7 razred - obradovic - geometrija - udzbenik

8/12/2019 7 razred - Obradovic - Geometrija - udzbenik

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PAAOJE OEPAAOBHh

TEOtUlETPUIJA34 VII PA3PEA OCHOtsHE I.IIKONE

[email protected]



fPA3PEA.OCHOBHE



Perle3eHrn

,{p llBAlf BAHAI4C, npooecop yHrrBep3rrT€ray Beorpaly.

Ap BEJIIM'IP nEHAB'IH, caB€|,rHnK llor<pa-Jrttciaof 3aaoAa y HoBoM caay, a

CI{MA UPEPAP, caBe"nrK Mebyo[urrnEcKor3aBOAa y HO8OM caay

Oqersrnaqxa lpyra 3a MarexarrEy SaBolra 3a r3aaBarseyuoeFrEa Coqrjanrcrrtfre PetryonrKe Cp6rj€:

,4p CJIOBO.4AH AJ[,AIiq,1A. npoQ€cop yHrB€p3naeray EeolpaAy; Ap tiAAH EAIIATIE, rpo$ecop vrltnEep3'rrera yEeoTpaly; Ap MI{JTOPAA BEPTOJII4HO. AoiIeITT IIptrpoAlIG'.

Oaxynrera y Eeo.paAy; TIUIUA IIEIIII(OB,(E,npocaersu caBeTsnE 3asoAa 3a cTpyqno o6pa3oBane Coqilja-rrcrrqKe Penyorrr<e Cporie y EeorpaAy; Ap BE'I]tldI4P [E-HAISnI{, rpocBerHn caserHr.rr no(palrEcxor 3aBoaa' y HoBoracaay; MrlJIoPA,4 IIETPiOBI{1, BacraDExx OcrroBEe ruxore,,Bypo SaKoErh" y EeorpaAy; Ap CJI-{BrIIUA UPEI1I'fE. Ac-qesr [pxpoAso-MareMarx'rxor OaKynrera y Eeorpa4y; EO-TOJEyE CTAIIOJEBII'E, rpocB€rr.r caBersxr 3aBoAa 3a oc-HoBHo o6pa3oBalle t{ obpa3oBaBe HacraBqrBa CoqrjanrrcrrrqtePeny6nrKe Cp6rje y teo.paayl PATOMTIP CTE<DAIIOBI,fE,npooecop Bxru€ reAaroruKe ulKore y KparyieBrry; Ctlf,OqPEE AP, nr,ocBernu

caEersrtl. Mebyorurnscxo3aao.qa y IIo-

BoM CaAy r MI4PKO lul4JAlIKr , lpocEe rtr caBernrE 3aBo-aa 3a ,3aaBa6e yu6est{xa Coqrjanrcr,,l'rxe Peny6rrrre Cp6r-je y EeorpaAy.

3a r3Aaaaqa qpazocJlaa Joxoa4h, TexHrqKr, ypeAEdK Bopbe llonosul,KoPutle Mu^oul Puctuh

npocBerH, caBe4 CoqzjanrrcrrqKe peny6r[Ke Cp6rie o,{o6prao je oBaj yu6eE K3a ynoTpe6y y ie,qMoM pa3peay ocHoBse luKore csojrM peruebeM fI. c.

. 6p,aj 271/7 oA 31. xapra l9?0. roArEe,

IIITauna: Ipaou.rKo [peAy3ehe ,,Pa.qrua TnMorrh", EeorpaA, JaKrtlr,llreBa 9,



IMHOTOYT'AO

I. TIOJAM MHOTO}TJIA

Tpoyrao r qerlopoymo cy, xao ETo cMo rtrIeJrn, 3arBopene tr3Jrolr:beEe

.wje, o4 Tpf, oglocgo verxpx ,q6r. Ap ff ua trexxM npe,uvrerrMa MoxeMo

3aIldtETf, 3arBolreEe t'JJroMirene :rnurje x ca Bnrre o.q rernpx ,4nxn? florure-lajrro ueleEe crxre: sa ttrMa cy nlreAcraEJreBr{ yna3 y Earop (or. l), rauenEsreca.f, 3a noctorre (c;r. 2) x npegrr geo jeggor rpora (cl. 3). Ha omxrnrr

f, ClItqf,trM npe4rfeTtrMa lrAe Ce 3aTBopef,e I3JIoMJbexe nIrEuJe II oA BtrUle OA

rGrxpx Ayxr.

3araopexa r3.'roMlbeHa fiulruia y paBnx oA l.pu utn Bl4rue Ay)ro{ 3oBe

ce rEomyrao. fiy>xu ns xojrax ce cacfoj, x3roM"'beHa .rr,rnrja jecy crpa-rnqe MHoroyrra, Kao turo cy Ha cJIxIlu 4 4y>xu AB, BC, CD, DE, EA.3ajegnr,rvxe rayKe aBejy crpaxrlqa cy re ena MHotoyr[a (4, B, C, D, D uac;sagr), a cBaKe ABe cyceAne crpaHxlte (rj. orre xoje uuajy :ajeaar'rvxeTexe) oApebyjy yrao Mnoroyrna (xa npuuep I e, I B urt.).

Ilocrr,rerpajrrlo MroroyrtoBe rrzl cr[raMa 5 r 6. Ilpar o,q nnx rnvra c;reAehy

m6my: aro ce Ma roja og rrerorrn crpaHfiqa npo,uJxrr na o6e crpaxe a

raro ao6rje npara rojoj Ta crpa {qa ripntr.aaa, oxra he ce qeo r'floroyrao

f,aJra:rmf, ca je.rye orpare re trpare. To je rcomercar .Mloroyrao. -Mxoroyrao

Cr. I Cr. 2



Ea clrqu 6 aeua ry oco6try, on xnje romercan (nerornercan je) MHoroyrao,V Aa;rer.a xsJrarary yanraherrro y o6szp Snrarxorra r(oHBeKcHe MxoroyrnoBe,rra ieMo noA pew ,,tfforoyrao., .- axo Apyr&je rre ncrargeMo _ [o,qpa-3)M€BaTE yBei roEBexcaE

ACrr. 4

BpojaEeM ce.3a&rbyqyje Aa cDam rrerBopoyrao sria. r{stz 6poj crpaurqarao I )lrnoD:r, olsocEo TeMeEa.

Gr.5 Crr. 6

Tpoyrao r llerBopoyrao 6y Mf,oroyrnoBx ca 3 o.6ocuo 4 crpamqe.Mroroyrao ca 5 crpar4a 3oBe c€ nerolrrlo, ca 6 crpaqa

- uecroyrao,f,4l. [perta roue, cryny MEoroyrnoBa [email protected] roAcryloBr rpoyrnoBa,qcTBopoyrnoDa, treToyDlora ETA,

.{co parrz orpaf,sqer srpaxqaMa MnoroyrJra je noupu Mxoroyrna, arcrrEa re noDprnf, je uorylurrr Mf,orojrrna.

4

cJr. 3

Crr. 6



MsoroFao o6erexaBaMo raroo6ruo xroJrtrrlM cJloBoM

AKDEF.

rtrTo r.fy sraro rer,re o6orexnrro je,runr

a6eqeAe. Taro je Ea cJ@ 7 o6eneles

Ctr. 7 Cn- 8

IIo4 4rjaroxa:roM MEoroyrna IroApa3WeEaMo ryx roja claja .ua rra

rerrexa xoja se npzlaAajy rcroj crpaqrr{u. Ocan y Tpoyrry, ,ryjarouanc

xory noryhr y cBaRoM Mf,oroy$ry, r ro f,3 Ma rojer reronor rcrrleua. Taxo,

EToyrJry ABCDE (cn. 8) troEysef,e c1 g4iaroxaae lD n AC B reMeEa l,Eecroyrny MNPQRS (c.lr. 9) ,ujarosane MP, MQ, MR url. f,3 reMega M.

CJL 9



2. YTJIOBTI MHOTOYTJIA

Ilor.naqereu pajarona:ra B jemor reMef,a noAelruJrx cr{o treroyrao Ea

3, rj, 5-2 rpoyua, a rucoroyrao na 4, 11. 6-2 tpyvn, Hacraanrajyhn xa ucrlEarrxE AeJbeE€ tr Apynm MxoroyrnoDa, yEtraenr 6f,crrlo .4a ce cBaKtr Msoroyrao

Moxe troAetrrr ,rgjaroxa;raIraa, rroDy{egnrr m je,qEor reMena, f,a osormo rpo-yrnoBir rormo usuocr 6poj TeMeEa Jrrr,raE€E sa 2, ro jecr, aro ca n o3saqnuo

6poj reueua, Ba n:-2. Ca6npajyhn caAa cBe yflroBe y cBrMa rnM rpoyrnoBtrMa

.qo6rhevo s6np yraona y Msoroyuy, xao rtrro noraoyje clnxa I0.

Cn. 1l

Ilom'ro je s6np yrnoBa y rpoyrrry 180', r6rp yrrroBr y Mxoroyrny roSf,iauo

xag 180' rroMrrorsnf,o 6pojem reruena yuarsenou ea 2' rj. axo ca S osHa'r,tlro

s6np ymora, a ca n 6poj TeMeEa Mtroroyrna, nr{ahevo:

,S:180' (n-2).

TaKo, rra upzruep, s6rp yrtoBa y neroyrny tt3nocr{ 180" (5-2)::180'.3:540', y urecroyrrly fiA. rc-2\:720', t4rq.

Kao mro je 6ro cnyuaj roA Tpoyrna tr qerEopoyrna, n roa MloroyrJra

yonmre; yrao rojr rpagr npotryxerar jeaxe ua rojc crpaanqe ca cyceaxoM

crpaMqoM 30Ee ce cnoJLol[I[f, yriro MEoroyrna. Ha clqn ll T^r'aE je LXLM.

Ko,q craror reMeEa MEoroyrra Mory ce HaIIprarI{ IFa crloJBarl6a yDra, r oEEcy xao yEaxpcm je4rarr. Ca Ecre cJrr{re je osnrre.uro u ro ga je cBaRE ctro-

JbarrrBr yrao ynopeA:rE cyceAEoM yEyrpsIIIEeM yrIy, Drro 3Easr Aa Cy IG:YtrJIEMEETEtr.

6



3AAAr[t 3A SE?I{EAGE

L lktc Ec nlpDrgre xt Ko-

c-l

Egrr soroyraorr, Cra*ou.r

-rrrn

ariit sre npeMr 6poiy

t El'r*rt Jeaas ror €Kcal a je-

cer{oyrao, 6) ocMoFao.

a.rF r F ctcytr Mfloro]rnoBr 6ec-

A EEr cy cl{yfloBx ncienxn|{x

-qF afr,pe{aMo reroyr,roB:1,

a Srro ce yKyr|Bo A{jarolana

-t-t reroyrry, a KoJrf,Ko y [re-

H 'rrire r rrft,ojre.

5, Elct|ljTe r6f,D yinoa. y Mf,o-

* F 6poj reMera 9, 10, 16, 20,

- a rre m ie t6rp yrto oc-

-f

taL oA 3frDo yrnoBs [e.

7. Ha{rraire renr rrroroyrao r JqAE 'Ma Koiy, taqry y rreroroj yfy4uuEocrrcnoiTe ca cEiM reMemra. IloMolty rrrco

ao6riese c'ure rrpacy{aiTe t6f,p yflrorr y

ToM iiiomyrny. (flpnlrcsire $EuLe o yrno-Bmi& rpoyrua.) Ao6ri€M pe3yJrr.T qloEe-

Drre troMofv f,tBeAenor o6 6c[8.

8. Jeari yqedl( ,ao6ao je orar<ar

razarar: 36np yr:rora jearor Mxotlyrnr

ftrccrl I 80(Fi x3lt6{y{rrn 6poi crpoEqg

Tof MlOmYr,'lA.Or je raj 3&Aar4r Peuno otaKo: tro'

'treflo ie 6Doi 1800 ca 180,m ie nont.wq

(lo).aoaro 6poi 2 x ao6{o 12 crp.tFr"A Jfl ie of, Ta.lto Eanrto penre*e? 3aoro

ie raKo parxo? II&Dtre r Ex 6poi crlamnuaoroyr,ra-rxjx 36{ YrJroEa rcloq:

a) 2340", O 28E0', E) 540f.

9. 36rp yrnora Jefror ndoroYrJtl

r3f,ocr 18 nprlE yrnota. KoJtxto grDa$qa

or m.

4. IIPABI'TJIAH MHOTOITAO

Itrolennro rpyxffmry Fa tr3Becrag 6poj jegnarux ,{eJroBa' peqw{o Ea

rr. To rorcr,(o yqxHtrTtr Ea oreJlehn EaqItH. IIyx yrao c reMeHoM y qeETpy

c ,[e:tf,Mo Ea 5 je,quarn AenoBa, 3600:5:72" (cn. l2). Kpaul orntm ctry rpyxn{qy y ralmaMa Koje ,4elre'xpyxnrlqy raxole Ea 5 je4naxrx

.rr- (3au'ro? KixaB o,ryoc uocrojl u:vefy qeHTpaJrEor yrna tr 3axBahe'

r rtB?) Arojom ctrojnrro

,uyxtrI{aysacronnb ge-

c raqrq Ao6uheMo neroyrao ABCDE. Ilpoy'muoa r r4i trcToyrao.

Ilouro je.qrarnv JryroBr{Ma xpyxlrnqe npfina-

.ojy imare rerffBe, crparffrle ror rleroyrna cy Meby

oo6or jegaxe, gaue:

AB: BC:CD: DE:EA.

Epan E ooayaapH{, jep cer rvajy jeaaare yrrorerNf }pxy, a Rpa4r cy r.rM rrorynpe{Hnln xpyxHr{qe.

Itrmro ce csaxx yrao uocMarpauor rreroyrna cacrojl I,rs ,qPa yrna Ha ocuoBltqtr

m je.qnaroxpaxfir rpoyrnoBa, sarryryjevo Aa cy tr yrtoBlt leroyrna Meby

co6or jemaru:

Cn. 12



Ha f,srtr Earrf,f, MorJrr cvo go6urz a Apyrre MsoroyrnoBe c jeggaxtrMcrpawqalia f, je,Earnr{ FnoBnMa.

MnoroyrEo wjc oy crpaxzqe jegare z ),rnoas jegaxf, E:xrtrBa ce trpr-lxJnH MHOfOyrao.

ABa trpaBaJsa rr,noroyr:ra aeh cMo npoyqaBaJrr: npar&rrau rpoyrao fiega-f,ocTpaBEqaE TpoFao) E trpaBxJraE qerBopoyrao (xBaApar). ffparnnan r*laoro_yrnoDE Mory ce yoqnTx sa EeKtrM rrpe,q}reTr{Ma. Taro, ra cJtr u 13 npeAora-anena je onoEra trjf, je tonpewr trpeoeK npaBruraE ruegroyrao, f,a cnqtr 14trpeAcrar:reEe cy r:rorrzqe ua noAy y o6nury npaBrurgnx tterBopo)rnoEa nocMoyrnoBa,

IIpz omcmany t ymcf,Es*y kpy*Erqe xotr TpoyrnoBa EetrEx qerBo_poyrnoBa (rye je ro Moryhe), Ea) ra:rr cMo .oa troA onf,cruoM KpyxgqoM

Eo,qpDyMeEaMo oBy rpyrsrqy roja npola:n rpo3 cBa reuena rn Qrrypa,a noa yrrf,ct[oM rpyxEqoM ory xoja aolFpyje sBe slpastrqe rux Qmypa.

Cr. f3 Cr. t4

lI3 caMor saqf,f,a qprata qpaBnrxor Mroro)rrna _gelbebeM rpyxrtqe

aa jeEare trenoDe - npotr3Jra3tr At ce oro cB (or trpaDf,rEor Mf,oroynro Moxcotrf,ctTf, rpltrErll4

. , Iloclratpajuo caaa tro,ryrapse je,uEarorpare rpoyrnoBe, Kao mro c- ,ABO, BCO trr.q. xa cJrrytr 15. Bncme rnr rpoyrnoBa (OM:r) je,qnare cy

uefy co6ov, cBe rolra3e tr3 qerrpa onrcare rpy-xxnqe f, cuajajy raj qerrap ca cpeAr.ruaMa crpa_srqa Mloroyrna (sanrro?). To guartr aa MoxeMotr3 trcror qexrpa O, .qfxugoM nonynperrHnra,,

ynscartr RpJxrrqy y Msoroyrao. 3aruby r: y cBarcf,

trInBXJrlf, uf,Onoyrao More ce ytracsTf, rcpy:rxxqa.

Bwrs.a (OM:r) jeJFaRorparor rpoynta ABO3ole ce ttror€mt npaBtrnlor uuoroyma. Ko:rmoaIoTeMa trMa npaBr;r:lx z-rpoyrao?

Cr. t4

Cr. 15



Omcana z y\mca&a rpyxEqa trpaBtrJIEor Msonoyrna cy xouqerrpnEe.

