bMElgkñúgbøg; kñúgemeronenH eyIgsikSaEtEpñkFrNImaRtsuT§ eRBaHfaFrNImaRtviPaKmanenAkñúgesovePAfñak;TI 10

emeronTI 15 rYcehIy . eyIgelIkykrUbTaMgLaymksikSakñúgeKalbMNg kMnt;)anbMElgkil bMElgvil bMElgcaMg rYcnaMeTArkkarsÁal;Bakü bMElg . cMeBaHbMElgqøúHeFobnwgp©it eFobnwgG½kSeyIg)ansikSarYcehIyenAfñak;TI 9 . I- esckIþepþIm

]TahrN_

etIrUb F' )anBIrUb F edayrebobNaEdlgayCageK? sisSnwkeXIjdl;karrMkil . enAelIkþarexon RKUlubrUb F'

rYcsYrsisSfa ³ etIeKRtUveFVIdUcemþc edIm,IGacKUs )anrUb F' eLIgvij ? cMelIy Edlcg;)anKW ³ epþImecjBIviucT½r . eKnwgGackMnt;)an niymn½y bMElgkil ¬esovePAsisSTMB½r 196¦

A

B

A'

B'

O O'

A

B

C

A'

B'

C'

rUb F rUb F'

]TahrN_ 2

etIeK)anedayrebobNaBI A eTA A' BI B eTA B'

BI C eTA C' ? ENnaMR)ab;sisSfa ³ O CacMnucsMxan;mYy EdlRtg;enaH eKmanmMu 90o

eGaysisSpþÜl)a:NUeTAdI ¬BITisQreTATisedk¦ . sisSGaceFVI)an 2 Ebb sisSGacpþÜleTAxagsþaM rW eTAxageqVg .

eKnwg)anniymn½y bMElgvil ¬esovePAsisSTMB½r 201¦

]TahrN_ 3

[AB] nig [A'B'] CaGgát; 2 Edl (AB) // (A'B') ; AB = 2cm ;

A'B' = 3cm ; (AA') nig (BB') RbsBVKñaRtg; O . KNnargVas; OA , OA' , OB , OB'

eRbobeFobpleFob ³

OA

OA' ; OB

OB' ;

AB

BA ''

etIsmPaBénpleFobenH eFVIeGaysisScab;GarmµN_ rWeT? ¬mincab;GarmµN_eTeRBaH OAB∆ dUc ''BOA∆ ¦ .

C' B'

A'

O A B

C

rUbTI 1

rUbTI 2

O

A A'

B'

B

O rUbTI 1

ABC nig A'B'C' CaRtIekaNBIrEdl ³ (AB) // (A'B') ; (AC) // (A'C') ; (BC) // (B'C') ; BC = 2cm ; B'C' = 3cm

(BC) nig (B'C') CabnÞat;RtYtsIuKña (rUbTI 2)

(BC) nig (B'C') CabnÞat;epSgKña (rUbTI 3)

eGaysisSsegáteXIjfa ³ RtIekaNTaMg 2 manragdUcKña yk O CacMnucBiessmYy enAelI (AA') pg nig (BB') pg . eyIghak;dUcCaeXIjfa (CC') kat;tamcMnucBiess O enHEdr . eGaysisSvas; AB , A'B' , A'C'

rYceRbobeFobpleFob ³ AB

BA '' , AC

CA '' ,

BC

CB '' .

etIGñkeXIjdUcemþccMeBaHpleFob ³

OA

OA' , OB

OB' ,

OC

CB ''? ¬rUbTI 3¦

eyIgeXIjmancMnuc O nigpleFob 23 .

enAelIkþarexon ebIeKlubrUbRtIekaN A'B'C' etIeKRtUveFVIdUcemþc edIm,I)anRtIekaN A'B'C' eLIgvij? ¬edaymancMnuc O nigpleFob

23 ¦ .

eKGackMnt;nUvniymn½yén bMElgcaMug ¬esovePAsisSTMB½r 203¦ .

A

A'

C' B' C BO

A A' O

B

B' C'

II- sBaØaNbMElg segátrUb I , II , III nig IV . rkviFIEdlGaceGaybþÚr)anBIrUb eTArUb ?

