BA = BE = AE E 214

@D .oI6[lk,c9 ABC |b’j , gryoj c[’j 0hk’|bj’0v’,aovvdgxaolk,lj;ogmqkj daoD zjko [aofkg,afc[j’faj’dkj ;F gryjoc8h,glAo-n0j t|koda[rNoD Ih6;kj rNopk; !@ cm. 9q’j 7yfw]j];’pk;0v’mjvo-nj0t|komuj4nd8affh;plv’0hk’0v’I6[lk,c9faj’djk;D #D gryjo.shmjvo-nj [AB], [CD] c]t [EF] D 9qj’c8h,mvj o-nj x g-yj’;jkG AB = EF CD x $D .shI6[lk,c9 ABC F (PQ)//(BC) faj’I6[D 9q’j -vdsk];’pk;0v’mjvo-nj x c]t y .oc8j]td=]tou8+wxoUD dD 0D 8 cm 7D A A Q 3cm P 4 cm 3,5cm 4 cm x 6 cm P 3cm Q 6 cm y A 10 cm P 4 cm Q 3cm 2 cm y Cx B B yC Bx C ສະຫງສ.ວວນ.ລິສຂະສິດ &D .sh Fc,oj g,afg7yj’dk’0v’0hk’ [AB] 0v’I6[lk,c9 ABC c]t (FE) 0t|ko (BC) S (FD) 8af (AC) 16j I D 4hk AC = 12cm, BD = 8cm c]t DC = 2cm F 9q’j 7yfw]j];’pk; FE c]t CI S *D lv’glAo-nj (RU) c]t (TS) 8afdao16jg,af O 55 44,8,8 lv’glAo-nj (RS) c]t(TU) 0t|kodaoD OO R U 33 9q’j 7yfw]]j ;’pk; OT c]t RS. 66 (D .oI6[lk,c9 ABC, .shglAo-nj (DE) 0t|koda[ (BC) T c]t8af0hk’ [AB] 1j6 D, 8af0hk’ [AC] 1j6 E I6h;jk AC = 5 m , AE = 3m , BC = 6m 9q’j 7yfw]j];’pk; DE. 215

)D .shI6[7k’\\6 ABCD, lv’glAogoaj’ A 3B C 9v,0v’,ao8afdao16jg,af O. 42 dD 9qj’[vdI6[lk,c9mujgIqklk,kf O .-hs^adgdo8kc]afwfh 5 0D 9qj’7yfw]]j ;’pk; OD c]t CD. D _D .shlv’glAo-nj (EC) c]t (BD) 8afdao1j6 A, (BC) 0t|koda[ (DE) 9qj’7yfw]]j ;’pk; AB c]t AE .oc8]j td=]tou]j5,oUG E 3cm 0D D dD D 5cm C 2,5cm B 6cm E 4,5cm A ສະຫງສ.ວວນ.ລິສຂະສິດ A 1,5cm C !WD .shI6[lk,c9 ABC, D gxaog,afg7y’j dk’ 3,2cm C 0v’ [BC] FM gxaog,afg7yj’dk’0v’ B A [AD] , glAo-nj (CM) 8af [AB] 1j6 F. zjko D, gryoj 0ufglAo-n0j t|koda[ (CF) F EM 8af [AB] 16j E. dD 9qj’-Uc9’;jk F gxaog,afg7yj’dk’0v’[AE] B D 0D 9q’j -Uc9’;jk E gxaog,afg7yj’dk’0v’[BF] 7D I6h;jk ED = 4 cm 9qj’7yfw]]j ;’pk; FM; MC. !!D .shI6[lk,c9 EFG, 0hk’ EF = 5 cm c]t EG = 8 cm F M gxaog,af|b’j 0v’ [EF] 3fp.sh EM = 3 cm F N gxaog,af|bj’0v’ [EG] 3fp.sh EN = 4,8 cm . dD 9q’j c8h,I6[ 0D I6[lujc9 MNGF gxaoI6[lucj 9-toyf.fL 216