EuoB qeFrap Jrextr oqEDreAEo y npec€ry csMerpara crpaErqa, oAEocxo

{nrrrpara yrnoBa. UeHTap oErDr xpyxErqa 3oBe ce tr qeBTrp MnoroyDr&

3a rsoroyrao ka:(eMo Aa je lucar y rpyxllr[y xoja je oxo E€ra olru-

cr-E, a .qa je omcar oro rpyxEtrqe roja je y Bera yntrcasa.O.EoBopn*{o jont xa rzrare: naro heMo Fparyxarn troJlntro {3Eoctr

ilpx oa ysyrpa Bux o.&ocso cnoJbamEx yrnoBa trpaBtrJIEor Mf,oroyrna.

Ilorro cy roA TarBor MEoroyula cBE yEyrparrlEE tr cBtr cIroJbarIIEu

lrrrn xely co6orra je4xarx, o,4uocur s6zp rpe6a uoge.nrru 6pojeu yrnora.

Er npnrep, npaBruIHIr ocMoyrao trMa yryrparrlsu yrao:

180' (8-2) _ 180'. 6 _ 135.,

88l crnsarrrnn (xojr je nevy ynopegan):

180"- 135':45'.

4. KOHCTPYI{qIIJA IIPABI{JIHID( MHOTOYTJIOBA

Koucrpyrqujol*r npaBn:uror treroyrra (c;r. 12) uora:a;rr cuo jegar ormrr

rw lorcrpyrurje rpaBltlsrD( MnoroyrnoBa xa4 je uoarat uotytrpetrxtrxmaxe rpyxurqe: rpyxumry rpe6a uo.qe;rnra Ea oaoJtffio je,lgarlx genora

DrIro uxoroymo uMa. crpanrqa, tra y3agrotrse aeose raqKe cnoju:m. V nernr

GrttatEsmaa oe oBo .EeJbeBe rpyxHrrqe xa je,unare genole uoxe o6arrrn catao

crapou x neBxpoM (6es yrurouepa nJrE Eetrmr npyrux onpana). BqqeheMo

GrrB larro ce To u3Bo,IE.

Ea rernpu jeasara aela Moxe ce Rpyxmqa noAetxru noBnaqegeM

.la refyco6xoEopMarHa trpeqlrEKa (AE s GC Ea cntru[ lQ' Cuajarerr ysa'

cmm rpajera oBtx rpestrma ,qo6njaMo KBaapar, Tj. trpaBrurntr qsrlopoyrao

(.ffi1.



Koncrpynmeuo ;rr ca,qa cuMerpane yrnora xoje rpage orx npequrrqn, AeJuMorpytFr.rty xa ocau jeEarax geaora (samro?), u cuajareu ysacrorrBux AeoHtrxrasara Ao6HjaMo npaBr-rnrrx ocMoJnao (ABCDEFGH).

Hacrarrajyhu oBa(aB rrocryna( tr ,qaJbe, yBtrbaMo ga ra raj EarmE MoxeMo

xoHcrpyrrcarx upaBrurrre MxoroyrnoBe ca 4, 8, 16,32 urn. crpasrya.Iloorarpajrr,ro ca,qa trpaBnm { [ecroyrao Ea cJruqfl 17. Ty je qenrpaJrxtr

yrao noAerleH na 6 jeAnarux AenoBa (360":6:60"), ua crarr o.q 6 Ao6trjeHtrxjeAxaxorpaxrx rpoyrnom nMa yrao nprr Bpxy 60' (xa npnraep IAOB). Kaxooy ro4 jeparorpaxr ( TpoyrnoBa yrnoBtr Ha ocnoBrrqtr je.qaru, n rtr yr:rolntr3xo@ tro 60' (aauro?). Orn rpoyruroru cy, naxle, jemrarosrpaurrytr, ApyrsMpeqtrMa: trorrytrpeqHnr re xpyxHtrqe je4xar je crpanEu trpaBf,rf,or uecroyrna.

I

I

Cn. 19

I,Is rora raxryryjeltc .qa ce noJrynpeqHux Moxe rao reruBa peusrl yKpytc@ rawo 6 nyra. Ory,qp oBo ytryrcrBo sa xoxcrpynlujy flpaBruigor

m€cToyrna:HaqpTa

ce rpyxHlrqa, na oe oaHexe

Eege TaqKe npenece noJry-trpe'nrrr Rao terrna 6 ryra, n go6rjeue AeoEe railxe oe cnoje. Axo ce nnarcuojx craxa Apyra aeona raqKa, oHAa ae ao6vja npaBuJran Ceguakocrpaxman)rpoyrao, rao IIrrc je A ABC (ctr. l8). (3amro cy srpaHqe AB, BC, CAjegnaxe? Ilocvarpajre :ryxoae HaA btrMa n o6jacxure).

,{urenev Jry(oBa EaA orpart laMa npaBrrnxor ru@Toyr:ra ua IFa je.wara

trera go6ujauo reMena trpaBr{Jrror ABaHaecroyrna, xao rrrro je norarauo xa

crxqx 19.

.Kao urro Br.rJIuMo, oBuM nocrynKoM MoxeMo Korrcrpytrcaru npaBIUrEe

MuoroyrnoBe rnju je 6poj crpaxr.rqa 3, 6, 12, 24 lrra.,

Iloor"rarpajlro caaa je,[an o.u, rroayAapr]D( jeruarorparnx rpoyrnoBa Ea

rojn je nogerer IIp:rBEnaH MHoroyrao, xao rrrro je L ABO (c;r. 20). O6e;

lo

,-l-/

-xr

Cr, 18



rrlllt ca 4,, EenoBy ocEoB6y, rj. crparfliqy 11paBllnf,or Mgoroyrna ca 11

crtll@E, a non)mpeqEsre ortrcaxe x yntoaHe rpyxa rqe ca .R tr t.

Cr. m

Yrao c, npu Bpxy ror rpoyrna rogxar EaM je snr.t 3HaMo KoJrf,ro crpa-ur (r) nra raj unoroyrao: q.:360"1n. 3a rosqpyrqujy ror rpoyrna no-

rpc6es j, jou cauo je.qar ereverr: R, wu r, s.nt an. KaA o€ TaKo rlpeMa

Jfa troAAqlll{a (oHoipyntr€ jegxaroxparn tpoyrao ABO, oxAa ooraje Aa ce

E renoror Bpxa O ollme rpyxHf,rla, noJrynpeqHfira OA, u.aa ce npeE@e

@@ AB xao rerwa n uyra. Ilorire,{ajMo ro Ea Eerr[r4 trprrMeptMa,

l) Koncrpyncrrr trprBf,n[f, .qeoeroyrsr, crl]aEf,[e ,ro= 15 nm.

Aa 6rcMo roEcrpyr{cilrrtr L ABO (ctr. 2l), na noryBpaBy lX upenecevo

ryl rare orpaurrqe AB:15 mm. Ilouro je 360': l0 : 36', yrao rtptr Bpxy

iparorparor rpoyrta rrcHocrr 36o, a yrnoau

- nf,{f, :ro 72". KonmpyrqtrjoM oBa apa yrta

@ u aa qra MopaMo rrprar[ yrnoMepoM) roA

t*n I u 3 .u,o6ujauo y npecexy Ibrr(oBlrx

flnlm qcxTap O onncaxe rpyxmle. Caa jom

{rarc ry xpyxxnuy n npuoonro 10 nyra

oqt, oaEocHo oTpaHtruy ato rao TeTtrBy,

rr m6xjarro rparreHtr npaBruIxu ,Eeceroyrao.

t) Xoncrpyf,carf, nprBf,.rrr ocuoyrao uJr rryacmc ymcaEe xpyxmqe r:25 mm.

y Eeroj ra$tr O (ueurap ocMoyrta) roEcrpynneMo yrao oi:45o (3alrro?)'

r E€roEy cfMerpiuy, r{ Ira By trpegeceMo ayxxEy Aarof rlonycp€qE ta'

\\\II

BOL 2l



OM:25mm. Y raw. M f,oEsrpynEeMo EopMany AB L OM, rge cy A u grpeoequ Te HopMaJre c rpaqtrMa yrta, lla, je AB crpaHxIla rpaxetror ocMoyrra

(ctr. 22), Oxo O ce oaAa ortr[rexpyxHuqa norytrpeqHtrxoM

OA- OB

r ocraje ga ce jom l8 npenece

xao rerma 8 ny.ra...

5. CI,IME'IPNIIHOCTTIPABI'JIHI,TX

MHOIOYTJIOBA

Korcrpynurn',ro xa xaprrjntrpaBlnEn neroyrao (ga .lrrr uo-XEMO TO II3B€CTtr C?MO III€CT&.

poM a neBnpoM t ,qa ntr je ro6am npara roucrplrquja?) tr npa-

ErJrf,E lrrecToyfao, rao rrrTo cy

Ea cnutr 23 neroyrao ABCDE,^Ea cJrqx 24 mecroyrao MNPQRS,

x f,celmllo rD( TaKo aa .(o6ujeMoMOAene TrD( MIIOTOyDIOBa.

Aro r,roge.r npaErJrxor treroyrna trpecaBxjeMo raro aa My trpetronoBtrMo,

aa trptrMep, lD, ro jwt npero oEMerp:ure D.F rora yrra, DHAeheMo ,qa he

ce ,qBe ronoBnue rot lr€ToyrJra raqxo rrourarrartr. Oao nora:yje ga je nparu.nutreToyrao oouo cnrrerpnura t$rrypa,

Ct- 23 c^n" 24

Kaxo cy ce Irptr oBoM orJre.ry noxJrorrure E troJroBtrfle AF v BF ctpa:EirleAB, \pu seuy je DF L AB, nponurast pt fD tr crpagnqa AB n4ajy 3ajeAll.nryctrMerpaJry (DI'). 3a raraB yrao n TarBy orpaEtrqr npaBrulHor Mr{oroyrrardKeMo aa ce f,aira3e je.uo uacuparlt gpyror.

Cn. 22



Ilorrfo cMo MoAen rrpaBrurEor neroyura MorJm npeqrBrrtr rrpexo ouMe-

rFne rra xojer yna (oguocuo rr.ra roje crpax4e), trpomra3s Aa os trMa trer

e cm,rarprje (ner onuerpala).

Aro cara Moaer rrp:rBuruor uesroyrna (cr. 24) npecarujer'ro najupe

TflD tranpenonorrnuo jegan yrao, peqnv(o

lM, asalrnv' raro

aaFopMaJroM

rp.[onoBruo jegHy srpaufry, peqxMo crpaH]rror Mlf, strrehe]ro oEo: y trpBoM

rr;rrajy rpeno:roalurtr cMo t lQ, y ApyroM tr crpanuqy QR, a y o6a uytajacs rxr:roBf,Ee rrrecroyrJra ravno uor:ranajy. ,{re crparnqe roje rurajy saje.urmy

Grpary sraher',ro mcnparrsrM crpaHqaMa, a ncro ra(o heuo z ,6a yraa

nii grajy rajemqqry oxMerpany 3Barff tractrptMdf,M yrnoBaMa.

- I[3 oBor orure,qa rtaoxertrd 3artb)nrnrE ga je nparu.naa mecroyrao ocso

o-rplwa rlrrypa u aa nr"rit mecr oca cnuerpuje.

Axo oBaroe orne,[e tr3BeAeMo I ca apyl'nv{ trp:lptr]rgtr}r,l MgoroyrnoBuMaa trapEnM E setrapHlM 6pojeu crpauua, gohn heuo Ao cae,4eher 3arbytlxa:

Ilprurr MtrorolrnoBn cy ocuo ctrMerpnsue {rrype, s 6poj cmterpana

bF n e 6pojy crpauqa" Aro je 6poj crpaqmta Eetrapag, ctrMerpate sy

-rorIEMeEo rr ctrMeTpane lrrnoBa n E ffMa Eacrlpatr,rnn< cTpau4a; a aro je

6pd crpasqqa napau, EeroBe cl{Merpalle oy trcroBpeMexo ctrMerPane BaonpaMFtr

tre, o.ryocEo ctrMeTpaJle Eacflp:lMEn oTpas4a.

llsrpmenr or.negu noraryjy jom r ro ,qa ce yonurE cBe ctrMerpanelFrrrFor MEoroyDra Cexy y jeruoj raqr. Ta rarra je qerrap npaBDlqor

rnroyrna ( e cnmaMa 23 n 24 ro cy rawre O z n.IIpEmoM npoyv:rB:rba cnaerpnje .napailenorpaMa (leouerprja :a VI

tr4'e,q) Hay{ M cMo Aa cy ,Be raq(e, peqn{o I r .8, cnvrerpnme y otrocy

n rlrhy ravry, peqnuo C aro je C cpe.nma gyxn AB. Tama d je oEra Bru(oB

-rrep cnrerpr{e. 3arnvr cvo straentr ,qa je nexa <frrypa qef,rpaJm cnMeTltr+ma

& cc 3a claKy rbeny ratrry Moxe sa ucroj ilurypn rahr gpyra raxa roja

i- rynoj cnuerprFa y o,uHocy sa trcrE qeETap. Tato cvo yaBpatrntr aa cy

Frp€rno ctrMerpr{rrue $urype rpyxrrua tr napaJleJlorp:rM: qen-rap cnrerpnje

ryfixrre je ner qenrap, a napanenorpaMa - trpeceK .ujaronana.

lcnrraheuo oaaa Aa JIU cy tr trPaBruIxtr MgoroyrnoBg qerTpaJruo cnMe-

4rw Qmype. Ogae vopauo ognojexo [ocMarpaftr rpaBtrnue Muoro)nnoBe

c ErraprxM tr oue c uapxmr 6pojeu crpaEqa. Ako y3MeMo y treroyl'ny

lrcDE @t. 23) pyx DF, u;m ua rojy Apyyrtrjtr cy rpajeuu ratrKe rreroyrrla

rroja npo.na:r (po3 Tall(y O, yrrgeheuo u 6es rrreperra,qa ra*a.O se

Enu re ayxl{, na sax.nyryjeuo Aa trpaBrJrEq treroyrao mje qeurpamro orue-

4ewa.Qn'ypa.Koa npasxnnor mecroyrJla, velyrnrr, qesraP Mnoroyua je r uerrap

*'rpde, nortrTo oH nonoBl{ cBary .ryx roja flponiu]R Kpo3 nera a rnjl cy

ryrrjcIll raqre Mnoroyrna. Oso ce Ivroxe yrBpAtrTtr n 6ee ueperra 3a ratrxe



MxoroyrJra Ea EeroBtrM ctrMerpanaMa. ffpem rorrae, npalnJrntr ureoroyrao jeqenrparso crverpnvua Qurypa.

I4cru sarJbjnrlx uory ce go6nru tr pa3MarparEeM Ma Kor Apyror MEoro-

yrna c EenapEtrM o.qHocHo c nap$l{}{ 6pojeM cTpanlrqa. 3aro sartyyyjerr,royotrIrrre: npegtrnru Muoroymo c trrpB

trdpojeu crparnqa je qerryrurxo

cnre-Tpxrf,8 Oryp{, ce qef,rpoM cnuerprje y q€f,Tpy MroroyrJra,

3AAAUU 3A BEXEABE

1l l&FaEyrajre ieApr lrao y rya-BxnEou: a) ocMoyrry, O Aecemyr"ry, r) rer-elqgToymy, r) DaAecEroyDry.

2. IloKar.rre m je jeA|x yrro rr q-

rEmor l2-royijra 3e 30' €nr oA jelf,orymr trDaBlljrEor 6-royrra.

3. JeAa[ ryo8xnfl MEonoyrao rMr

24 crp6nue. I{lpaqynalr€ lelAs 6eror tEy-TpqruEx yfao: a) lacyraiyhf, rairpe ymore

ieasor oA 24 Tpoyrm rojr srieno merEj nlarf,,m uBo.oyn|o, O rpGte6yjynto6pa a{ t& r6f,p yrnora y Mroroyrnt.

4. AIto trllaBf,JEo nrscroyrry nDo-

.4/rN@ro

4rrEecyceaf,e cll}ltrEte, clar(y Ea

odc grl} Ee, orA Ao6qiaMo rDa npecsar,

xojr oapcbyjy rpoyrao. flol(r*rte aa Je

rai rpoyrao learaxogrparncrf,.

5. Iferoyray ABCDE (c'.2s)f,@qpr s

Je xoA craror reMetra caMo ro jegar cno-

JioIlrrr yr'&o. Ilorylaire aa uqlaqyf,are

r6xp rl[ ctroJDatll&D( Fnola, (yri,B.rc yo6 rp Ir cy r3oA claKot tlcugEr qroJEuerI CyCeIf,t yqrrlnutEn yrao q/'Jr€Mef,rrr,n& 3pasyrajre 3ff,p Tu clloJralllEta f,

cr@. yrjnlnu&a yruolq tere,uo. mTs JouTF6r ysxul'?)