I II

III IV

viFITaMgenHehAfa bMElg . cUrKUstarag nig R)ab;Fatu sMKal; bMElg EdlmanenAkñúgrUbTaMg 4 . cMNaM ³ bMElgqøúHp©it O KWCabMElgmYy ehIyCabMElgTaMgp©it O pleFob -1 rW k¾CabMElgvilp©it O mMu 180o . III- lkçN³Edl)anGMBIrUb - rUbEbøgénbnÞat; )(∆ KWbnÞat; )'(∆ . kñúgkrNIbMElgkil bMElgcaMg bMElgqøúHeFobnwgp©it eK)an )'//()( ∆∆ . - rUbEbøgénGgát; KWCa Ggát; . - rUbEbøgénrgVg; KWCa rgVg; . bMElgkil bMElgqøúH eFobnwgp©it bMElgqøúHeFobnwgG½kS bMElgvil KWCabMElgEdl rkSacMgay EdleKehAfa {bMElgb:un } bMElgcaMg minEmnCabMElgb:uneT elIkElgEtkñúgkrNIedayELkBIr ³ ¬TMB½rTI 20¦ - rUbEbøgéncMnuckNþalrbs;Ggát;mYy KWCacMnuckNþalénGgát;Ebøg . - rUbEbøgénbnÞat; )(∆ tambMElgvilmMu 90o rW -90o KWCabnÞat; )'(∆ EdlGrtUkUNal; nig )(∆ .

IV- sMNg;rUbPaB sg;rUbEbøgénrUbenHtam ³ - bMElgkilviucT½r

ADtAD =

- bMElgqøúHeFobnwgG½kS (ED) = s(ED)

- bMElgqøúHp©it D = sD

- bMElgcaMgp©it D pleFob 21−

−=21

,Dh

- nigbMElgvilp©it E mMu 120o = s(E , 120o)

enHKWCarUb nig rUbEbøgrbs;vatamkrNInImYy .

A B C

D E

F

A B C

D E

F

A' B' C'

D' E'

F'

rUbTI 1

A B C

E

F

F'

A' B'

C'

D

rUbTI 2

A B C

E

F

D

C' B'

A'

F'

E'

rUbTI 3

V- Gnuvtþ ³ lMhat;TI 1 ³ ENnaMeGaysÁl;bMElg eKGacKUrrUbelIkþarexon ehIysYrsisS . ABCD CakaerEdlmanp©it O . I , J , K , L CacMnuckNþalerogén [AB] , [BC] , [CD] nig [DA] . cUreGayeQµaH nig FatusMKal;énbMElgxageRkamEdlkMnt;eday ³ 1- A a I nig O a J (bMElgkilviucT½r AI )

2- B a B nig A a C (2 cMelIy = s(BD) nig r(B , 3O))

3- K a C nig O a B (bMElgcaMgp©it D pleFob 2)

rkrUbPaBén L tambMElgenH? 4- A a B ; D a C nig O a O (bMElgqøúHeFobnwgG½kS (IK))

rkrUbPaBén J tambMElgenH ? 5- A a B ; B a C nig C a D (bMElgvilp©it O mMu -90o)

rkcMnucEdlmanrUbPaBCacMnuc A tambMElgenH? ENnaMeGaysisSsÁal;Bakü FatuedIm rW rUbedIm . rkFatuedImén J ? rkFatuedImén K ? rkrUbEbøgén L ?

B'

A'

C'

D'

F'

E

F

A B C

D

A B C

F

E D E'

F'

A' B' C'

rUbTI 4 rUbTI 5

A I B

J L

D K C

6- A a C ; B a D nig I a K (bMElgqøúHeFobnwgcMnuc O)

rkrUbEbøgén O tambMElgenH ? ENnaM ³ eGaysisSsÁal; cMnuc\tERb 7- A a D ; K a I nig B a C (bMElgqøúHeFobnwgG½kS (LJ))

rkcMnuc\ttERbénrUb ? lMhat;TI 2 ³ kareRbIbMElgkil ABCD RbelLÚRkammYy . M CacMnucmYyénbøg; . bnÞat;RsbeTA nwgbnÞat; (AM) kat;tam D RbsBVnwg bnÞat;RsbeTAnwg (BM) kat;tam C Rtg; N . 1- etIeKTayfa bMElgGVIEdlEbøg RtIekaN AMB CaRtIekaN DNC ? (cMelIy

ADt )

2- RsaybBa¢ak;fa ³ ADMN = ENnaM ³ eRbI

ADt

AD

t )(∆ = D nig (AM) // (DN) ; dUcenHrUbEbøgén (AM) tambMElgkil AD

t KW (DN) .

ADt (B) = C nig (BM) // (CN) ; dUcenHrUbEbøgén (BM) tambMElgkil

ADt KW (CN) .