7D I6h;kj FG =12 cm 9q’j 7yfw]j];’pk; MN. !@D grnvj -vdw]ptskj ’]ts;kj ’g,af A c]t B B A g-y’j [+vkf9t;afcmd3fpdq’wfhD D gryjog]nvdgvqkg,af C .f|b’j c]t cmdw]pt C CA S gmy’glAo-nj CA cmdgvqkmvj o-nj CD E .f|bj’ c]tzkj o D goa’j glAo-nj (DE) 0t|koda[ (AB) D fa’j oAoF g,nvj cmdmjvo-nj [DE] c,jogIqk9t-vdwfhw]pt ABF phvospa’L 4hk CA=1,8 km, CD=90 m c]t DE=150 m. 9qj’7yfw]j AB. !#D .shlv’glAo-nj (TS) c]t (RM) 8afdao16j A, c]t (TR) 0t|koda[ (SM). 9qj’7yfw]]j ;’pk; RM, .oc8j]td=]tou]5joUG dD 0D S ສະຫງສ.ວວນ.ລິສຂະສິດ A 9 7 7 M A T R 15 5 R S M 6T !$D .shlv’glAo-nj (BD) c]t (CE) 8afdao1j6 A. D dD 9qj’-vdvaf8klj;o AD c]t AE 24 B AB AC 16 0D 9q’j 4vo;kj lv’glAo-nj (BC) c]t (DE) 0t|kodaoD A 33 C 11 E 217

!&D .shI6[lujc90hk’0t|ko ABCD muj,u0hk’ AB=12 cm c]t0hk’ AD=9 cm, E c,jog,af|bj’0v’mjvo-nj [AB] c]t M c,oj g,af|bj’0v’0hk’ [AD] I6h;kj AE = 7, 2 cm c]t AM = 5, 4 m dD 9qj’c8h,I6[ 0D 9qj’-Uc9’;jk lv’glAo-nj (EM) c]t (BD) 0t|kodao. A B !*D .shlv’glAo-nj (AC) c]t (BD) 8afdao16j O. .sh OA = 15, OB = 25,2 OC = 7, OD = 54 O C dD 9q’j d;f7nog[yj’;jk 7 × 54 = 15× 25,2 0D 9q’j 4vogvqk OA = OD D OC OB 7D 9q’j -Uc9’;jk I6[lujc9 ABCD gxaoI6[7k’\\6D !(D .shI6[lk,c9 ABC, M 1j60hk’ [AB] A N 160j hk’ [BC] c]t P 16j0hk’ [AC] M 3fp.sh (MN) (AC) c]t (NP) (AB) ສະຫງສ.ວວນ.ລິສຂະສິດP I6h;kj AB = 6 cm, BC = 8 cm, AC = 5 cm c]t AM = 2 cm . B NC 9qj’7yfw]]j ;’pk; MN, AP, CN c]t NP. !)D .oI6[c8h,0hk’oUF glAo-nj (BE), (CF), (DG) 0t|kodao8k,c8]j t76jD 9q’j 7yfw]j];’pk; BC c]t FG. 12 C9D A8 B E6F G 218

  [qfmu $( I6[lk,c97hkp7ndao   !. dyf9tde dyf9tdemu ! A .oI6[lk,c9]j5,oUF gryjo.shglAo-nj (DE) //(BC) c]t (EF) //(AB) D 9q’j 8nj,[vj o9Eg,af dD .oI6[lk,c9 ADE c]t ABC F gIqk,u D E Aˆ gxao,5,Ij;,S B FC ADE = … S AED = … vu’8k,s^adgdo8kc]af gIqkwfh AD = … = … AB … ... gryjog;Qk;jk I6[lk,c9 ADE c]t ABC 7hkp7ndaoD 0D .oI6[lk,c9 EFC c]t ABC F gIqk,u FEC =…S EFC = … S ECF = … ສະຫງສ.ວວນ.ລິສຂະສິດ { vy’8k,s^adgdo8kc]af gIqkwfh EC = … = … AC … ... 9kdoAo 9b’j wfhI6[lk,c9 DDDDDDDDDD c]t DDDDDDDDDD 7hkp7ndaoD 7D gIqk4vowfh I6[lk,c9 DDDDDDDDDDF I6[lk,c9 DDDDDDDDDD c]t DDDDDDDDDD 7hkp7ndaoD dyf9tdemu @ .oc8]j td=]tou]5,j oUF 9qj’8nj,.lj[jvo9Eg,af I6h;kj (DE) //(BC) c]t (EF) //(AB) D A ED C A B CF DE FB I6[ !D I6[ @D { XI6[ !?G I6[lk,c9 CEF c]t CAB 7hkp7ndaoS phvo,u,5, DDDDDDDDDDDDDDDDDDDD 219