6, 3a,qaraK 5 D€nrrre raA le lorc- 'rao ca 6, 1, A, 9 . .. crDsmF, a lsrrnllEler[lro ollltrtx laKrLjFlaK o g6rDy cro-JlSiIltEX yrnotr MBofo 'fJtt Kr,q ce Koaqll((r relref, paryra cauo rc le4an

7. .{a rn mcroje nF f,Jrsr ulono.yrto qJ yryrDqeEf, yroo x Eo.u: a) 16(F,

Q 115", r) 179? (Korrrs 6f, 6{0 ctrorE$nErtrdo o$olrlstyier roomyraa? IIITa A.JEe

rD€6a lcFrr? BqMr€ pellleEe r.q. 6.)

8. Aro je ieAaE cuoJbauof, yrrorDalxltxor uBoF0ynnr 15o, xonsro rtloatr6rp beroru yf,yrt|sreEr yrnora? (BrIsIrpe|tteee srn. 6.)

9. Ar(o cDdKy crlnrdqy xrsAII0TrtroAe,rnro Ea rpx ieArqr.i Aexr, oEAe cjt

Aeo[e rsqr(e reMef,a ienrcf ocuoyflrs (cJr. 26),

llcruralte ,qa ru te m qrarf,Jtaf, rtxoruyrao.

c.n. 26

CJr.. 25



L, Csar 27 nDercrti.Ea csefl e

Di oe x().Y Etrrgrt rcla ce na'

a-r rreaajy rqo3 ry[y, y Dehxll{

a ct o6ruca npoEtr tle uecro-

-,F.

Koie ce orrype Aoof,ialy ra.4

cria lrScfom EpioBr Kpa(oaa?

lL I(ffrtryrrrlrle rraalrar qrlr trony-

r) orncaEe rqyr*xue rcEocx 23

D tErc Kpyxrue rcso{rl 15 Dm,

tZ l$re, noJDrrtpe{EEcr 25 mm,

llndlf,re xBaaFtrT, TaKo ,qr uq o-C--r 6yae ocro cererpxqna.

t3. A-Ta ie Kpy)ftmta r-3cm, Ot(o

-tey Ey yni[rrre jeArarocrpq-q4tr.o.

CJr, 28

, IcoEcrpyEllr're trp€tDurrtr urecro-

tE, crF |e 25 cm. KoJnr{o oE rrMr oca

cr-at

TpoyrD. I{sllfrsirc Koid{- c€ iorrl npqBrr-

n&nl f, rolryaaDxrM rtrloroyrnolxMl MoIae

nornytro noKprTt DaBaE troar. (Konf,tar uoDo

6rlll 6xp yrnoss oKo claRot 3ai€Frqrcor

reMeaa rf,)( r,fioroyrnoDat)

16. IlaqDrajre rltloErJru neloltrEo rcra

My je: a) nony[p€qmra otrxcnEe Kw?rtruc3 cD, O crpaEolr 2, cn.

17. AeoSoM trerrpatnor yrns troae,rt-

Te (Py/rf,x{y na 5 ielnaru ,[eJro a, n{aulrraiTe nsmryrary ,,3 eoay", raao lrlro je

np€.{eraBJLetra Ba cJrEttr 29. Koie aeoxe

TAgrCe rpe6a Ge"rmdqno) cnaiarf,?

C . 29.

18. IIoMo[y ,rBa no.qyrapf,s JerEaKo-

cfpaf,Irltlr Tloyrna traqprajTe ltleclox|laKy

3Be ,qy, xao uro roKa3yje c.'ItrR& 30. (tlalnpe

troAeJnrc lIplrNqu{y |la 6 je,qm$d ,[enoBa.)

19. I(oEcxplquiire trpaDunrn ociro-Jaaao raa ,ry ie:\ r) fio.ryrlpeqxm{ oficsreKpll'rExle 3 cm, 6) crlraMqa 2 cm.

20: y Kpltutruy npelxnKlr 8 cm yflri'

utre rlllltf,JlDf, aB{traecToyrao.

Cn, 27 - KpricrajDl csexsl[ Irary.lEErla'



2L Haqpratre qlaDEmu lerEaecr1o-yrro: a) qloqe 2 c0, O mJrytrpeq$raomcsre reyr@c 35 cD,

22. Korcrpyrurre lDqrdJq oqro-yrao trDor3toJDte te,rFrxte f, no4EITe TDf,

IalrncJEe .qrarof,sle. I& naroe aa iDu.Jpeqene re,ryftroraae? As rN ra f,eKN Ae.rxwsrpruot

K. Crr, 30

23. Y rp,'@y r-3 cur yultrro

rta'q$rl r nlrllxJ@ moct\0yfao, ta (o aaIelroryma cJmaa 6yae oc8o amlg4lFrEg.

KoEqrly@re, rarnr, raJcqrrrre oce cr-uenaie- KoJsKo D€rneEe mra Tlj sraarsr?

24 V l1tara.rm rcrloyro yrur.rexDrrmEr, c aarrr cmiTe leaoM aomlrf,etacre, mrr ce ,qofria? Aa m le qeJol(yEa

c.ttrtar oqo cme4n ra?

45, Kof,cDylnrrre trpsrxli& Mlono-

yrto y Roxe co rr le,6or rcu€B Mor(erolyif, 9 AriaroBurr, a troqruletam ollrcgleKpyr@e x3xocr 3 cm.

ft.31

Itolurx {ef,TplJrm Fao r?aAe cr.re AlGyracroue cnoJmqe?

27. I(of,crpyrtrEre oce cr erprle Jeg.tror tlptlrutf,ot urggToyrra. Jeay Jrf, cEl o,q-

seqqioaf,r oca 3 ety TaqaKa lD€ceKa crInocroyrnoM MeDy co6oM iegxaft? As JL

je rgrx crlgal r rog npaaraaor aeroyraa?

28. Je&oJ 4nrrroltroJ rr.nq srcrpaqt trIldrf,JlEror trrecTottJlt o.4rcar[ .

q€frIEIXO C0MCTDXEy Tatlrv y O,qtocy rrq€frap trlecToyrtt

29. Koru(I yrro raxraraJy 4p yra-clome .grleqlljte rlrgtf,.rtrlort e) treroyDrr

O rlecroyrra?

30. IlartllraJre trpara,lrn MEoroyrto,crpsqe 2 cm, rrla oe lqa & os rrrayKyro l0 ocr cE{erpdc,

3f. Hs'q e,.r MIV Kao ra,q srps-@olf' a ta Da3mM crysaSvr oI Ee, Ea-

apraiTc cro6oIpor pyxoM trIlitf,Jtaf, trcrolraaoMNOPQ n qErBJlrr rrrecrollllro MNABCD.

O4p[rc, 3srrM, rernwy rr;I8 ONA.3L EJl,f EC sr cJEtr 31 ,e daro-

Eajla rlnEr.rf,or trerc'':M, ABCDE. Vl94pqf-f,alrc yrao lEC.

Cn. 32

33. IIaA f,croll g*t AB (ca" 32)

KotrClpyrcam cy nDelljltrl MsoroJrJrotf, ct



Io&i€se pey.rrraie tr Mepe&eM (yrno

:4. IIan clpatrtruaMa (BaApara (or, 33)

cy upaalljrull Mnono].rnoBf, @

r t grpaErua, I{tDaqyEajre seJrx{unya. b, c, d, t 3arut{ Ao6niese pe3yn-

-Fae r MepeEeM.

Cr.33

35. Aerf,eEeM xpyxf,sue Ea je.uaRe

trrsrre y r$y trparnnlrl qeTBopo-

r dEyfEo, tcro rxro mr.a3yie cJlxl(n 34,

a) r.oserpl'Busre cre occ ctrMerpF

p-yee 0E lTe, O Bpac 'f,aire''raosa cJmE

:f, Ac.6aeeM Kptrorue tra ie,qf,r(e

rEtPtruD're apaErrtry 3Ee3Iy: a)

F) aeccroKpa(y. (Koje ,ueone

4.i orFrr?)

:'7 Aro orxy crDanqy je,usaxo.

tlxryf&r troae,rnMo [a Tps ien.

(rtr rasKe rDe.[crua,Eaiy

TeMeEa jeJEor ueicTolTna (or. 35,), Iloraxrre

Aa je ro trpaBuJmr lrlscToyrao.

38. tla omqn 36 r BAcraBlbese qy

$trype Roie cy ,qoErjetre Ea ocroBy troAene

raplr(xrue dajealane

AeioBe, trcaRo cy ore

Cn. 34

raupraf,e? IloK '[ajre t caMB ,qa Eaqprarg

tarcBe f, cnFme otrI:?e.

Cr. 35

39. Oalrcaxre 6er Mepe*a ]'rtoDo

/1DB t DMN (c.:l,. 7). Ileroyno ABCDE

[paErnar.



Cn.37

6. O6I,IM MHOTOVTJIA. AYXI,IHA O6HMA

Cae crpaanqe MHoroyrra qr.rre rieroB o6[M. Moxe c€ (oHcrpylrcarn Ayxje,qxara o6mry rpoyua. y3MuMo, Ha npr.rMep, neroyrao ABCDE (cl- 3g). AxocBe rleroBe crpaEHue rrpeHeceMo xa jegny norylpaBy (AX aa cnwlu), oHAa

ie ao6rdeuu s6lrp crpanuqa, ri ryx AM,6uru jegnar oOurrry ror MHoroyrra.Ca6uparsev Ayxl{ta cBHx crpaHnqa MHoroyrna ao6[ja ce AyxEHr o6rrMa.

Ha rprirMep, 3a MHoroyrao ABCDE (cn, 38) lrravo;AB-2,1 cm, BC:Icm, CD:1,5am, DE:1,6cm, EA:1,4am,

na je .qymfia o6uua O:7,6cm.



Ymre, aro ryxnqe crpaEqa Muoroyura o6e.nexrruo ci a,b,c,d.,.,o6n 6the O:a+b+c+d..,

D

. , I . I I t ,ilNPQRSXCn. 38

lm i raxoroyrao upaBr.rJraH, Ayxrny BeroBor obtrMa Hahu heuo raay i.ue crpaHtrue, peurrMo d, rroMHoxr&ro 6pojervr crpanrua, peqlrMo n.

ayrajy gyxuna o6naa 6rhe:

O: n. a.

hlo xexe 3aaar(e y sesu ca o6llMol'4 ruiroroyraa.

f) Aro ie ,4nrrna . odnnra rpoyrrna O - 76 cm, a ABe crparulqe trMajy

g'try a:1,9 dm r 6:250 mm, l(orrma r,ry je rpehr crparnqa (c)?

Ilmro narrl je ,4a ce jegan ca6upar 4o6aja xa.u ce og :6rpa oay:uy

-aJr ca6upun. fia 6ncwo ro ypa.alrn , caegnuo uajupe gare 6pojeae

-rmaeEe.

Axo xohelro ga go6ujeuo rpehy crpanr4y, Ha npxMep, y qeH-

rorpe6nr cy oBx rlorau[: O:76am, a:lgcm u b:25cm, na

.lm rlraxexe crparrr4rle

c:o-(a + b):76-(19 +25):76-44:32,

irr ryrma rpehe crpanaqe, c :32 cr'i..

4 nlm y odnrKy mecroyrna ABCDEF, ca crpaHf,{aMa AB:25 m,

.'r*r's39 upeAcraBjbeso

yuarexo), rpe6aaa

ce orprAf, Tpocrp''r(f,M

lfmrr Eo,q;rmaae xf,qe. KonEKo [e uerapa rwe 6llTrl trorpeouo?

farpc6m ryxma x qe 6rhe rpnrryr reha oA ayxlrxe o6IMa. 3aro lrajtrpe

Drauo.ryxmy o6nr'ra: O:25m+ l0 m+20 m+ 18 m+ 9 m+ 8m:90 m,

rre, 6rhe norpe6ro xnqe



3) Koma je Ayxf,m crpltrnqe trPlBtrnHor treroy$ra qlljl je gyrraa odmaa

r0,05 m?

Ci. 39

Ilourro je o6rzM 5 tryra nehu og crpaurfle, Ayxtrsa crpauuqe 6uhe:

10,05 m:5:2,01 m

7. IIOBPMHHA MHOTOYTJIA

Bern, rrra noBplmr Mgoroyura (rj' Ae.na paoxu xoju je meroaxlr crpanlt-

Uarvra o6yxaahex) Ha3ma ce troBpltrf,tll MHoroyrna' Mu cuo rch r.t3paqyHarlt

rroBprrrr{Hy Tpoyrna rleTBopoyua.

,{a 6r.roIuo .ao6nntr troBprulluy luflro-

royrna yoflrure, rpe6a ranajnpe gn'

jatonanava l ru seKIlM apyrhM .Ily-

xnMa noaeruTll ra le.nore vnjY

noBpmxry 3lraMo ,qa r:pavynauo (ro

cy rpoyrnoBtr u serBoPoYHoBx) a

3arlrM IbI,D(oBe nonpurNue ca6patl.

Ha cneAehLu np[Mepr.IMa BlraeheMo

raRo ce To pa.4{.

1) ll4raryrarx troBplrf,f,y rcrolrna ABCDE (at' 40)'

Iloryun"ro r"ry gnjarouane n: je,qror reMena' Rao mlo cy 'nnjaronale

AC , AD ra c:rmln, quMe cMo ra fio,[eJlflJltr Hl rpoyrnoBe' (TaKo3BaHa"Meroaa

TpoyrnoBa".) Oslrne,4ro Aa je s6up troBplrtrEe oBId( TpoyrnoBa jerrax nolp-

A

Cn. 40



Etror Mfloroyma. 3a norpmnry rpoyua, vefyruu, rpe6a Aa 3HoMo

(EEoBxue II BlrctrHe (cerrre c'e o6pacua :a troBplrrl'rHy rpoyrura). 3aro,

t - GEaraMa Ea ctuuu 40, MoxeMo nltcaru:

p : L eo. t, + ,tc. n. * lec' n,,22'2'

aro xajupe u:parynaMo ABocrpy(y noBplnlffiy rtrx rpoyrnoBa' a 3arIM

troAeJruMo ca 2. 6nhe:

r : @o' t,+ ec. h2+ Ac. 4).

Aro noclte 3aMege .ILarI{x

6pojera tr3BpuuMo Pa-

F'IIEe npeMa oBoM

ao6rievo rpaxexY

|uEt treroyrna.

2) R:prparr troBpmf,rry

rrrtr" ABCDEF (u. 4l)'

Ilonyur.ro xajupe arrjaro- A4 AD, a f,a By cnyc'rr.IMo

lFai.c Et ocrarl{x TeMeHa:

nroymo rroaeJbelr Ea qeTupll

Nt -rF ? rpoyrna, AFF., EErD,t\ rcaD f, Ea ABa rpatrea,

ffraE s SCCt 8t (,,ueroaa

.rrcr1. lI orge he oqurreago

-FrrruEoroyfna,,uarne:

c.4l

s6np uorpma oBIt( aeJIoBa 6nrujeguar

r :f, ,r,. r,r + | r,o - az,* ta,' r'n + | c,o' cc, +

+f, @n, + rr.')' z,F, +

+(BB, + c c,)' B,c,'

g;'rctlo

tra ca6nparrlo noBprlrage rnx ,4eJroBa' r'loxer'ao oa6parr rr.rxoleDgtP'TC trOBpIr[Ee, a aarmvr go6ljenu s6np no4enuru ca 2' [ar:re:

Ip- IAF .4F'l Ep.EEL t A4. BtB + clD ' cct +



KaA u olAe r.BlprrrriMo 3aMerry MepHnM 6pojeuua (xoju cy .qaru fimrce llory go6uru MepeBeM), a 3arlrM rr3BpruuMo Ea3naqeue paeyacKe pa.q;lenpelva go6njenorvr o6pacly, go6theuo troBprrrxHy Mf,oroyrra.

Karo xa. TepeHy qecro MepuMo

rroBprnruty sev;lnmra xoje lola o6lurHeI(OHBeKCHOT MHOTOyIJIa, nOCMa-

rpajuo uuoroyrao ABCDE (cj'.. 42),

y roue je EALAB u EAll DC. Axorpac[paMo CF y npogyxerry DC,

BnAxMo ,aa ce oto aeurmrrre cactojrr|r3 jegtor npanoyruror rpaneza, AFDE,tula je r'ucuxa AF, u jegnor rpoyr.na,

FBC. IlorpniNuaolor

seMJLtrrura,

nperua roue, 6uhe:

r t^-

{ez + ro1.AF + + . FB.Fc - | gee + ro1.ei, + FB. FCl.2- 2 2"Kao mro @ tr3 oBor nptrMepa Bfi1r., npv acJlerby Mtoroyrnoaa rpe6a

xacrojarra Aa rrrro Ma6e Mepr,rMo, a Aa npn roM go6fijer*to ,Eenone rnjy noap-rmHy 3HaMo .4a. g3paqysirMo.