M sßitenAelI (AM) nig (BM) ; dUcenHrUbEbøgrbs;vatambMElgkil AD

t k¾sßitenAelI (DN) nig (CN)

Edr . snñidæan ³

ADt (M) = N nig ADMN =

lMhat;TI 3 ³ sg;rUbEbøg B' éncMnuc B tambMElgcaMgp©it O nigkMnt;eday A manrUbPaB A'

C

N

M

B A

D

1- B minsßitenAelIbnÞat; (OA)

etIcMnuc O , B , B' manTItaMgdUcemþc ? etIbnÞat; (AB) nig (A'B') manTItaMgdUcemþc? eGaysisSsMKal;fa Cajwkjab;kñúgFrNImaRt cMnucmYyRtUv)ankMnt;edayExSekag 2 RbsBVKña . 2- B enAelIbnÞat; (OA)

ykcMnuc C NamYyEdlminenAelI (OA)

ehIyeK)andUckrNI 1

lMhat;TI 4 : karkMnt;p©itbMElgvil

eKeGayGgát;BIr [AB] nig [A'B'] Edlman RbEvgesµIKña . eK]bmafabnÞat; (AB) nig (A'B') minRsbKña . kMnt;p©it O énbMElgvilEdlEbøg A eTA A' nig B eTA B' . ENnaM ³ eRbobeFob OA nig OA' ,

OB nig OB' , O sßitenAelIemdüaT½rén [AA'] nig [BB'] .

kñúgrUb 1 : emdüaT½rTaMgBIrenHRbsBVKña eK)ancMnuc O .

kñúgrUb 2 : emdüaT½rTaMgBIrRtYtsIuKña O CacMnucRbsBV én (AB) nig (A'B') .

B

B'

A A' O

B A' B' A O

C C'

B A

A'

O

B'

rUbTI 1

B B'

A A'

O

rUbTI 2

lMhat;TI 5 ³ edayeRbIbMElgcaMgrkPaBrt;Rtg;CYréncMnuc ABCD CactuekaNBañy Edl (AB) // (CD)

eKtag I cMnuckNþalénGgát; [AB]

J cMnuckNþaléngGgát; [CD]

E cMnucRbsBVrvag (AD) nig (BC)

F cMnucRbsBVrvag (AC) nig (BD)

segátrUb ³ E , F , I , J hak;dUcCart;Rtg;CYr . RsaybMPøWfa E , F , I , J rt;Rtg;CYr ? ENnaM ³ KiteTAdl;kareRbIbMElgcaMg tag k1 bMElgcaMgp©it E EdlEbøg A eTA D . rkrUbEbøgén B tam k1 ? rUbEbøg B RtUvsßitenAelI (EB) nigenAelIbnÞat;Rsbnwg (AB) kat;tam D . dUcenH k1(B) = C . rkrUbEbøgén I tam k1 ? I CacMnuckNþalén [AB]

rUbEbøgén [AB] tam k1 KW [DC] ; J CacMnuckNþalén [CD]

dUcenH k1(I) = J

eKsnñidæanfa E , I , J rt;Rtg;CYr . eFVIdUcKñacMeBaHbMElgcaM k2 p©it F EdlEbøgBI A eTA C edaynaMeGay)an F , I , J rt;Rtg;CYr . CaTIbBa©b;eKsnñdæan E , F , I , J rt;Rtg;CYr . lMhat;TI 6 : kareRbIbMElgvil ABC CaRtIekaNsamBaØmYy . xageRkA RtIekaN ABC eKsg;RtIekaNEkg sm)at BAD nig CAE EkgRtg; A

¬emIlrUb¦ segátrUb ³ (AI) hak;dUcCaGrtUkUNal; nig (BC) , BE = CD nig (BE) hak;dUcCa GutUkUNal; nig (CD)

• eGaysisSKitdl;kareRbIbMElgvilmMu 90o

ebIbMElgNamYy ?

B

C J D

A

E

I •

•

F

B

D E'

I

E

C

I' A

1- RsaybMPøWfabnÞat; (AI) GrtUkUNal;nig (BC)

edIm,IeTAdl;eKaledA eyIgRtUvecaTsYrCasMnYrdUcxageRkam yk r CabMElgvilp©it A mMu 90o

a- rkrUbEbøgén C tam r ? rkrUbEbøgén D tam r ?

b- sg;rUbEbøg E' én E tam r . etIcMnuc C , A , E manTItaMgdUcemþc ? c- sg;rUbEbøg I' én I tam r . KNnargVas;mMu ?

d- ehtudUcemþc)anCafa I' CacMnuckNþalén [E'B] ?

e- etI [AI'] CaGVIcMeBaHRtIekaN E'BC ?