  c]t CE = … = … CA … … I6[lk,c9 DDDDDDDD c]t DDDDDDDD 7hkp7ndaoS phvo DDDDDDDDDDDDDDDDDDDDDDDDDDDD c]t DDDDDDDDDDDDDDDDDDDD gIqk4vowfh I6[lk,c9 DDDDDDDDDDF I6[lk,c9 DDDDDDDDDD c]t DDDDDDDDDD 7hkp7ndaoD { XI6[ @?G I6[lk,c9 DDDDDDDD c]t DDDDDDDD 7hkp7ndaoS I6[lk,c9 DDDDDDDD c]t DDDDDDDD 7hkp7ndaoD 4vowfh I6[lk,c9 DDDDDDDDDDF I6[lk,c9 DDDDDDDDDD c]t DDDDDDDDDD 7hkp7ndaoD @. .97;k, !D oypk, lv’I6[lk,c97hkp7ndao d=8+g,nvj [aofk,6,dq’daoc8]j t76j0v’r;d,aogmqjkdao c]t [aofk0hk’dq’daoc8j]t760j v’r;d,aogxaovaf8kl;j orq;raodaoD A M { lv’I6[lk,c9 ABC c]t MNP ,u G A = M ສະຫງສ.ວວນ.ລິສຂະສິດ NP B=N BC C=P c]t AB = AC = BC MN MP NP fa’j oAolv’I6[lk,c9 ABC c]t MNP 7hkp7ndaoD @D s^adgdo m5dMglAo-njmuj0t|koda[0hk’|b’j 0v’I6[lk,c9 9txtdv[da[lv’0hk’gxaoI6[lk, c9|b’j muj7hkp7nda[I6[lk,c9gfu,D A { I6[lk,c9 ABC F ,u (DE) //(BC) I6[lk,c9 ADE c]t ABC 7hkp7ndaoD DE { m5dMI6[lk,c9mugj mqjkda[I6[lk,c9 B C ADE ]h;oc87j hkp7nda[I6[lk,c9 ABC D 8q;1kj ’ !D .shI6[7k’\\6 ABCD muj,urNoohvp AB = 28mm c]t rNo.spj CD = 35mm D zjkog,af M 1j60hk’ [AD] , gryjo0ufglAo-nj0t|koda[rNo8af (BD) 16j P c]t 8af 220

  0hk’ [BC] 16j N. 9q’j 7yfw]j];’pk; MP, PN c]t MN I6h;kj AM = 3 AD, MD = 1 AD 4 4 [qfcdhG A B 0=hlq,,5f I6[7k’\\6 ABCD F AB = 28mm F MP N CD = 35 mm F ( MN) // ( DC) F AM = 3 AD, D 4 C          MD = 1 AD 4 0=hlts^5[ 7yfw]j MP, PN c]t MN D { 7yfw]j MP. gIqk,uI6[lk,c9 DMP c]t DAB 7hkp7ndaoF phvo (MP)//(AB) F gIqkwfh MP = DM s^n MP = DM = 1 AB DA 28 4DM 4   4vowfh  MP = 28 = 7 mm     ສະຫງສ.ວວນ.ລິສຂະສິດ 4 { 7yfw]j PN. gIqk,uI6[lk,c9 BPN c]t BDC 7hkp7ndaoF phvo (PN) //(DC) F gIqkwfh PN = BN s^n PN = BN = 1 DC BC 35 4BN 4 9kdoAo 4vowfh  PN = 35 = 8, 75 mm   c]t MN = 7 + 8,75 = 15,75mm   4 8q;1kj ’ @D .shlv’I6[lk,c9 ABC c]t DEF 3fp.sh CB = FE c]t C = F. CA FD 9qj’-Uc9’;jk lv’I6[lk,c9 ABC c]t DEF 7hkp7ndaoD [qfcdhG C FA B 0=hlq,,5f I6[lk,c9 ABC c]t DEF A1 B1 CB = FE c]t Cˆ = Fˆ . CA FD 0=hlts^5[ -Uc9’;kj lv’I6[lk,c9 ABC c]t DEF 7hkp7ndaoD DE 221