8. UOBPItrI,IHA TIPABIT.IIHOI MHOTOYTJIAAro qenrap rrpalltnror Mxoroyrna cnojur.,ro gyxrnra c reMeHrrMa, oHAa

ra aertrMo Ha oHoJrr{Ko iio4ygapanx je.auaxdrparrx rpoyrJroBa noJrmo nMa

oTpal{mla. Ha cllrqu 43 TaKaB jeAax Tpo:'yrao je.ARO. If,eroaa octolnqa je crpa-EXua MHoroyr-Jra (a), a BncnHa uy je nony-

trpelmxr ynlrcane r(pyxnrrqe (r). IIoBpEraEa

.tTOI Tpoytra Je

-ar, a floBpulJ4Ea rrpaBxjl-

zBor MHoroyrJra rojt ce cacroju oa rTaxBr{x rpoyflroBa je n rlyra Beha, aaxne:

IP : -1 nar.

2

KaKo [pou3Bo.q na 3na.l;x Ayxt4ryo6rora unoroyrna, 3aMeHoM,qo64jaMo:

'|

P:: Or.2

urtr peqr{Ma: troBprrrtrHa trIanlrntror Mfloroyrfla jeAtrext je troJroBtrrn fipotr3Boato6uMt f, trorytrpesHf,Kr ytrf,c&rc Kpyxuqe.

AFBCJl. 42

Cr.43



(Mr helrro tr Ea,[a:be, rparrohe paarl, qecro ynorpe6rararu rupase o6rr*,

, troJrynpetGax, rrA. yMecro gyxuxa o6nr"ra, AyxIrEa crpanEe ' Ey;rlaEL

aJIE, pa3yMe c€, caMo y clyvajenn''ra rae ce uehe jaarartr

Yo'rnro y Tpoy ry rra ctffiUv, 43 eJreMeI{Te: cTpaHrqy IIpaBtrJIEor MHoIo-(a), trotytrpeqHtrx onHcaHe xpyxEEe (.R) r norrynperxltK ytrtrcaHe Kpyx-

(r). 3a roxcrpyrutjy npaBnjlHor MHotoyrJla 6fiJro saM je .qoDoJbHo aa

GaMo je,aax oa rux ereMesara. rIDM cMo ,qo6nnu jeaas' ocr-ana AEa cy

6f,na ogpefexa. To sHash ,qa u:uely oBa rptr eIIeMeETa nocrojn oape'

eaarcxocr (ran je aar jeaau oa BIrx, Ap)'ra ,4Ea se MoxeMo y3uMara

Kaxo ra tpu eJIeMeB-ra :aance jegar o4 gpyror, uoxalyje

Ta6era.

ql@EsEt'yrao

xtronlmpeqlIE(

odtcare(pyxrflqe

IroJryllpeqll fxyrlllcttEe

t(pyxrl,lqecapaurIta

PnoBprrufia

qqt .o 0,577 a 0,289 a 1,732 R 3,443 r 0,433 d

EIET 0,101 a 0,500 a 1,414 R 2,@O t 1,000 4'

-ro:Fe

0,851 4 0,690 a t,176.R I,453 t 1,725 a2

cmtrao 1,000 a O,a66 a 1,000.R r,155 r 2,598 a2

@tr-do 1,307 a 1,208 a 0,7 65 R 0,828 r 4,82a a'

-toyrao

1,920 a I,866 a 0,5181R 0,536.r ll,l9} a'

Ilora:ahevo Raxo ce MoxeMo xoptrcrtrTE osoM ra6rtr[ol{'

Aro je, xa npflMep, aara crpaxrqa trpaBtrJIEor neroyrta, d:4cm' noJry-

orlf,care Kpyxssqe (R) ao6sheuo r<aA 0,851 nounoimiro ca 4 (rpehu

npBa xonona), naje:

R:0,851'4:3,4O4,ro jecr R:3,404 cn (npu6-

jcp ce tro oEoj ra6etm rc6niajy nptl6nmlcHn pe:yJrrartr)' Eneuexr rq69O'4 cm:2,760 cm. (Hera yrren4tr ro o6jacxe')

Eajaa.q uocnegna xoJIoHa - 3a x3paqyEaBane troBplnnre -- ao6ujeua je

sa cnysaj trpaBtrnHor treroyrlra:

IP- :' 5' a' 0,690 a:1,725 a2,

2

trrflroAara( 4: 4 cm ao6ujavo P : 1,725 ' 42 :27

'6,

ro jecr 27 ,6 cm2'

Y:c$evo jom jegau npxMep: n:lpaqysarr noBplrrlrHy npaBlltxor ocMoyrna

no-lyrpeqstrK otrIlcage rpyxglqe .R:10 mm.

Iloum je y ra6;ruru troBpnrtrffa n:lpasyrara nouohy crparnqe a' ro

npBo n3paqyuartr. V uerou pe'4y u rpehoj I(oJroHu crojrr Aa 'R

DNcortrm ca 0.765 ga 6u c* Ao6mz srpasgqa' Aase:



3AAAt{I,t 3A BE,I(BAbE

,,r,, \X;f;H "oy r roEpu,Erv rc-

BCn. 44

%rsxffi;"Flf#rlffitr

BCn. 46

3at&n:

AB-12m, BC: S n, CD:tOra,AD:8,3m, EE:6 m?

ffir,,;:r;;**:**-:#,fr"#

*m:;;*5:-*;drgpr**a,i#lf"#x

Cn, 4i6

,.To*o apr nrra Bisa y o6nr$,rr€"royrra (cj. 40 raq je:

ABIBC, ADrCD \ EEtl.tD,

*-t. Kof,ctpyn'me npa.sxrtrEr .,ecro-

:rl'-lrEcaEy rtej/'t euv mrynpe{srxa 3 cD,

;,mTH:##"''"Tffi

7, C,rpamta

En ry,nrEy 2 ce.tor ruecioyf,Ia.

4paB{,taor u€f,?o}TJleI,Irpavyuajre norplrnuy

CJr. 45a

Cn. 47



r sotr-_ra fv4 Behx o6nu, f, 3a Ko-

-.a n* xMn Behy troDprfrtrry, rr 3a

r-a?9. IioEcrplrurrre npaBBJrn{ npasae-

qF-r ctF-mte 2.5 cm, na [3paqyrair€

10. lla c,rErn 48 npeacrarruena ie

lr-rra

xeia, yMabeaa y pa3Mepx

-:Lf,odro mz nrla ra .rcja?

ll- Jera ,uaaaa rtrrra o6nlrl. nero-

trIrcDE,

rye ieDCIIAB

<6.49).

E+rc a'm ie trolr€6rc s Bspa{Fajre

Er _-t-q-Y y ap'}la rta,[ ce 3Ba .ua

Er- (a crur) lpeMa nllDanll croitr

t r-tr l:1(x)0. (IloBy$rre rl reMesa C

tsqr .Eri, a 3atxM Meprrre lxyrno .rerupr

.fEllryrtEyMsnwerpnra.)

liL Y- rcty rpyntrxuy ynnurre je-

EIE tpoyrao i fipaBf,Jrx{ urocTo_

tE r rrx rrcrsuffiTe - 6€ Meper6e ItrpBa oDnx tora

s&qdor rloyrna f, nocn|'rrpaiTe cBe ao6lrieEe

rpoyrnoBe...)13. CBars crpa{rqa ielEaKocTpatrriq-

Eor rpolrna MNP (cn. 50) notrerLefla ie f,a

3 ie,{xaKa aena, lla iecrlaia|6eM

neofllxTa{ara f,scrao',,ie(Ioytao

ABCDEF. Koj

aeo noBpurt rpoJrrna o6)r(Bata noBpul olesro-

yrna? (He ape6a e&ofla Mepsrrr sx paqrartr.)

P

Cn. 50

14. Ha.q crpaE uoM AB tyaenrlf,'ornerotrna ABCDE (oltcrpyncaf, ie ieaf,a(o-

crpatrrqA,| rpoyrao ABO (cJt.51). Vlsp '{y-BajTe lloBptrutry oceEqeEe noBplu[. (fla3llTei

Epx O urio llenap rcroyma ynorpe6fiTe

rabnuUy 3,r u3paqyDaDaEe troBplllnne npa_

Ef,nlrG treroyrnoBa.)

Cn. 51

15, tlspas$ajre odnrr r troBpllltrry

trIraBr.[qor aarf,aecro]trra Kaa ie: a) clpa-

f,{qa a:5 cm, mtytrpeqsK orucarc Kpy-

N

C,

B

Cn. 48

Cn, 49



ilAYXIIIIA KPYXHI,IIIE. IIOBPMPIHA KPYTA

1. IIOHABJbAbE O Kpy)I(HItUr4 I4 KpVTy

Axo ce jegna ,Eyx orpehe y paBHx oKo jegHor caor rpaja, rojra ocrajerlerroMrrr{au,

rbeH Apyrn rpaj he ouncarN rpnay rrrpljy _ Kpyxn[qy. Taxo jeHa cJrlrqu 52 rpyxruqa Hacrana oxperaseM.qyxtrCM oxo HenoMr4rrHor rpaja C, cre 4or gpyru xpajM ue rc\e y npno6rrul no,roxaj.

Ha ocnory oBaKBor rrocryn(a 3a(Jbyqnnn cMoga je xpyxxuqa cKl'tr r&qaxr y paanr xoje cy jegrnrco

yaubeHe oA jeAue cra,rne rarxe y rcroj paBm-enTpa (cpeAmrra) xpyxn[qe,

.{yx roja cuaja ua xojy ra.rry KpyxHtrqe caqelrrpoM jecre nolynpeuaux Kpyxtuq€, Ayx Koja fiajava xoje aae rasre xpyxHrue je rer[Bar a rernna roja

trpona3x r(po3 qeurap je npetlHr.rr (pyxqrrqe. Cxyuoau HonylpesHrra, Termaa nperlHuxa cy HeorpaHuqeHl{.

Aeo xpyx[nle Ea3rIBa oe n]T xpyxlr.rqe. ,{ae ra'rxe na KpyxEr{qr.r aene

rpyxHr{rly Ha ABa Jryxa. Axo cy re aBe ravre xpajeru uperxura, nyxoBlr cyjegxaxn u 3oBy ce tron''Kpyxtrmle. V c:rylajy gpa nejegaaxa nyra yrulrahelro

y o6sr.rp uaru, aro ce gpy1urje He Ha Iacu,

. Cnarou- lyry upurraAa (ogroaapa) jegna rerura: ro je ora vnjn ce xpa-

JeBX noxnarrajy ca (pajeBnMa JryKa,

Caaroj rerunu npunagajy (oaroaapajy) ,@a nyxa.

Ha lptrMep, rerunu AB (cn. 53) upuna.{ajy nyKoBH

AB rt ACB. Axo ce gpyrwje rre Hauacfi, Troapa3y-

MeBaMO MaI$U OA TI-IX lryKOBa.

fleurpaaur yrao rpyxnnue je yrao rr.rje je reue

y II€HTpy rpyxHr{qe. Cnaxou qeurpanaoM yrJry npu-

ua.qa (oarosapa) jegan nyr r.r jegaa reruaa, Taro,

yt:ty AOB (ca. 54) uprznala reruRa AB n JryK AB.

O6pnyro, cBaKoM Jry(y oAnoclro rerfiBu np[na.qa qeHTpaJrut yrao.

Lbuely uearpanr 4x ynrola u rluua npunagajyhnx ayroaa n reruBa,

c'n. 52

Cn. 53



rxE nyxoBf, f, jeAmr(e rerlBe, a BeheM qemparnoM yruy trplmrat Behf,

o.uocHo rcha rermlHexa yqexlrqu ucraxy x o6pnyr o.quoc, rj. nexa

oA jeAuaKnx oAsocso sejeAnarnx nyr(oBa.

Kpyxunqe roje avajy no.rynpevrure je.araxe

rroayAapne cy. One ce, EalrMe, Mory Ao-y uoaoxaj 4a jegua Apyry raqHo troKrrara.

aarc.4eur.r oglroc u:uefy 4rxoaa, npnnaaajy-

rerusa u qeHTpanHux yrnoBa Baxu, trpeMa

He caMo 3a jeAny l{cry rpyxHr{uy reh u sa

KpyxHrue.

Axo cy,qare rpu Tar{re roje ne npr.rnag,ajy rcroj npanoj, oHAa ce MoxereyxHnqa, u ro caMo je4na, roja [poJla3x xpo3 Te rpu TarIXe.

Ha cruuu 55, rpn rarre xoje ue

:rexe na ucroj npatoj jecY A, B,

C. H€Ka yqeunqn o6jacne xaxo ce

t(oncrpy urc qeHTap O I.I KpyxHtrIla.

Kpyxnrua ie oaPetem cr

Tpf,ra.rne xoje ne upnnagajy ncroj

trprBoj nf,Hnjr.

floqesurll og w roje ra'xeKpyxHr. Is, MOxe ce I]o rpyxHEqu

xperaru y .qDa cMepa. Ma :a rojlroa ra aBa cvcpe xaxevo aa je

cynpoTaE oHoM ,4pyfoM.

Kpyr je aeo paBux orpanrF

qeu xpyxtrnqoM.

Ilojrrroau Kao ruro cy flony[peqnr.tx, TerI{Ba, rpeirHlrK Ir rIeHTpaJrEE yrao

raxo 3a KpyxErrqy, TaKo n 3a rpyr.

3AAAIII4 3A BEXBABE

- FrryraiTe xplr(Enly, florynD€q-

r: 45 @, na rs jelxe rbeBe ra{xerlrFc .* rsrse, qxie cy ,qyarfle 2 cm

rl5 c.

2, LctDymrre ltpyrr(|l{qy Koia IIpo-

-rF rF raql(e Koie ne npf,Ila,qaiy

Ei .uir.

je oA Ta.rKe A, ltao on raq[e ,, yAaJL€Ea

2 cm. Korlfxo roo peruena?4. I{3 jeAEe ragKe (pyxsxqe lroBy-

qlne aB€ Terf,re xoie he 6n-rr Mebyco6[o

Eop arEe. lllTa ce npnnehyie rt(o ce cnoie

,lpyrr Kpajels rrd rc'rsBa?

5. XoEcaDyuIIIrIe ltpyna$nuJ, nony-

rDerHfiKa 2 cm, raro aa nponatn xpor jeafiy

Crr- 54

Cr. 55



2. AYXI{HA KPYXHT{IIE

Axo noclrlarpauo lra xojy rpyxuuuy u je,qas BeH npetrgr{K, oAMax HaMrlaaa y osrr 4a je gyxlrna rpyxnr.rue sHarHo seha oa gyxl,rne npeqxrrra, 3a

IaJbL paJI y reorr,rerpr-rju 6[he HaM Bpno Baxr{o Aa r{C[r{raMO, npBO, Kont|KOje nyra 4yxnna Kpyxurqe neha o4 gyxrrc rbeflor npeqnr{Ka }i, apyro, aa nf,ie xolrrulr rrrx Ayx[Ha 3a cBary rgyxtrrqy ncrn. y rov uxr,y o,qa6epfirr.Io

6. O3saqme IBe raqre, A n B. apaooiarsy 3 cm. lllesrapoM rr reBrpoM aa_

l)rre Taq$y roia ie o,q A ll€'r6esa 2cm,aoA B - 2,5 cm. Konf,Ko peureba {Ma oBai3a,[aTaK?

7. Onnulrre KppxEnuy oKo rpoyrur:tABC xa4 ie 1Io31ara crpa6 ta c:3cm {tranealDr yMoBr c.:60", p - 75..

8. PacrolaEe uBeiy raqa*.a, A r B,f,$rocu 4 cm. Koncrpynurrre rpyxrorqy/:3cm raKo ,qa Dpona3x Kpo3 Ta.rre,4 n -8.

9. 3a6o&rTe ujecrap y ieAoy raqray

l(pl*a{qe, oTaopxTe m ao qe[tpa n on{-Urxre ftp}?erxqy. Koja ie ry'noaput laieA-axqxa mro nacr&'rxM l.T:trgBxMa? l4urnap_TajTe ay noBprrr.

.qyxt,xta*ojlllqllqI(

xpyxr rqe trpeqqrrxa

39,3 cm 12,5 cm 3,14s

8,7,, 3,137 . .

8,5,, 3,t53..

47,l ,, 15 ,, 3,t4

28,3 ,, 3,144. .

pa3r{e Balbxacre npegvere (ao-Eaq, qa[y, rorraR I{ Ipyfo) EE3Mepr{Mo .tly)Kr.rHy npyxH[qaMeTpoM, (ao rrrro nora3yje crxra56. 3aruu r.r3Mepr.tMo n Ayxxue

IIPCiIHIZXA TIIX KPYXIII4I{A, A IITOMOXeMo yrltrH,lTr_t Her},lM YBpC-TI'M (MCTATISI'M SJII,I,4PBCHI,IM)

MeTpoM, r€o mTO CMO paE[jer]ayqrurr (cn. 57). Paldf' rc6u_jarra mro rasuujr pe3ynrara,

rpe6a nperxo.{no ytrope.qBTr Ana ,,rr,rerpa.. rojrcra hervro Meprfir,l, jep ce or{tr He

cMejy pa3nuroBaru y .qyxlrHn o,qrorapajyh * .qyxurrcKnx jeArnuqa.

Caa je jom norpe6uo troAeJrr.rrrr Mepne 6pojeBe cBaKe (pyxnqe n oAro-rapajyher npeqall(a.