snñidæan ³ )()( BCAI ⊥ 2- bgðajfa CD = EB nig (CD) ⊥ (EB)

ENnaM ³ edayeRbIbMElgén r dEdlniglkçN³BIrrbs;bMElg r CabMElgb:un nigrUbEbøgénbnÞat; )(∆

tambMElgvilmMu 90o CabnÞat; )'(∆ GrtUkUNal;nigbnÞat; )(∆ . lMhat;TI 7 : kareRbIbMElgqøúHeFobnwgG½kS ksikrmñak;manpÞHenARtg; A' Ek,rRbLaymYy nig cMkarrbs;Kat;enARtg; B . Kat;cg;dgTwkBIRbLay eTAeRsacdMNaMkñúgcMkar . mat;RbLayenHtageday bnÞat; (d) . etIpøÚvNaEdlxøIbMputsMrab;Kat;ecjdMeNIrBIpÞHeTA dgTwkykeTAeRsacdMNaMkñúgcMkar . ENnaM ³ eRCIserIsykpøÚv AMB NamYysin rYceyIgRtUvKitfaetI M RtUvenAkEnøgNa edIm,IeGay AM + MB xøIbMput ? . eRbIbMElgqøúHén B eFobnwg (d) (rWbMElgqøúHén A eFobnwg (d) ) . B' qøúHén B

eFobnwg (d) . enaH AM + MB = AM + MB'

AM + MB' xøIbMputkalNa A , M , B' rt;Rtg;CYr. dUcenH pøÚvEdlxøICageK KW AM 1B .

IAI'

B

A

B'

M M1

(d)

(d)

B

A

lMhat;TI 8 cMeNaTsMNg; [ox] nig [oy] CaknøHbnÞat;BIr . A CacMnucmYysßitenAkñúgmMu xoy . 1- sg;rgVg; (Γ)b:HeTAnwg [ox) nig [oy)

ENnaM ³ p©it S énrgVg; (Γ) sßitenAelIknøHbnÞat;BuH [oz) énmMu xoy . eKeRCIserIs S NamYyelI [oz)

. tag H CacMeNalEkgén S

elIknøHbnÞat; [ox) . rgVg;p©it S kaM SH b:HeTA nwg [ox) nig [oy) . ¬cMeNammancMelIyrab;minGs;¦ 2- sg;rgVg; (TMB½r27) kat;tam A b:HeTAnwg [ox] nig [oy)

eyIgnwgeRbIrgVg; (Γ) b:HeTAnwg [ox) nig [oy)

viFITUeTA ³ k- ]bmafa (...) CargVg;sg;rYcehIyepÞógpÞat; nig lkçN³énsMnYr . eKsegáteXIjfa (...) CargVg;p©it I b:HeTAnwgknøHbnÞat; [ox) nig [oy) Edlkat;KñaRtg; O ehIy A CacMnucmYyenAelI (...) . etIp©it I énrgVg; (...) enARtg;Na ? eyIgRtUvsg;rgVg; (Γ) Edlb:HeTA [ox) nig [oy) .

O

H

S

x

z

y A

(Γ)

O

A1

A

I

S

H x

z

y

(Γ)

I RtUvsßitenAelIknøHbnÞat;BuH [oz) énmMu xoy . (TMB½r28) CarUbEbøgén (Γ) tambMElgcaMgp©it O . A CacMnucEbøgéncMnuc A1én (Γ) Edl A1 sßitenAelI (OA)

I sßitenAelIbnÞat;EdlKUsecjBI A ehIyRsbnwg (SA1)

x- sg;rgVg; (...) - KUsknøHbnÞat;BIr [ox) ; [oy) nigcMnuc A mYy - KUsrgVg; (Γ) b:Hnwg [ox) nig [oy)

- KUsbnÞat; (OA) Edlkat; (Γ) Rtg; 2 cMnuc A1 nig A2

- tam A KUsbnÞat;Rsbnwg (SA1) . bnÞat;enHkat; [oz) Rtg; I1 . rgVg; (...) p©it I1 kaM [I1A]

CargVg;b:HeTAnwg [ox) nig [oy) ehIykat;tam A .

- tam A KUsbnÞat;Rsbnwg (SA2) . bnÞat;enHkat; [oz) Rtg; I2 . rgVg; (...) p©it I2 kaM [I2A]

BitCargVg;b:HeTAnwg [ox) nig [oy) ehIykat;tam A .

K- BiPakSa GtßiPaBén (...)nigcMnYnéncMelIy ebI A sßitenAxagkñúgmMu xoy enaHbnÞat; (OA) kat; (Γ) )anBIrcMnuc . cMeNaTmancMelIyBIr .