  { .oI6[lk,c9 DEFF gIqklhk’I6[lk,c9 FA1B1 3fp.sh FA1 = CA; FB1 = CB { .og,nvj CB = FE gIqkwfh FB1 = FE ltcf’;jk FB1 = FA1 CA FD FA1 FD FE FD { vu’8k,s^adgdoxUo0v’8kc]afF gIqkwfh (A1B1) //(DE) F ,5, A1 = D 9bj’4vowfh B1 = E { .og,nvj I6[lk,c9 ABC c]t DEF ,ulk,,5,dq’daogmqkj dao r;d,ao9bj’7hkp7ndaoD #D [qfg/ydsaf !D .shI6[lk,c9 ABC S 0hk’ AB =15cm; AC = 20cm. 16j0hk’ AB gryoj ;k’g,af D 3fp;jk AD =10cm c]t 0hk’ AC gryoj ;k’g,af E 3fp.sh AE =12cm D dD lv’I6[lk,c9 ABC c]t ADE 7hkp7ndao[+L phvospa’L 0D 1kd.shr;d,ao7hkp7ndaoc8j AE =12cm 7ngdqkj F AD 8hv’pk;gmqjk.fL @D .shI6[lk,c9 ABC S 0hk’ AB = 4cm; AC = 5cm. g,af D 16j0hk’ AB 3fp.sh AD = 8cm c]t g,af E 16j0hk’ AC 3fp.sh AE =10cm. lv’I6[lk,c9 ABC c]t ADE 7hkp7ndao[+L phvospa’L B’ ສະຫງສ.ວວນ.ລິສຂະສິດ #D .shlv’I6[lk,c9lkd ABC c]t A'B'C' mu,j ugonhvmuj 4800 cm2 S 7500 cm2 c]t 0hk’ B |bj’0v’,6,lkdgmqjk 40 cm c]t 50 cm 50cm 8k,]efa[D 40cm 9q’j -Uc9’;kj lv’I6[lk,c9lkd ABC c]t A'B'C' 7hkp7ndaoD A C A’ C’ $D .oI6[c8h,F gryoj .sh (AB) //(DE) c]t (BC) //(EF) dD 9q’j lq,mP[ OA c]t OC D OD OF AB 0D 9qj’-Uc9’;kj (AC) //(DF) E C F 7D 9q’j 4vo;jkI6[lk,c9 O ABC c]t DEF 7hkp7ndaoD 222

[qfmu $) d=]tou7hkp7n0v’I6[lk,c9   !. dyf9tde .oI6[c8h,F gryoj .sh (OE) //(BC) c]t (OF) //(CD) D 9q’j 8n,j .lj[jvo9Eg,afG dD { I6[lk,c9 AEO c]t DDDDDDDDD 7hkp7ndaoF A gIqkwfh AE = … = … gIqk4vowfh AB … … 2,4 3,6 AE = 3 ×… = 3 ×… = … E3 F 4 4 { I6[lk,c9 AOF c]t DDDDDDDDD 7hkp7ndaoF B D O1 gIqkwfh AF = … = … gIqk4vowfh AF = 3 ×… =… C c]t AD … … 4 I6[lk,c9 AEF 0D gonvj ’9kd AF = … 9b’j 4vowfh (EF) // (BD) ltoAo AD … DDDDDDDDD 9b’j 7hkp7ndaoF ສະຫງສ.ວວນ.ລິສຂະສິດ @D .97;k, !D d=]tou7hkp7n0v’I6[lk,c9 d=]tou7hkp7nmu|bj’ 4hklv’I6[lk,c9,u,5,|b’j gmqkj dao c]t 0hk’0v’,5,oAogxaovaf8klj;orq;rao daoc8]j t76j c,oj lv’I6[lk,c9oAo7hkp7ndaoD   A A'   C'      D E B'   C B A = A', A'B' = A'C' I6[lk,c9 ABC c]t A'B'C' 7hkp7ndaoD AB AC 223

d=]tou7hkp7nmulv’ 4hklv’I6[lk,c9,u,5,dq’daoc8]j t76gj mqjkdao c,oj lv’I6[lk,c9oAo7hkp7ndaoD A A' D E B'   C'    B C A = A', B = B' I6[lk,c9 ABC c]t A'B'C' 7hkp7ndaoD  d=]tou7hkp7nmulk, 4hklv’I6[lk,c9,ulk,0hk’dq’daoc8j]t76gj xaovaf8kl;j orq;raodaoc,oj lv’I6[ lk,c9oAo7hkp7ndaoD A A' D E B' C'ສະຫງສ.ວວນ.ລິສຂະສິດ B C A'B' = B'C' = A'C' I6[lk,c9 ABC c]t A'B'C' 7hkp7ndaoD AB BC AC @D zqogonvj ’ !D lv’I6[lk,c9ltg\\u pk,.fd=7hkp7ndaoD @D lv’I6[lk,c9mPj ’mu,j u,5,9v,gmqkj dao Xs^n,5,rNogmqkj dao? pk,.fd=7hkp7ndaoD #D lv’I6[lk,c9lkdmuj,uG B { |bj’,5,cs^,gmqjkdao pk,.fd=7hkp7ndaoD { 0hk’dq’lkd c]t 0hk’,5,lkd0hk’|bj’ D B' gxaovaf8kl;j orq;raodaoc8j]t76j c,jor;d,ao7hkp7ndaoD A E C A' C' 224