OlrcaNn onreg n3Bpmnnu cMo 3a lrer pa3rtrrlsrax rpyxxuqa x.4o6trjeHenoAarrc yHenx y ra6;ruuy (or.58).

IIIra ce npuwehyje? - IIocneAFaroroua noraryje ga je gyxnra cBare rpyx-Er{qe neruro Bnrue oA Tpu tryra Beha oA

,4yrr(IlHe fiperrHuxa.

Axo ce osaraB orneA BprIItr cacse sehoM Taqnomhy, 3a nornsul,rKe hece noorjaru 6pojerz: 3,1; 3,i4 3,141;3,1415....

Ha raj uavrut Aorurtr cMo Ao je.ForBpJ'ro BaxHor 6poja y uarevarlrqlr, xojl

Cn.57n. 56



c d..E-aBa rp.rKIIM cnoBoM ft (urraj: ntr). Aa 6II yno3HaJIIr npupoAy

q trqrxarrr MareMarl.Iqapu jom ns anrumor .4o6a na coe ,{o HoBlrjer

Eplmlrx cy Mriora vctiu'rvBat*,a, Tatro je c.naaxu rpqKI'I MareMarllqap

,ro5ro :a 6poj :r apeaxocr: : ], xoiu y o6.nrmy reul'IManHor 6poja

3,14... Kacsde, y ro(y BeKoBa, nprrMerbtBaJlll cy {yBeHI'I MareMarll-

qr a roan : I*leroAe u ao6ujasm sa 6poj r cse gehu fi aehn 6poj aeruMara

b i oro 1600. ro4. xorran,qcxll MareMartllap Jly,aolr ran Uojlen (Ludolf

n f-st [) npaqyHao 6poj n ca 35 .qeqlrMana:

: 3,14159265358979323846264338327950288 . . . ,

.:t' E- E€ra cy HalqHr{qr ao6wlann :a r jou lrmoto sehlr 6poj ,ueu na:ra.

.I- ce =a la je raj 6poj aeurnrana HeorpaHlrqeu u .qaje

6poj?r Henepxo-

.EO 6pojy a uopavo :narz c:reaehe:

Epj -- noxaryje xo,rnxo je tryrr Ay,ftf,ua rua rcoje xpyxnnue B€hr oa Ay-

-

Lr(lr trpeqxltxa, 1LJIff, mro je rcro, ,4yxrny Kpyxsrne Mepery H'ellLM

t-ox x:o je4nnuqonr. To je nenepnogural AeqlMaJIIrr 6poj ca reorpann-

-

Qojew AeqnMaJra, a FeroBa npr6.ruxua BpelHocr cr ,|Ba raqHa aeq[MaJra

Far{Aro ce Kpyx-Hxua nocMarpa rao rpaur{qna lnrnja xpyra, ou.aa je oxa

-arp:/ra- Cse mro cMo ao caaa peKJru o ayxuHu rpyrcIl'Ile Birxfi rr 3a

gruy o6rua rpyra, jep je ro y cymnxu ucro.

III ror craanor d,qHoca ayxfiHe l(py]Krrl,trre (nau .ryxnue o6nl,,ra rpyra. O)

r E€Eor trp€rrHn(a (aujauerpa, /)o--:T no6vjaMo O-T.d,d

Er nortrTo je tpeunur d:2 r.O:r-2r:2rr.

,{aEre: Ayxnsa Kpylnrurqe jeguara je trpotr3Bo,4y 6poja ?r ll rpequlrxa

r|4lr5le.Aro je uo3liara Ayxr{Ha Kpyxurrue, a rre 3HaMo rreH fioJrynpeqHtft, Mo-

rr{o ra tr3pasyEaT[ oBa(o:

ttl. O:2rr 1616niat;1o r-9 --2r

Taro cuo ,qo6u,ur o6pacqe 3a ayxr4ly r(pyxHlrue rag je noluar oex no-

-qttrpeqgrx, tr 3a [oJry[perrHrrK Rpyxgdle xaa je no:uara BeHa ay)rsiHa.

Henouera. .- Baxno je 4a ce :analrrlr npu6luxra npe4uocr 6poja22E:= s npu6,rnxna o6pnyra rpegnocr 6poja r ca 3 geqnaa:ra: l/r-0,318,



Jep ce lbtlxoBoM npuMeHoM y HeRr.rM 3a.qAUr.rMa 3HaTEo ola(rrraBa I{ cRpahyjepaqyHabc. uTo heMo Bn,qeTtr y cneaehnM rrpr4MepHMa.

llpuuenuuo ca.4a re o6pacqe y Herrrv 3aAarltrMa.

l) I{spa.ryuaru Ayxruy xpyxflxqe v{t je no.lynpernuc r:12 cm.3aueuorr.r r:12 y o6pac\y sa Ayxnsy KpyxHl{qe Ao6rjauo:

O:2 n r:2.3,14.12:75,36, ro jecr O:75,36cm.2) Kon (tr je no,nynpernlr xpga rnja je syxrHc of,utt'- O:62;g cm?l4z Aarnx noaaral(a xMaMo:

o 62.8 6280 20

21 2.3,14 2.314 Z '--'--'- "

3) Koluxn nyr Hr qrcoBHnxy npefp rpx Mflryrxe rc$aJEKe gyxure 3,5 cm,3a 10 gscoBa?

Ilomro Bpx Mr.rr{yrr{e (a3aJbrte orrume 3a I rac rpyxurrgr fioryflpeqlrKa3,5 cm, rahlr heuo najupe rbeHy AyxuHy:

o:..a-21.2:22, ro jecr o:22 cm.7

3a 10 sacona Bpx xa3iubxe onucahe 10 TaKBrrx ftpyxnr,rr{a AaKrre10 .22

cm:220 cm:2,2 m.4) Axo xrnry, gyxnne 65cm, carqieuo y xpyxHtrqy, romrxu he 6rrrrrp€qrtrln Te rcpyxnnqe?

lI: o6pacqa 3a AyxnHy rpyxHr le: e:tc,d, go6uherr.ro npeqHax xaa odyrrorerruMo 6pojeu r (sanrro?). .{arure:

.o 654:

tr 3,14

65 ca 3,14, uoxeuo65

rroMHoxtrrrro6pnyroM

lr' :- ' :0.318.tc 3,14

IUrO Je HeCyMllI-IBO JtaX e.

TaAa uuauo: 65.0,319-20,67, ro jecr raj uperutr je d:20,67 cm, usmnpr,r6nuxno 207 mm.

5) Caarn rortaK HeKf,x rtoro oSpxe ce 100 nyra trt crtrf, oa 15? m.

Komrxr cy tronytrpeqH[qtr rrx roqrong?Aro ce roqax \a cta3la oA 152 m o6pne I00 nyra, suaqx Aa c€ rryxrora

beroaor o6 na cagpxt 100 tr)"ra y Ayxustr oA l5Z n, Ha,,ra3nro, Aanlreirajnpe o6mr ron<a:

o_157 :t.s..

YMecro ,qa Aen[Monpe4noruhy:



E] tryxr{Ee o6nMa .ao6{jaMo noJrytrperrHuK:

3A,[AIII'I 3A BEXBAIbE

Y q 3a.oaqnlr axo mie ,[pyKquie

ysarc d-3,14.

L llrryraire rpyal|xry, Irluepf,re ioj

r 4oqyEaiTe ,lirr(f,Ey.

2- Ilebrre noJrynpeqruK rcpyra qnju

E .E Ey: a) 3,14 m, O 15,7 cm,

rt I (o, n) lm 5ds 7cm.

:L flaaelrre npeqtrtrKe sauer Mera:r-

-,

E rrr mpaqyqaiTe ayxrore oErIMa

o6rMa je.qlor rcpy1r(sor

F 62,8 dn I(o.'r Il$r MY je troJry-

i.aEor Kpyra je 5 cm.

F noaynpe.rEK Kpyra {f,ir ie 0616r

.&?

C ll4aqyf,arf, ayxnrYKPYr.f,i e oflf,-

-ia) jeasaKocrpasuqEor rpotr"'r4,

5c, O rsaapara, Aniatotrane 6 cm,

[r*l. ut€croyfnat crpa{fiIe 8 cm.

7. lfupa'r tsaiTe EyrNrEY Kpyxnsue

t: a) rca,|lst, crpamtqe 5 cm'

-ame 4 cm.

I O,t ,mejy KotrqeETPaqtnrl I(PYxmqa

E .ryrolFy 94,2 cm, t aPYr. 3l,4 cm.

-t)sy*pyramr rDcresa. (o,ryoBop:

t Atr ie LTyr, r-3cm. Ytopearre

&r ca o6ruoM KPYroDa slrj{ ie

2, 3, 4. . . n].ra nyx{ (xpshx).

lC. Ala l(pyra lrlll4iy lo[ynpeqddKe

r.:6 cm, lfupaqFajre flonynpe{uxl(

pye wjr je ofttrrl ie4raK 36rpy o66da

{tt{o.a-

o 1,57 157 11Ztr 2.3,14 2.314 2.2 4

I

Eje r:-

m:25 cm.4

{

11. Itonnl(tt rr,'T {a qacoBury npefeBpt MnEyfie KaraJLKe, qf,ia je ,qy4$rBa

2,5 cm: a) 3a 1 qac, O 3a 1 Aatr?

12. Mrtr]'rna xanal6K u ieAtro* .ra-

corBtl(y lMa .qyxtrry 5 cm, caTfia 3 cm.

Konrxo nl.Ia Eehtr rJ.I npebe alrx M[q].Tue

xa3aJ6Ke oa Bpxt ctrlre raujl're ra ieAar

aa{? (OaroDopi Z) tryra BehB nyr)13. JeAaE otqrymo clo f,Ma ony-

nDeqExK oa /5 col. t(oJlal{o ocooa Mo*e

c€sl|l 3a rai cro ako Ge 3a cBa(y oco6y

pa{yaa 78.5 cm? (Oaro8op: 6 oc 6a)14 Jeaas rpKaqxf, xor6. xoji y m- -_-----

notry trpe,la3i 314 m 3a MnFyry, odllDe re]rr.ny cra3y 3a 6 Mxtryra" Koauna je uoay-

rryevro* re cr-a:e?

15. JeAEr Kpyrqa cra3a f,Ma noJty-

npeqElK /-250m. Koru|{o MErapa nperasr

y MnI|yrE TltraaqR{ RoIr RoiII Ty cra3y o6rDe

:Irrtryr 3a 15 Mrryra? (OAroDop: 314 m)

16. I(or Kn r|yr fipeDe 6iqur(n aKo

ce cBaICll beaoB Toqax, rlonynpe.lHnKt 30 cm,

odpue no trpaDoi cra3{ 100 n}.ra? (O.qro-

Bop 188,4 m)

17. Xonn(o he ce q"Ia o6px 'T T oqrrq

ipe.r8xrca 80 cm, {a tryry on 2512 fr?

(OaroBop: 1000 trr"ra)

18, I43paqysrjTe nonynpeqs * 3ei'rj6e

xaa ce 3tra aa je I Merap 4o-Mrrf,oErrr

Aeo Mepm{iatt19. llrMepme ruro ie noTpe6uo x

rupaqyEsjre ,[yrrr{y tmr&ja aa rfiin'paaua

cnf,Ke 59. Itoscrlryusrme raKEe onrype. (Oa

eera ce olle cacToie?)

@ Q7@ @n. 59



20. y paanoEneMr r,Eoru upe.ryteiaMOr(eMo aeTx .rya seiemrl(t Toctco oKD

roitr ie ooeDf,jef, rrx|U, tqto orro troKmyje

olxxa 60. To je ,,6ec$paixe ftaf,ur... flonrohy

MoTopr o6phe ce ,eAax To.rar, a Ktr{rx orAa

o6phe f, Apyrr roqaR. HeKo JcAan on rrrTortaorg rMr npcuK 50 or, a qlyrn 12 cm.

Ar.o lrlrlx rupura 132 oEpraia y uuyrr,KOJnrco he ce rryTa o5plryIf, alrtrIrf, a rcroEDe[re? (OAroDoD: 550 ry]&

cll. 60

3. AyXIIHA KPyXHOT JryKA

Kpyrcn nyr je, rao mro 3s:rMo, Aeo xpyxnqe. fiyxluy rpyxxor ;ryra

qair4 heMo oBaro.

Caarorrt qerrpa;rxorvr yrury (a na

cn. 6l) oartiaapa oAp€DeE Jryr KpyxHEe

(AB ta a.61). Ilenrpa;norvr yrrry o4 looAroBapa 360-ra aeo rpy)Krlfle, a ro je

-axo nyr o6erexrnro cnoBoM /:

, 2rr rt

360 180

I-lexrpa:rxor't y ry o,( 2' ogroaapahe 2

rryra rehn nyK, aarne t-i .2, ynry o,q180

3o o.qroaapahe 3 rryra nehz nyx, rj.i--

I:"' .3 rrA., raro ..(a he va xou180

rIeHTpaJrHoM yrny oA ct creleur{ oAroBaparrr nyx o( n}'Ta aeht, 19 jecrl:n'.n,

180

To je o6pasaq 3a r.BpaqyHaBarbe ,uyx[ne RpyxHor nyra roju oArolaparlerrrpaJrHoM yrny oc I(aA je nolynpeunur xpyra r.

clr. 61



Ha npullep, ayxqaa rpyxnor ;ryra unju je noaynpernn r:6dm n oa-rorapajyhn qeHTp:urnx yrao c:15o tr3noctr:

r:t' .::6 .,s- 3:t4:1,57,.qagre t:1,57 dn:r dm 5 cm 7 mm.180 2 '

Vst"rruo .qa je yrao gar y MrfiyraMa.Ha npnrep, Ayxntra nyra, noJryrpetrHrrKa r:l5dm, roMe oAfoBapa [eE_

parrutr yrao c:10"40' 6uhe (nouro je 10.40,:640,):

t- 3'11J5 '-640 :2,79, ro jecr t:2,79 dm =2 dm 7 cm 9 mm.180.60

Hexa yren4n caux nspavyrajy ayxr{Hy Jryra xag je r : I 0 crn n 6 ::25'3s',20".

Ilocuarpajuo uoxoao o6pa:aq t: * .",

Mtr ra MoxcMo yuorpe6rrn n180

a rtpaqyHaBarbe tronynpeqstrKa raa je uosHar lyx r oaroaapajyhn qcxrpanRntao, a HcTo TaNo Ma t3parlyEaBa6e lleETpajuror yfJla raa cy aaTL lony-

rpcrHr{K rr o.qroBapajyhl nyr.IlocMa'rpajMo lpar cayuaj: .qaTo je / t1 a., rpa}{i(. ce r. y rou cayrajy

rperxorHa jeAaaxocr trpeAcraEJba jeanasuuy y xojoj je Eetro3xaT rronynpetrnrrr. Pcurmo

ryje.Exaqusy

lo xenosnaroj r, na le 6ntn:180 . /: rc ra, r:180 ' /.

V gpyrou c:ryuajy ,qaro je I u r, a rpaxr ce a. V nperxoguoj jeAsarrunt

n',rahevo ueno3Hary c., na go6tjar*lo:

l8o./: arcr, 4- l80 ' /.

1tf

Ilpnr"renrrr,ro osa aga o6pacqa y c;regehru 3a,[aqr.rMa.

l) I4sparyrarn ronytrpeqnm( rrpyxnllqe rrjl ayr, ryxxxe /:15,7cm, og-roBepl qeHIIlanHoM p;ry cc:30".

3aMeuol.r aarr.rJ( Bperptocrr{ za I s q. y o6pacrry ea r ao6nhello:

.- l8o ' l5'7- 30, ro jecr r - 30 cm,

3,t4 .30

2) Ko.rnu je qenrpannr yrso Kplra, r: 6 cm, xoMe oFoB4Ila nyr AtrDue/- 7,85 cm?

I4 onge herr,ro 6pojne apegnocra ga r r / saMer ,rrs y o6pacu5r sa yrao,BA NMAMO:

180.7.85/). To t€CT ct: /)-.



Vnorpe6a m6:rna. - fiyxmy rryrona aahr he*ro Iunoro 6pre ynorpe6ou

ra6.nnla y rojnla cy re .Eyxrrue u3paryEare npt6lumo sa 5 qewMana,

3a xpyxr{Ey mjn je norryupeynm je4anua ayxm{e (r: 1). V osnvt ra6mraMa

cy ayxnne rryKoBa ,care 3a cBaRtr creneH, oA 1' lo 360', 3a cBary Mmyry' o,4

l',qo 60', x 3a cBa(y ce(yn.[y, oA l" ,ao 60".

,{a maruo na jeguolvr npxuepy ynorpe6y rux ra6axua. lllpaYyuajuotroMohy o6pacla AyxuHy nyra rojn ogrooapa yrny oA l5o y epyxat&tr' trony-

rpeqlrlrKa 4 im. Ifutahelro:

n.4.15r:

-

: 1,0 4720, ro ject 1,0472O cm.180

yrmrlajyhlr r-3,1416; rrrro qecro o,qy3I.IMa .4ocra BpeMeEa aot( ce tr3paqyua.