O A2

A1

S I1

I2

x

z

z

z (Γ)

A

lMhat;TI 9 : kareRbIbMElgvil ABCD CakernwgmYyEdlmanp©it O nigRCugrgVas; 3cm . OIJK Cakaercl½tCMuvij O

RCugénkaerenH manrgVas; 4cm .

bgðajfaRklaépÞrYménkaerTaMgBIrenHmantMélefr . eGaysisSkat;RkdasragkaercMnYnBIr EdlRCugman rgVas; 3cm nig 4cm ehIyeGaysisSbgVilkaer OIJK CMuvij O . eKeXIjmanTItaMgBiessBIr ¬rUbxageRkam¦ kñúgkrNITaMgBIrenH RklaépÞEpñkrYmesµI

41 RklaépÞénkaerEdlmanRCug 3cm .

krNITUeTA KMnUrbMpussisSeGayeRbIbMElgvil r p©it O mMu 90o .

eKehA P cMnuckNþalén [AB] ehIy Q CacMnuckNþalén [BC]

1- rkcMnucEbøgén P tam r ? 2- rkcMnucEbøgén E tam r ?

B

K

J

Q

F

C D

A

O

P

I

B

C D

A

O

K I

J

D C

O K

A B

I J

ENnaM ³ E sßienAelIbnÞat; (AB) nig (OI) . cMnucEbøgrbs;vatam r RtUvsßitenAelI (BC) nig (OK)

EdlCarUbEbøgerogén (AB) nig (OI) tam r . eKsnñidæan r (E) = F

3- rkrUbEbøgénRtIekaN OPE tam r ?

cMelIy ³ RtIekaN OQF RtIekaN OPE nigRtIekaN OQF manRklaépÞesµIKña 4- rkRklaépÞEpñkrYmrbs;kaerTaMgBIr ? EpñkenHmanRklaépÞesµInwgRklaépÞkaer OPBQ

KW 2

49

cm rW 2,25 cm2

VI- bNþak;énBIrbMElg ABCD CakaermYy . I , J , K nig L

CacMnuckNþalerogén [AB] , [BC] , [CD] , [DA]

ehIy O Cap©iténkaer . yk t CabMElgkilviucT½r AI yk s CabMElgqøúHeFobnwgcMnuc O yk S CabMElgqøúHeFobnwgG½kS (IK) yk r CabMElgvilp©it O mMu 90o

yk h CabMElgcaMgp©it A pleFob 2 1- rkcMnucEbøgén L tam t ? rkcMnucEbøgén O tam h ?

COL ht →→

eKfa C CacMnucEbøgén L tambNþak;énbMElg t nig h eKsresr ³ h [t(L)] = C rW (hot)(L) = C

hot CabMElg : CL k→

2- rkcMnucEbøgén L tam h ? rkcMnucEbøgén D tam t ?

KDL th →→

t[h(L)] = K rW (toh)(L) = K

bMElg toh tageday : KL k→ '

*** CaTUeTA 'kk ≠ rW tohhot ≠

R

R'

B I A

L J O

C K D

3- cUrbMeBj a- ...... →→ rSB (roS)(B) = ... b- ...... →→ srI (sor)(I) = ... c- ...... →→ ssJ (sos)(J) = ...

cMeBaHcMnYnsMnYr c etIlT§plvaBitcMeBaHRKb;cMnuc rW eT ? ¬)aT¦ d- ...... →→ SrB (Sor)(B) = ...

cMeBaHsMnYr d etIlT§plvaBitcMeBaHRKb;cMnuc rWeT ? ¬eT¦ cUrrk]TahrN_pÞúy ? 4- cUrbMeBj (tos)(I) = ... (sor)(A) = ... (Sor)(I) = ... (Soh)(L) = ... (rot)(D) = ...

5- a- k1 CabMElgEdl k1(D) = A

bMEbk CaBIrbMElg ¬bMElgEdlmankñúgbBa¢IxagedIm¦ cMelIy 1 : = sor cMelIymYyeTot = Sos

b- sMnYrdEdlcMeBaH Edl (B) = D

cMelIy 1 : = ror cMelIymYyeTot = roS

c- sMnYrdEdlcMeBaH Edl (A) = A

cMelIy 1 : = SoS cMelIymYyeTot = sos

d- sMnYrdEdlcMeBaH Edl (A) = K

cMelIy 1 : = tor cMelIymYyeTot = sot

R1

R1 R1

R2 R2

R2 R2

R3 R3

R3 R2

R4 R4

R4 R4