#D dkorq;rao]ts;jk’d=]tougmqkj dao c]t d=]tou7hkp7ndao0v’I6[lk,c9D d=]tougmqjkdao d=]tou7hkp7ndao ! A = A', AB = A'B', AC = A'C' A = A', A'B' = A'C' @ A = A', B = B', AB = A'B' AB AC # AB = A'B', BC = B'C', AC = A'C' A = A', B = B' A'B' = B'C' = A'C' AB BC AC 8q;1kj ’D .shI6[7k’\\6 ABCD F rNoohvp CD = 5 cmF rNo.spj AB =15 cm D 0hk’ AD =12 cm F glAo-njzkj o C 0t|koda[0hk’ [AD] 8afglAogoaj’9v, [BD] 16j I c]t 8af0hk’ [AB] 16j E. dD 9qj’[vdm5dMI6[lk,c9muj7hkp7ndaoD 0D 9qj’7yfw]j];’pk; EA, IE c]t IBF I6h;kj BD =16 cm D ສະຫງສ.ວວນ.ລິສຂະສິດ 7D 9qj’7yfw]j];’pk; IC c]t IDD [qfcdhG 0=hlq,,5f I6[7k’\\6 ABCD F AB =15 cm F DC CD = 5 cmF BD =16 cm; (CE) // (AD) F I (CE) 8af (BD) 16j I. 0=hlts^5[ dD [vdm5dMI6[lk,c9muj7hkp7ndaoD AE B 0D 7yfw]j];’pk; EA, IE c]t IBF 7D 7yfw]j];’pk; IC c]t IDD dD gIqk,u (CE)//(AD) F fa’j oAoI6[lk,c9 BEI c]t BAD 9b’j 7hkp7ndaoF g-aoj fP;dao (AB) //(CD) F I6[lk,c9 ICD c]t IEB 7hkp7ndaoF 0D 7yfw]]j ;’pk; EA, IE c]t IBF { .oI6[lk,c9 BAD F gIqk,u EA = CD = 5 cm c]t EB =15 − 5 =10 cm gIqkwfh IE = EB s^n IE = 10 ltoAo gIqk4vowfh IE = 10 ×12 = 8 cm AD AB 12 15 15 225

IB = EB s^n IB = 10 4vowfh IB = 16 ×10 = 32 cm BD AB 16 15 15 3 7D 7yfw]]j ;’pk; IC c]t ID I6[lk,c9 ICD c]t IEB 7hkp7ndaoF gIqkwfh IC = CD s^n IC = 5 4vowfh IC = 5×8 = 4 cm IE EB 8 10 10 c]t ID = BD − IB = 16 − 32 = 16 cm. 3 3 #D [qfg/ydsaf A !D 1kdcmdw]ptc8j A sk B g-y’j [+vkfmedko ;afcmd3fpdq’wfhF gryjogoaj’mvj o-nj [BC]F wfh BC = 90m F cmd,5, B c]t C D 9kdoAo gryoj c8h,I6[lk,c9 A'B'C' .lgj 9hpg-yj’,u B'C' = 90 mm, B' = B, C' = C F 9kdoAo B C gryoj cmd A'B' D 4hk A'B' = 225 mm F 9qj’-vd];’pk; AB. ສະຫງສ.ວວນ.ລິສຂະສິດ @D A c]t B c,jolv’g,af0v’glAoc[’j g7y’j ,5,0v’,5, xOy 3fp;jk OB = 2×OAD 9kd A F gryoj 0ufglAo-n8j A’lkdda[0hk’ [Ox) 1j6 H D9kd B F gryjo0ufglAo-nj8A’lkdda[ 0hk’ [Oy) 16j K D 9qj’-Uc9’;jk BK = 2AH c]t OK = 2OH D x H B O A Ky B D A C #D .shI6[lk,c9 ABC lkd1j6 A D [AD] c,jo];’l6’muj 0uf9kd9v, A D dD 9q’j -Uc9’;jk ABC F ABD c]t ACD gxaoI6[lk, c97hkp7ndaoD 0D 9q’j 0Povaf8klj;orq;rao0v’0hk’mudj q’daoc8j]t76jD 226

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