V ra6rlrqn (crpaua 36) uafmuo 4llperrxo Aa je ga 15" ayxtrHa fiyra

0,26180. Ilonno je ro sa r:1, a HaMa rpe6a 3a r:4, IxvlaheMo ro caAa,qa

troMnoxr{Mo ca 4, Aarure:

l- 0,26180. 4 : 1,04720 (rao y rperxoaHoM npnvrepy).

Hafulro y ra6:ruuu ayx roju oaroBapa ymy o,q 59'36' 17" razje r: 10 n-

Ha crpann 36 y ra6;ruuu rurar''ro:

3a 59' .[YxIrEa ;rYra je 1,02974

sa 36'gyxNua

;rYraje 0,01047

ta L7" lYxttxz:ryra je 0'00008

yxynro:1,04O29

rrro rpe6a nouuoxnru ca l0,jepje y garolr c.ryvajy r: 10, na je /:1'04029 ' l0::10,4029, ro jecr 10,4029 m'

3AAAIIU 3A BEXBAIbE

. Ha rpyxfilrr1rr. r: 1 cm, rrxpaqy-

Eaire ayrrorsy rqtyxflor 'qyxa KoMe orroBtpa

[eErpnnun yrao: a) 60", 6) 90" 8) 150',

r) 2;2: 3O',.q) 10' 20' 30".

('fr rcor*ror ,tryt np'.naai uen rpa,rrfloM

yuy &t/75" y Kpyry qxia ie ayxuaa o6rua

31,4 cm?

3. l,Irpa,rJ'f,aire ny'xroy n5n<ora alely

K{tB{esrp&qrflx tapyrLll[uar a r:6cmi

t-4cm, aKo nM Ilpxalaaa lcrq ueFrparlrll

lno a=22" 3O'. (Koi[ cy aenoBlr Kpyx{l'ue

oBr ryhoBi?)

4) Y JrpyxBrqy, /*5 cm, YrucaE ieig) ieaf,axocrpaturfli rpoyrao' O npaBlnnrt

uesroymo, ItoJ4l(f, ie JryK. rpyt {mqo xrA

cBaaoM crpaE[loM rora Tlpyrne oatro_

cgo ule'croyrra? (Oarotop: r) 10,47 (m'

6) 5,23 cm).

fs. y ie$oir tpyry' {e'rpar'oM y.,ry

oa 75V.qro apa JryK Ay:lentre 15? cm. Ko-

Jtrlr(r| je nonynpeqmr, a Kotfi(f, o6mr rpyrS

,6. JI n(, rtorynD€qmKa 4 ctl, KoMe

oAmEapa. qetrTpardx yrao or 120"' ieABat.

ie o6xMy te,umr xpyra. Konxlg ie nory'

npe.rinr ror rpyra?

n,Y Kpyry r-20 cm, llaDFre lt€tr'

tt|aull trao KoMe oAroEalla JryK Airrsue

/-31,4 cm'



8. Iloa Kojht ce lTnoM i3 rerrprnql ,nyr /: l5qb ra,u ier: locm?

yrao je oapeter qonlmpe.rnlqrira ttoirKpaie.aM{ ror nyta.)

9. Korrrtr je ierrparu trao r$yraie nyK ieruaK nonytrp€qrmq/? (I,I3pa-

ra y cTe{emtua, Mrml.Ianr:r tr ce-

10. Y loAEoM qyry, nyK oa 60" roaa

15 cm. Kojrtu(a je Ay)rars4 .rl{a oAr20' 180' 15", 730,? (y r(aKBoM cy

.'ryr n oAroBapajyh[ qerrpa,Etr ymo)?

11. KoJE&r tryr npeh€ jelrm ra,rKaexraiopy ycneA loTarutje 3€M,,r€ o{o oce:

3a I MnfilTy, 6) 3a 4.raca? (Io;yryecExK

ie fiprl6nn^To /-6370 km.)

12, qerEprnm Rpyr*nfix|e 3oae ce rcDa-

a urecrrrtra ce*craar, Aro y jearoj

xs{x A}"irma KDaApa{aa r|3{oc{ 54 cD,

ie nyrftsEa ce|(staFr4 Te l{pyn(Efi$ ?

13. Il:paqJlllajre y,qaJLesocr ,qBa Mec-

A a B, Bi xcroM Mept|.ttljal'y 3eMrbeo ie Mecro I Ea ceBepuoj rllxp&un 35.20,30,,,Mecro .B Ba ceBelltroi ulryrrl 27.15,30,,.

trptl6rD(so E58 km, 3a 'r:3)14. AKo 3eM,%y cMaTtonio rronroM,je yAa.aetr no,t 3eMJre oA €Rasropa?

3€njf,e r - 6370 k&.) (O,ryo-

npx6rrxuo 10 001 krn)

15. Ha ,aaroj. Kpyrcrqn f/) oAlr€,unre

leAlaK rreroj: a) urecrmn, O r?ehnmr,

ABaEaecrffix, r) qcrsp,r.{Et A) ocMxrfl,

lt) rlefi{tl, e) AeceTlc{. Aa n{ ce BeKf, oAoBlt( JryKoatt Mofy oap€ltlTti tra arlllre tra{r-aa? O6jacmre.

16. OEo ieaIre raqne otrxrrrre rDf,(rosueDrp{qse) Kpyr(Err{e c flonyupeqBlllf,Ma

r-1afi, r,2 cm. r=3 cm, u uaupraire

rGnxoE 3aieAstr"R[ qeqrpanutl yrao d-36".llrpaqytraire caAa At r(nlly tryr.os^ It, 12, I$xoje oa tnt (pyxEtua o.uceuaiy Kpaqu qe -Tpa:rlor 'rna, IIa ocnoBy oBor pa4a oaro.oopllfe tra riTatSe: otr qera x Kalto 3gBf,cE

.qyxnra n]'roBa KojrMa olroBapaiy Jeatrarq{euTpursfl yrnoBtr?

c

C\. 62

17. JelxaKocrpaEn {roM rpoyr,ry,4.BC(or. 62) onncaEa je x yiarcada rrTyx{{qa, aMN, t P c Ao,qllprc raqre yflrcaf,e KpJ.r(-

Etrle. Kontrrao je nyra nyxuEa nyr€.4, letaoA .uyriofie nyx MN? Obiaclrtfle 3auro.

4. NOBPMIIHA KPYTA

ffoll norpnnasou xpyra noApa3yMeBaMo Be:lrqfiHy noapurN rojy o6yxBarao6r.rM. O6pa:aq sa rl3paqylIaBa6e noBpur.rHe Kpyra Moxe ce Ao6fiTtr Ha

uaqull.

Aro y je4nou Kpyry noEy{eMo 12 no;rynperrnxa raro .qa cBa(a,4Ba y3a-rpaAe yrao oA 30o, noAenuheMo ra Ha 12 noAy,qapHlrx .qenoBa _ fice{aKt

63-I), llcrqnvo caaa cBe r€ roeqre r.r ncpebajMoux

jeaaHy3 apyrrr, Kaonora3yje cnuxa 63-II. ,{o6turu cuo Qurypy ABCD, Koja no.qceha Ha napa,

qrje cy aBe cTpaHffIe cacTaBJbeHe r-l3 nyKoBa.

Axo Ha cnzqaH Haqnu ro,qeJruMo xpyr xa 24 noAylapga ncer (a, ra q](

r€o paHxje, .qo6theMo $urypy MNpe (cn. 63-III): oHa, Kao rxroBx4lr, jour rume Jrrry[ Ha napanenorpaM, r{ To rra lpaBoyraoHu(.

N



TAEJI14IIA

,UYf,rrc KPYRxd JrYr(olr 3A t-l

CTedeat MnEyre CexyEAe Crenesr Muayre CJxyrl,qe

010,00000r I oor ras

z lo,orlgrr I o.os z:o

+ lo,oessr

s I o.oe zzr

o I o,lo+rz

t I o,pzn, I o.,r ru,

,: L:;;,,, I o,,n tt,, I o.ror*

,, I o.r, urn,o I otroo** I o.iu,ro

,, I oerr*,, I ornu,,, I o,r, o,u

rs I o,r: ter

,n I o.,o rn,

,, I o.ruu'

,, I o.rr rn,

,, I o,oo,o,

,o I o.o, ,r,,, I o.o, u*

,l o.ort,,, I o,.o', r^* I o.la aus

,, lorou,,

0,00 @o0,00 029

0,00 058

0,00 087

0,00 116

0,00 154

0,00175

0,00 204

0,00 233

0,00 262

0,00 291

0,00 5t2

0,00 320

0,00 349

0,00 378

0,00 407

0,00 436

0,@ 465

0,00 495

0,00 524

0,00 553

0,00 582

0,00 640

0,00 669

0,00 698

o,N7n

0,00 7560,00 785

0,00 815

0,00 844

0,00 000

o,oo o0l

0,00 001

0,00 002

0.00 002

0,00 002

0,00 003

0,00 003

0,00 0oi1

0,00 m4

0,52 3600,54105

0,55 851

0,5? 596

0,59 341

0,61087

0,62 832

0,64 577

0,66 323

0,68 068

0,69 813

o.71 559

I

|0,73 304

I 0,75 o4e

| .o,re zrs

I o,re sno

lo.ror*,I o.r, o'10,,,,,,u

I 0.,,,,,

0,00 873

0,00 902

0,00 931

0,00 960

0,00 989

0,01018

0,01 047

0,01 076

0,ol 105

0,01 135

0,01 164

0,01 193

o,ot 222

0,01251

0,01 280

0,01 309

0,01 338

0,01 367

0,01 396

o,ol 425

0,01 454

0,01484

0,01 513

0,0t 542

0,01 571

0,01 600

0,01629

0,01 658

0,01 687

0,017l6

0,00 0150,00015

'q00 016

0,00016

0,00 017

0,00017

0,00 0 I

0,00 018

0;00 018

0,0t 019

0,00 019

0,00 0m

0,00 020

0,00 021

0,m 021

0,00 022

0,m 022

0,00 023

0,00 023

0,00 024

0,00 024

0,00 025

0,00 025

0,00 026

0,00 026.

0,00 027

0,00 027

0,00 028

0,00 028

0,00 029

30

31

32

33

34

35

36

38

39

40

4l

42

4144

45

46

47

48

49

0,00 005

0,00 005

0,00 006

0,00 0060,00 007

0,00007

0,00 008

0,00 008

0,00 0o9

0,00 009

0,00 010

0,00 010

0,00 0t1

0,00 011

0,00 012

0,00 012

0,00013

0,00 013

0,00 014

0,00 014

50|

0.87 26?

st I 0.89 012

,, I o.ro r',, I o.rr ro,

,o I o.ro ro,

,, I o.r, nn,

,u Io,nr

r*,, I o.n, oro

,, I ,,o, rrn

,n | ,,ot rro



TAEnUUA

AD{'Se r(Dyrrr'lr n:yror& 3 r: I

60

6l

62

63

g

65

6667

68

69

70

7t

73

74

15

76

78

79

80

81

82

83

84

85

86

87

E8

I,O4 720

I,06 465

1,08 210

1,09 9s6

I,ll70t

I,t3 446

1,1.s 192

l,l6 937

1,18 682

l,2o 428

t,22173

1,23 918

I,25 6641,27 4@

t,29154

1,30 900

1,32 645

2,34 390

1,16136

1,3? 881

I,39 626

I,4t 372

t,43117

1,44 862

1,46 608

1,48 353

1,50 098

1,51 844

t,s3 589

90

9t

92

93

94

95

96

97

98

99

100

t0t

102

103

104

105

106

107

108

109

1t0

I

tl2

113

ll5tt6

tt7

118

1,57 080

t,58 825

1,60 570

1,62316

1,64 061

1,65 806

1,67 552

1,69 297

1,11042

1,72 788

1,14 533

1,76 27.8

t,'t8ou1,79 769

l,8l 514

1,83 260

1,85 005

t,867 50

1,88 496

l,90ut

t,9t 986

1,93732

1,95 477

1,97 222

1,98 968

2,W1t3

2,02 4s8

2,04 024

2,05 949

120

l2lt22

123

124

125

126

127

128

129

130

t31

132

133

t34

t35

136

131

t38

139

t40

141

lat41

t44

145

146

t47

148

2$9 440

2,1t 185

2,12 930

2,14 676

2,16 421

2,18 156

2,19912

2,21 657

2,23 402

2,2s 148

2,26 893

2,28 638

2,30 384

2,32 t29

2,33 874

2,35 619

2,37 365

2,39llo

2,40 855

2,42 60t

244 346

2,46 09t

2,47 837

2,49 582

2,51 327

2,53 073

2,54 818

2,56 563

2,58 309

I50

l5t

152

153

154

155

156

ls7

158

159

r60

161

162

63

164

165

166

167

t68

169

170

l7t

172

173

174

t75

116

177

178

2,61799

2,63 545

2,65 290

\67 O35

2,68 781

2,70 526

2,12271

2,74 0t7

2,75 762

2,17 501

2,79 253

2,80 t98

2,82743

2,84 489

2,86 234

2,87 979

2,89 725

2,9t 47O

2,93 2t5

2,94 961

296706

2,98 45t

3,@ t97

3,01942

3,03 681

3,05 433

3,07 198

3,08 923

3, 0 669



flpoAyxl.r'ro xv u 1aJbe pal oBaro (ne:rehn rpyr xa cse sehu u Dehx

6poj uo,Eyaapxux rceqatra rl pefajyhr HX Kao Marosac), ao6rrjaheMo OrrypeKoje he ce no o6nnry cBe Burre npn6,rrxanarn fipasbyraoHuKy. Ono nac yuyhyje

Ha 3aKlbyrrax ,qa je Kpyr - npu aoBoJbHo BenHKoM 6pojy orn ucrvara -je,q-.

EaK npaBoyraosrfiy rojn u: rpyra nacraje na o rcaru Har rx.

IIICn.63

floapuruny (pyra, .qaftnej go6[heMo afto lnpatryHaMo troBprxtrly rleMy

o,uronapajyher oBaKBor IrpaBoyraongxa, Vqeusqnra ca4a xehe 6l,rrx relxKo,{a

Bx,qe Aa je ocnoBnqa raxBof npaBoyraoun<aje.uuara noJIoBnnH

xpyxuor o6llMa,rj. nononunu rpdxuwe rpyxuuqe, R aa je lbercBa nucnna je4uara nony[peqHl.{fty

(pym. O3HaqnMo Jrri lloJrynpeqHl{k rpy[a ca r, a noBpurHHy Ipyra ca P, nvaher"ro

llP- :- O.7:1)v7.7-v72,

22rlnR peyr.rMa: troBpntraa rqyFa jeAHaKa je trpon3Bory 6poja r r KBa,qpara-trony-

trPeqstrx&

Ha npnuep, Kpyr, noJrynpeqHr{Ka r : l0 cm, rMa noBplrrttrry (sa n npr.r-6nur(Ho 3,14)

lP:r'r2:3,14.100:314, ro jecr P:314 cm'z.

O6pa:au sa noBprnnay Kpyra MoxcMo Ecropltcrltrlr A 3a rli3pavylauat e

aeno3Haror no,Tynl}eqHtrI(a ri?yra je noaprurea 4ata,

AII



VIg P: r'r2

Llcraxn're u

^PMaMo /':-, a oAaBAe Je /:

omj o6pasaq pewnra.

l3AAArruj 3A BExBAEd

tr ;'

1. lllDocyraiTe uorpury xpyrc wjrje m,ryrp.@r:s) 15 co, 6) 1,2 dn, r) 3/4 n, r) I m 3 dm.

@ xo*o Je noDprrrrna. Kpyra qrjs

F npeqmn: a) 20 cD, 6) 7,2 n?

'i5. tc,rto ie o6rol Kpyrar a) 3r4 cm,

O rs,l-m, ttonBrcr My le EolDt rgr?

4. Halrro [oJtylDe,rDxK, a larruo6mr rpyra da Je rcrpurma: {) 12,56 cm',

O 0,2826 mr.

5, I{3Dosyrajre xpyEy nolprr@y

oKDyrnor croJra, qa@arEca, .uaa ]lotrrla,

rlEMoooxcr€ ruto{e, MerarBor loBqa.

6. Koncnwrnf,re rllsAprt Bdt je

o6e[ 10 cm u y E€rN ytrrum€ npyN$r4r.

.l(onrl(a ie noDpttrlEa ao6tr €eor Kpyra?

Je norpuruna [oJryKDyra Beha otr noDpttrxre

mr"poyira?

(OarosoD: 3a 5,13 cm,)

L IIpaBoyiaoER, Aj/rl(xu€ 8 cm,Irrtrpale 4 cm, n iena[ rpyr riraiy ie,uaraco6nMe. ltoia or orl[ of,rlTa qMr leiytroBp[rrEy n la xo,ruxo? (O.qroBop: &pyr, to13,8 cd np 6,ruxxo)

9. Jerar npotop cacloin ce xr nparo-

lnllosf,Ka rl tro.'tyrapym {a.q rb€roBoM Kpaiom

crpqntuIoM, ralo lltro noRa3yie cirKa 64.

nDeira ,uEl|eslriaMa otrarqcsBM ra cnBumDsryraiTe o6tnl f, ooEpuruBy roi [po3opo.

(O.uroEop: O * 6,37 ei ? -2,2955 tr)10. Axo troapu{ra cBar(,t (p ra aa

cnFru 65 rlsocs 314 md, Konlna ,c troBp-

Cr,65

uxEa a) otrype olototot M '.cro,6mrrr, O ruaeby rprrola?

ll. tlr po\rba oa r(ApmBar ocaoBnic

scm n Bic rc 3cm rpeda mpelaai aaioehE

iiyr, Kor Ka norpurfisa oqr"aie? ([lra te

trpeqnrK xplrr(tlxrc ynrrcatre y poMd?) (O.qro

Cn, 64

7. Y lonyxpyf npe,'dl8s 6 c[, Yrm-

lttr:re icaiar(oKpar{x nps3oJfn[ Tpoyrao rarao



12. Yrop€Arre fiorprrme rqyroBrqrix cy mn 4newtrx 1 cD, 2 cm' 3 cm,

4cm...., na o'&noBopxre sa [xraEe: r{axo

3a {cB loapr Ea Kpyaa oa trqlynpesllr€?

13. O6nl,l ie.qtr€ xpyrftEe ,[Boparc ie43,96 m. KontRo ie m' naprera rorpe6ro

a4 c€ nolcprie llo.[ alopaBe?

14, l(a{o hEre y farnfli f,atrpaBrrf,

.r(pyx(sll qBerrsa|., roBpuxxe 20 m2?

. 15. I&,q cEa(oM crp.nEtoM: r) r aA-para, o6nlla 16 cn, O iefiar(oqrrlAtlrqrcrTpoyrna, o6{Ma 12 cm, otrf,&rre ctroria

nonyxptr/fEr|uy, trs- rqEsylaiTe rcspIIrlIEIr

uercrq/flne cn .e. [O.aroropr a) 41,12 crr'.

6) 25,76 cDl'1'

16. Y Kpyr, nolpilme 12 cNnt-56 lnm,,

''nxcas

je npabllnf,{ uecrolnao. KoEqfpyfi.

ufare cJ &(y. (y pa{yEy ym4re&fle 7f -:3,14.'

5. NOBPIIIT{IH KPY?I{HOT I{CEqI{A

Kpyxsu qceqa( je .qeo Kpyra orp3nurleH ca aBa norynperlgtfrb r EfiMa

3axBahcHr.rM rpyra{uM nyr..M (OAMB, c;r. 66). Itrerory rroBpurtrHy ao6uheuoHa oBaj HautH:

Kpyxxn uceuar KoMe,npurraAa qen

Tpanur{ yrao oA 1' nua nonpurxry 360 uyr-

MaBy o.q noBprun'e *py"u, gu*" P: {360

Aro ruy cglorapa uenTpaJrHr.r yrao og 2'noBplrrrura he r'ry 6nru 2 ryra rcha, rj

--1P-'l'_ 2, HT,[., TaKo ,rla 3a rlefl rpirntfl yrao360

oA c[ crenerr{ troBplunsa uceyra nocraje:

P:n'" ' n.360

Ha npmrep, aro je y rpyry, flonynpeq-

urxa 10cm, qeHTpanHI( yrao ct:18o, noBp-

un.raa ogronapajyher race.rxa 6uhe:

p:nr'.n:3j l-99.r4-15.7. ro iecr t5.?cm,.

360 360

(I,1 osae, Kao It npfi ltpaqyHaBarby ryxnge Kpyxgor nyxa, MopaMo

BoAI{Tr.i pa{yga o roMe y rojtv je je,annuqaMa ,qar qctsrpallH[. yrao.)

Kao ruro ce u3 Ii3Be.qaHor o6pacqa BuAu, 3a ll3paqyuaBarr e rloBpllturre

xPyxHor Hcrqxa norpc6no je, Aa. sualro- ncryrp€qHr.rx, rpyra r.r ogrorapajyhn

rrcHTpa.lrHlr yrao. Caaa ht lro alrr"ru aa ce ra noBpu[Ha uoxe go6nru r.r xa.4je yvccro I(HTIXTJHoT yrra no3Har nyr aceq(a,

Cerur'ro ce aa cMo 3a rnpaqyHaBarbe , Ayxuuc rpyxHor nyra IBBentr

o6pa:au.

, rcrd,

180

C,r. 66



Ha ocnoay rora Moxc.Mo Ea\yrcartn:

+:t. : .u, rj. noapmnu Kpyxnor nceqxs jesnara je360 180 2 2 2

trpoU3BOAA lLerOBOr Jr]'xs ll tronynpe{llnKi Kpyro.

6. NOBPMI,IHA KPY)I{HOT IIPCTEHA

Kpyxlrn npcrcn je, f,:o trtro cuo pannje Blgenl,t, Aco paaxlr raslrelyKoHueHrp[qHe rpyxH]rqe (c.rr. 67). rlog

KpyxHor npcTeHa no,apa3yr{cBa-

pa3nr.rKy nonynperrnu(a glejy rouucnrpu-

xpyxu qa: d na cnut\l, ro jecr d:r-1ca r u r, o6c,rexrauo noryupc{HuKe,)

Ovlrnegno je 4a norpurnry rpyxnor

Mor(eMo Ao6urn KaA oA noBprxxue

rpyra o.Ey3McMo noBprxr4Hy Marscr,

Eax.nei P:Er2-trr2, unn p..rtr4Ma: no-

rcpyxHor npcreua je,uxaxa je parnnql4rajy xorqenrpxruf,x t(ptroB .

3MAUI,I 3A BE2I{BAEE

l. Konnl(e ie noBDulrra raDyxxor

ar{o je nonynDe{DlK Kpyra (r, tyrao Kojx My o,uroDrpr (d):

a) /- l0 cm, ct:75';6) r-2 nl 4 dm, cr:40";r) /:2 dm, d:20"20'ir\ r:2,4 n, a =20" 20"1

2. rl3pa{y{rjre noEprtrxy Kp:},xror

Rrlq ie nonyrtDe.rfirK xWtu (r) rnt'rc f/,/:

8) /:?l cm, /-45 cm

6) .- 0,5 m, /:7,5 dm:

32a) r:- ,yKa, ,: l- dm.

43:. Hlpralre xpyxnqy x y rboi iealy

(MaBer), O noEpurtrsy onroEapaiyler *pyx-

l|of toeqka.

f. Ko.rnoor f,cerar oAmBnpa ny\T oI12 cm y rpyry, odtrMa 62,8 cm?

,5 Y *py^t*y y,,f,u,xre npaBxnxrrrscroyrao, tra Epa{yaaiTe [oBp[rrry r(pyrft-tor oAceqta uait ie.,|lrdM tbenoBoM crprEn-uoM. (AKo oI (nyx&or r|oe{xa oAylMereom,raaDaiyhn rFryrao, Ao6[here I(pyxEI{o.qce{aK.)

G llo,nynpeqnrllrr .qEeiy l(oE{erffps-rrElx r(p}?l.qxqa cy /:21 cm u /:11 cm.It3pasy[ajTe r ptlEy f, noBpurxfly taD],llaortrpcre.Ea.

f-ooxux 4mjy KoxuefiTr. rx r$y-

C\, 61



8. I(oa F;uo_ xpysor qpcrena [o3-oara ie urpa" d : 4 w, t o6xM Beher

kpyra, 62,9 <r. llalrre troBplrtxtry Torl(p .n rror rycreaa.

9. Koscrpltxrrre KBa,qp,tr, crlaruqe

4 ctll, o|lxrrr':le My r ymllxxie Kpy)tcllrqy IttIa{yxajTe troEprxrry Aobnienor rpyxnortrDsrea&

10, 3anara( Rao 9. 3a jensarocrpa-.

stlqr rloyraq cTpalllrqe 6 cm x 3a trpa-

arullfu uecroyrao, crDa$sqe 2 ccl.

Il. rlrplqyDajre'tospfiruty

KD]',asor

lceq$a 3a r-4cm, d-20o x r:4cm, cL-

-40', na u lmoDeAxre. Illra.rr rora

lrcrKeTe 3aKJLy9tft?

Cn. 68

Cn. 69

12. It3pa{y[aiTe rlospulr{f,y Kp].,otoriceqr.a 3a /:2cm, cr:30' rr r-4cm, d=30o,Ira sx ylopeAlTe. IIITa tlt Tora MoxeTe

3arc,D ArrT'l?

13. ORo orplTror px6Ea8a, qrill jeo6nr 50 m, tTe6a sa.rll@rfi sra3y srrpoKy2 m. KontrKa je troBpuuEa re sra3e? ([Ipi-6Jnrlrtro 113 m?)

14. Ko.r t(y [oBpulrsy 6pmxe dpnceq

Ea lalllTmloM cmxny MoTopF x Bo3rnaaKo My ie 'oyxru,r 16 cln, a Bpr My y Kpe-

r€I}y ourcyie n}'t. oI 120" (tlprl6fi,r(f,o2,68 rtf)

15, Ha $rypaua [percras&esnM f,cJrrqn 68 lllMeDxre Erro te norpefxo sEprqyDaire t€nll.tuft oorBqeEor,Uem trOBIltE.

16. IttMepf,re uro te norlrcEf,of,3p6{yEaiTe troBpunBe {eJtorytEf,i oryDaEA CJrr{f, 69.

17. IlpolopcKo or(Eo rd{a o6ru( Dpe -

craB,'Ldlr Ea cnliur 70, r,Ee Mepfix bpoierrayr$x o3tra{aBajy MeTp€. xonnra je noopurr-m npo3opa? (O,uoBop: rlpr6n. 3,? Drr)

+



18, Hallrtajre ,[ e t o{qetrDFrxelojtlmpeuf,&r l0 0 x 3 cD, crFIOU OI z0o, na hEre no6rTx

Kx[nl'or rlETer:L r{ao uTo ie rrpea-

(yraEeEo) sa c,jts'I[ 7l^, Ilc€qa$eHlrc]Eqla r cnalaE€rr oForapaiyhxx clto-

KDalers Ao8d. ce 3ar.noE ta sroryry (cn. 716). lblawrajre llosDrmysl(ri f, t. (O.qrorop: 190 c0r)

19. H{ c.rE[ lZ r<py7m ry* ,fl--3 co r O,{ -3 cro llt'4gryrsitez LAOB

mrqDuxEy rce tr.a ,4O, (yJerr r -3,14).

\r---Y/

Cn. 70

Cn. 7l

c.n. 72



UI

nPonoPq[oHAJIHOCT Ay)I(U. CrntrHocTMHOTOYrJIOBA

I. AEJbEbE AY)I{PI HA JEAHAIG AEJIOBE

,{o caga clro oltMcrpanoM ryxl{ Morntr, xolrcrpyxunjcru, tro,qerrlrrr{ aylr

sa'2, 4,8, 16.... nr.q. jegxarnx .[eirosa. Caaa heMo Hayquru KJro ce Ayx

Moxe no.qeJntrtr xa 6rano rojn 6poj jegrarltx aelom'

flp:rnocraaulro aa ayx AF (c:r. 73a) tpe6a nogearlu Ra rplt jearaxaAesn. vz rpajre rarxe ,{ noBy'IlMO [onynpaBy ,4x y npourronHov npanuy'

CJl. 13

aJrr noA yrnoM MarbLrM oa 180" npelta AF (3ronlo Je ga raj yrao 6yge ourap)"3aruu ra ry nonyupaBy, norlcBllll{ oa ri.qxe ,{, npexeceuo jegxy nponuo:ury

ayx (AM) rpunyr. Taro ao6ujar,ro Syxtt AM, MN t NP' flocnegrcy rarry

1i) cnajar.ro c apyrr.rM xpajcu (fl) ,4are .aPKt{' a L3 raqaKa M n N ryrer'lo

lapanene ca PF. Taro ao6ljau: taqxe B u C, roje gary Ayx acne Ha rprl

je.{uaxa ge;ra.

\a je AB:BC--CF, norasahcl'{o osaro' I4: ra'tara M x lV noBna'Ir {o

[apanene-

ca AF, na ac6njauo rirvxe M, I N, (cl' 73 6)' Tpoyrrcau AtsM'

MMrN n lf?flP noAyAapHn cy (:z.urro?), Ha ocHoBy trcm rpou3ta3n aa jeAB: MMt: NNt. Icnrro cy tlerBopoyrnoBl'I BCMiM u CFNtIV napa'nenorparoa

(npoxsnaru l€ caMe Koltcrpyrqnje), BC:MM, u CF:NN1 lpeua rorrle'

saxruyvyjrr,ao aa je saucraAB=BC:cF.



Ha IIcrH Haq]rfl Moxe ce noAeJ .rr[ je.qHa Ayx r{ Ha BuI[e jeAr{ar [J( AeJroBa.

Ogpe,quMo, npuMepa pa,u,fi, rprr ce,qMHHe Ayxu MN (cr. 74). Cntrslro

nperxoAnoM trpuMepy, Ha nonyrpaBy MP rlpcnecnlz.o npox3BoJbr{y ryx 7 rlLyra,

Cn. 74

n xpaju noAerbax P crmjauo ca 1y'. Ilosro ce rparKtr * **, MN, w rpe-.I

ter noge.rxa -R roBaaquruo R^SllPl{, na raxo raqra S genr gyx Ha AEa aeta:

'taMS:1MN U SN: '- MN.

H(Ka yqe{ruI o6jacue sanrro oage xnje 6r.r:ro norpe6xo ayhu napa:reno

ca P,V m cBrD( troaiJbar&

2. MEPEBE AY)I(II. 3AJEAHI4qKA MEPA

II3MepnrE jeary ayx 3xaq[ oApealrrrl xonlfio ce nyra y noj calpxn

npyra Ayx y3era rao jeaunnqa. Maby Ayx (CD Ha cL 75) rPeHocrlvro Eec-

TapcM na Behy(lB), l,r axo ce oHa caapxtr y rchoj

ueo6poj uyra, orga je

oHa Mepr re AyxI{. flpcua o:xa-

traMa Ha clsqlr 75, ,ryx CD je

Mepa Ayxlr AB, jep re y rroj ca-

AP]Kfi TA:IHO TPIIIIYT.

Kao rrro xav je nosnaro,

6poj rojn noxaryje roamo ce uyra

caapxn jeaxnnsna ayx y vepexoj

A#B

cl+-rD

,[yxR 3oBe ce Mepur 6poj tutpcxe ayxu (xojrje

y oaorra cnysajy 6poj3). To

je yser nenr"renoaan 6poj. .{yxuuy Mcpcxe .Iryxrr ao6xjauo rag ya r''lepnn 6poj

(,3Haqr{Mo n jepmn'ruy Ayx. Y nareacucM npnMepy, ayxsaa ryx. AB je

3.CD. y reor"rerpnjcruu MtIxExMa, rao ll y nparct' uajvcmhe cy je4nrnrue

ayxn Merptr n rehe r uarrc je,unHlrqe r3Beacre oa Merpa' Ha npuvep' crpar qa

xEa.4paTa 4:4 cm, pacTojabe usuefy .qDa Mecra je 15 km, ura'

c.n. 75



lI

,{a ;ru he uepnn 6poj Ayxlt ocrarr.r HetrpoMerleH aro .ryx MeptrMo pa3HHM

uepnr. vr jeAnxuuaua? O6jacurre ro npnMeptrMa.

9ecro je norpe6no Aa yropeAnMo

M# AyxfiHe,qBejy AyxI{. HeKa cy raKBe ABe

\yxa MN u PQ@1,

76).

[yxa, noja

Pr-r.-.+-r Q ce y nproj cagpxt 5 nyra,.a y 4pyroj

,-g{'

I rryra, jeore rejegnn'ma Mepa rlt( .uDe-

' cn' 76 JY ayxn'

Ifocroje n raxBe Ayxu :a roje ce

HxKaKo He rrloxe nahu Ujegnurxa uepa, rj. roje neuajy 3aje,4Hmry uepy. Tarae

.ayxn 3oBeMo EecauepJblrBru, 3a pa3nuxy or onrx roje nvajy:ajeaxnrxy r*,lepy

n roJe @ 3oBy cdMepJLnBe AyrKlt.

3. PA3MEPA ABEJY AY,KII

Ha c;rlrqu 77 ftare cy .ue ayxrr, AB n CD: I4:r*lepmr.to rD{, peqnMo,qeHrrlMerpoM (xao jegrHr.qoM), na 4o6njauo: AB:6cm t CD:2cm. llpevaToMe, trpBa. Ayx je rptnyr Eeha oA Apyre, [nu: rpyra ,qyx ce y npDoj ca4pxfiTpr{nyr.

Ko:rrqufir ABejy fry)rrl rra3r.rBa ceBr. (oBa prtMepr (llrrr-r oAtroc). Pa3Mepa Ayx{AB a CD o3riaqaBa ce ca:

AB : n"tco.CD

A r---------r----- B

Cr-rDCL. 77

ruro qr.rraMo: Nx A8'wMa

Fyxn CD.

Ifonrro crrro re Ayxtr rrMep[nu, MoxeMo sanucaru:

B -3, -,u AB-3 ' CD'CD

.{arne, paruepa rBejy nyxrje HerMenoBau 6poj (3) rojn noxasyje xorurxoje uyra npra Ayx (AB) reha oasocro MaEa oA ryyre (CD).

Pasrraepa glejy Ayxr{ rre 3aBr{c[ oa rc6opa jeglHfiqe ayxr{He xojoM o(

MepIrMo: rnaBxo je ga ce rltxorl. uepun 6pojean oAHoce Ha f,cry .(yxnxcKyjeruauuy. Ha npuMep, aa cMo y trperxoanoM ctrytajy yte:ln 3a Ayxfiicxyjemunuy Merap, ao6rulu 6lz.c:ll.o 48:0,06 gmlu CD -0,02 m, rr rbtrr(ola pa3Mepa

6wa 6u

AB 0,06 6^

cD 0,02 2

46



Koja je pasr"repa ayxu MN :5 m PQ:1 m? Ilpeua ouolrl lrrro cMo

perrr o paruepu, 6nhe

MY: 5- . rc iecr MN: L.Po-PQ 7' 7

qrro 6ncuo pevrura o6jacnx.nr.r E oB{ltro: Ayx. MN tta;aocv 517 lyxn PQ,

lI yonurre, aro je xo:rN{trur rrejy .ayxr, a n b, leauax 6pojy t, rae je

k Her.rMenosai 6poj, *uxona par.repa ie a:k.

lgxu a u b cy vranonu pa3Mcpe, u ro d npBr,r u.nau, a 6-gpyrlr vlan,

Pa:lrepy cuo y[o3r{anr p:xnje, npn uprarby nnaHoBr repcra, qnra:Ilt

crrao je ra reorpaocrnrr,r trapraMa, IrT.u, OHa HaM je trora3xDara KonHRo nyTa

cy pacrojarra y nprpoAr{ aeha o4 ogrorapajyhax pacrojarra Ha firaHy, xaprrt

(l : 100, l:10000...).Peruuuo caaa jeaan gaAarar,

Crpanrue .[Ba KBarpara croie y pauepn 2 : 5. A(o je crparuua reher

Kaaapara 15 cm, xornca je crparnua ueirer?

Osgawurro nlr crpan[Ile IGaJIp3.Ta: Mar$er ca a ra seher ca b, rulaheuo:

o*u* o:2 b-L.l,s.=e, ro jecr 6cn.

4. nPOnOPqI,rJA. ITPOnOPqUOHAJTHE Ayxrr

Pa3Mepa ayx AB s cD Ha cnuutt 78 j" +-1, jep ce usnecxa- cD 5''e.qunuqa (m) :a uepe*e Ayxn ca1pxu y nproj 3, a y apyroj 5 nyra. Pa:uepa

B

A

a2b5

Apyrux ABejy A1':x.u, MN v PQ (Ha cnro$),

, MN 6 3lecTe

-: -: -,noIuTO ce HeKa Je.q -

- PQ l0 5'Hr.rqHa ayx p caapxr{ y npaoj 6, a y apyrojl0 nyra. floruro cy re ABc paruepe je4-

HaKe, MoxeMo r,lx llsjeaHaqnru:

+:y lrnx AB;cD:MNipg.CD PQ

Xsjenuavyjyhr.r,{re jegnare pasuepe

ao6r"ru cuo je4narocr, roju ce soae npo-trop[rja ]lrlll cpa3Mepa. 9errpu ayxu xoje

wme nponopuzjy 3oDy ce H,eHn qJrtHoBtr

xlu nponoptrona.le.

LC

Crr. 78

N

M P

47



3a rernpr Ayxtr KaxeMo Aa cy npotropqrroxaJrne xag ce o4 ronx, rj.oA IbIrxoBI{x vepnnx 6pojera, r"roxe o6parorarn nponopqr.rja. Ha nproaep,qerltplt Ayxu cy nponopquorra:rne aro cy rrM MepHn 6pojeau (pa:yue ce, y

oAHocy Ha trcry MepHy jeaurrlry) 9, 15, 2l u 35, jep je

9 321 3 921na le -:-5 s'35 5' " 15 35

Pemuhelvro caaa Eere Ron-

crpyKur.ljcKe 3aaartre y Be3[ ca

pa3MepaMa r npouopqujaua.

l) Aary ryr noAeJrnrx yqrrol prruepn.

. {.. HeRa je glra FXx AB

(ctr. 79) a ggra pa:vepa J- .

4

Ha ayxr AB rpe6a uahr lavryC raxo .ua oxa Aern ayx ld

lb xpaja A syrm. ABnoByqtrMo npor.r3BoJLHy tronynpaBy,{X noa omrpnu yrnoM rrpeMa AB. Ha

rylonynpaBy, fioqeEru]r o.q ratr{e ,{, npexecxrvro (ueorapoM) jegny npousro.rnoy3ery Ayx 3+4:7 nyra, na ao6tjar'ro ravxy M. Oay ravry cnajar,,ro c Apyrr.rM

RpajeM (t), a ns rpeher no.qerbra (N; oA ra{re ,4 ryreuo NCll MB. Tawa C

.qenrr Aary Ay'x. AB y pa:vepn 3 : 4, r rr.r: ,qyx[ AC s, CB nponoperonajrHe oy

6pojenur'aa 3 u 4. O6jacxure :anrro. (Bugerll- uperxoAHrr ogorar-)

ACCa. 79

2) lary gyx tro.qerrrrn lrtIBA AeJrA npOtrOprUOHanHO

' cMA AArtrM AyXnMa"

Hera je aara ayx,{A (c:r. 80),

rojy rpe6a noAeJrr,rrn npotropqflo-

na:rno Aynon4a c r.r 6.

Ha uo.rynpany, IIoDyqeHy

npo r.BBoJLHo (nog ourrpuu yraou

npcMa lB) us ravxe l, npeHecr.tMo

(mecraporra),,aare .ayxr a u b, na

B na aqrose nponoptuoxa:rxe 6po-jennrvra 3 u 4.

A,cCr. 80

Ao6njaMo raqxe M u 1{ oAnocHo: AN:a, NM:b. C;aga cnajauo M ct B, t

ry*uo NCll M8. 'tasva C saje peuere nrcrxa, iet F ,49*:f#:+

48



3) .{ary,qyr ymharf, y parnrepr 3:4.Oraj oagarar pentrheMo roxcrpyrqajcxn (Ee M€peBeM). Hexa je aara

ryx AB (c:1. 8l). ffo,qemnao je Ea rpf, jegxara gora. Aro je4ax raftla aeo(rpehnuy) npeseceMo Ea rronytrpaBy OX tetupa nyra, go6rheuo raury CrraKo aa je

oc -4, z.tru oc:4-,A8.AB33l

'rj. Ay:x OC n3nocH- A1:vr AB, xaje raro ryx AByrehanay pa:vepn 3:4.3

4) Asry .(yx yMorbrrn y pa3-

uepr 3:2,

Kao r.r y nperxoanoM rrpr{Mepy,

pa,qlheMo xoAcrpyx4rjcKll.Hera je aara ryx AB (ct.82).

Ey rpe6a rroAeJrurn na 3 je,Frara

,(ena, na EaqpTaTr{ Iyx oa,qDa TarBa

.qera (.qBe rpehune gare gyxr).

flpeua osaaraua ud cnuuu:

oc *2, sr' oc:L..aa,AB33

. ryx AB 1n,l.aarrns. cMo y pa3Meptr 3:2, na, a:ll.o ao6r,nn.Nx OC,

Hera yrexr4u peme 3. z 4. 3aAnrar MepebeM (y unmnrerprva) garxx

r.rTA.

Crr.81

I fe6.e1'I'&Ja 3a VII paslrqA ocrsBae trBore 49



L JeaEa ,qIra rMa rqnN.u{y 7 m.lllra

ie Eea MepE Spoi y o,Focy a qerrrusrap?

a Aro cy uepm bpoieDr ABeiY .ql/r(trE n 12, m n{ ce Mor.e peh{ xoia ie or

llu ,rqnxa? 3aurro?

3. Mepu{ 6poi ie.urc .4Jxx Y oaaocY

sa leurf,Merap ie 15, a Mepd 6poi apyre

all,l.n y onsocy sa ,le{rMersp ie 4. Koia ie

o.n Ed ,{}rxs? 3a r{onnao ie ,4/tas? Ko,IE(o

q'm ie ayxa?

4 Y Koioj cy PalMePx: r) qef,Tr eraP

np€lwo xeKmMerPy' O MxnllMerap llpeMa

,qerlnMerpy, B) aeqf,Mcrap lpeMa Kxnorrerpy,r) 50 Merapa lIPeMa 2{) KxnoMeTapa?

5. Aary nyn no.qerxrx f,a 5' 7' l0

ietrsalcld AenoDa.

6. Aary gyx no,qenilx Y PasMepll:

a) 3:5, 6) 2:1' B) 4:1.

7, Aary ayr. flo,rejrxre Y PosMePx

314, na ua,q cBaKrM neJroM Lotrcrpyfilnfle

ieA{rlrocrTal {fr rpoymo. Y r{oioi pa3Mepf,

je sscuf,a Ma6er rpoyrta [peM Bnclulr

rehdr? (ll3Mepnre Brcl|{e y mo Ir aaDrre

Da3Mepy.)

8. tla ayrrE ,4.B oapeaf,Te raqry C

rar.o Aa tEBMeDa ACicB 6YAer

11a) 1:5' 6) st7, 3) Tti.

3AAAI{,I 3A BE'I(EA,bE

9, Aara ie iea[a ayx. HauPTaire

apyry ry,t Koia he c,aar:oM 6rra y pasMep :

ltz, 2t4, 2t7, 3:5' 3:1' 4:3.10. Hatute DatMepy ie.qse Katere l'.

lral|oreqy3e cEor lrpaBoyrtor rpoltraoExxa.

tl, I4c raile aa nx.cy oBe nPoIloD-

Utjc laqf,e:

21 35 2Q 25

-:-- 15:40 12:32. - --.27 45 2a 42

12. casraBf,Te rnonopqriy oa qettrp]r

'['.i qnix cy Mepn{ 6poieB (y oaaocy na

f,cry ,n/rKIrEcKy ieAf,8E{y): 8, 2t, l2'14.

13, Iftr- AB-a cn troAejlrre ua IBa

asra, Kojr he 6rrtl llponopqtlolraJm{ c ABeMa

AArIM Jlf)I Maf a : 3,5 qt t b : 2'5 crJ.

14. ABTy a]'tta lroaenlrre xa,{Er acurt

' Koj he 6rrs nponopulotratm c ABeMa aa-

ruM aYrriMa.15. AarY $.r.: a) YBehaire Y Pa3MePr

2:3, 6) yMaExre y pa3Mepx 5:3. Pa'lllre

6e3 MeDe&a, caMo (oEcrDyKullicKf,.

16. ,AollyfrTe npaalra Mecra (o3xrqeRac.noBoM t) 6poieBlMa, Tario aa cneaehe ieA-

{aKocrt trocraFy trpotroD{de:

8i I'9 x J

r) 8:.x - 36:9, t\ 3:2-x.4'

5. CJINqHOCT. CJII'IIIHOCT TPOYruIOBA

3a npe.uvere Ha cJIuIln 83a, xao u 3a oHe Ha cnrug 836' xaxeMo ' a

cy cnll9flH. IIIra je y I5EMa I{CTO' s6or qera ao6l.',tjaMo cJlrrqaH yrl{caK n y qeMy

ce oun pasluryjy? Aro r'rx MaJIO naxJblrBnje Dre.tr3MO' yoqflheMo aa cy f,crd

obn €, a pa3Jr[Tlre Be'rxqltHe.

gryry VVE6)a)

Cr.83



lI y reoverptjn, caMo ca Mnoro BehoM Taqouhy Eero y cEaRoAEeBEoM

(,,cur cnr,r{an o[y"), rra3uBaMo clrnrnr Qrr5rpe roje cy ncror o6.nmapa3nFu.rre Benr.rqrure. Ha lpnuep, cBfi rBaApaTr, a trcro Taro tr rpyrsfiq:,

xoItre (cn 84).rvrefyco6no cy cnrnre Qurype.

r r Iooonflfl'I. 84

Caga herro aera.nxnje npoyqnrtr cntrErocr aeronr reouerpujcrux nypa,6ucvo yrlpaxrm roju ce reouerpnjcru ycnoEr y. roj cnrwocru jarurajy.

teuo EcrrtrTlrBarr{ cJtFrEe ryoyrnoB€.Hauprajuo npoa3BoJLHo jegax rpoyrao, Er trptrMep AABC (ca.851, a

Apyrt q)oyrao, ArBrCr, ca Farryr nefuna crpasqaua, m jecr raxrrna gaArBr:2.A8, BLq:2'BC

CrAr:2. CA, rlrro MoxeMo

n y o6nrxy:

. BtCr o ,. CrA, _"AB BC CA

A.B, B.C, C,A,--r--1: I ': I '.

AB BCCA

Taxra gpa rpoyrm rnvrajy

flpoflop4ronamre. AroyrnoBe y TrrM TpoyrroBtrMa yflroMepoM, rum ynopemi{o br oBe

ruecrapoM, yrepnher'lo ce M c-y yr.nosrr jennor rpoyr.na jemarr pe-

ca yrnoBrrMa .qpyror, To jecr .qa je:

LAt: LA, LBt: LB, LCr: LC.

Kaa ,4ra rpoyr,na xcuymarajy oBa aBa yc.lora, rj.4l

cy ntr crpaulrqe

rr yrnoBtr jegxanr, uaonaajy ce clrF r. (Om :azcra ruraajy

o6arr, a pa3nr ry EerustHy.)

Ilocuarpajrrro rroaoEo r?oyrnoBe Ha cJmqa tl n ynrnajuo ce xoje cy

trpotropqtrouaJme. Crpatrrrrla AB y rpoytty ABC, nex.t r.acnp n

C, a crpaHrrqa d,8, y rpoyrry ArBrC, texJtr nacnpartl yrm C,, rojr

A ,Arclr. 85



je jeAEax yory C. Crpanr.rqe AB n Arq cy oFosrpoiyhe crpasrqe. I4cro

TaKo cy oAroBapajyhe crpannrre u BC n BrCp jep lexe xacnpau jegnamr

yurora I u l, a ranole It crpaHl{qe AC t ArC., jep nexe nacnparnr je,axaxnx

yuoBa .B r.r Br. Vr.roae roju 4exe nacnpali npouopullouir-mrrx crlxrrnqa E:BBa-

heuo, rarofe, opoaapajybru yrnoo Ma. peua roue, KoA cntrqxrl:r lpoyrtoEao,4ronapajyhe cy oHe crpaHrqe xoje .nexe tracnpaM je.qHalqr( yrJIoBa, a,oaro-

rapajyhu yrnoBu cy onu xoju nexe Hac paM npoflopqlronaJlulD( crparuqg.

. floc.ne rora MoxeMo pehu: c:rurnn rpoyrtoBll cy our roir f,Mrjy ]rnolepenonr je,qmxe r ogporapajyhe crllrmqe trponopqnolraJrHe.

Pa3Mepa rBejy o.{ronapajyh[x crp:rrrxqa roA ,qBa cn[sna rpoyula 3oBe ce

pa:rMepa rnlr roer[xqnjeur c.lrrrtlnocrn ra nFa rpoynra. 3a c;ryraj sa cflr4r 85

pa3Mepa cntrtlHocru ie 2, o$rocno ].2

One ase oco6J.rHe cnniIxtrx rpoyrtora -jegxaxocr yrnoBa tr trpotropqno-

xamocr oaroBapajyhux crpannta - nay yB€K:ajeguo, mro 3gaqlr: a(o rpo-

yraorn ruajy yr.nore jegrtaxe, btrxoBe oarooapajyhe crpalrr'qe uopajy 6lrrtrponopq olrarffe, u o6pnyro, atco cy EM crpaHRIIe peroM npo opquonaJllre,

oarogapajyhN yr:roau vopajy 6nru je,urraxn' (Vvex4g HeKa ce y oBo yBepe

rrprajyhu cnFII{e rpoyrJroBe ca pa3JruqHTItM Pa3MepaMa crrrtluocrlr' Ha trplrMep

jegan 3 nyra aehnx crpanrua og gpyror, ur,4.)

Oro olroryhana Aa najeAqocraBan

Haqrff uctrllTaMo M t*. "y rparpo-

yrna cJrlrqna: AoBoJbIro je yrnp.uffx Aa cy lIM no .u.ra yr,ra je,qrara' na oar,Ly-

qnrfi,qa cy otrrr cm{qHlr, jep cy llrll onga u rpehu yrurorr jeguarn (aauro?),

a x o4rorapajyhe crpanlrqe nporopul{onaJlae. I{s rora :axryvyjeMo: lno aaa

rpoyrnr trMaJy tro pa yrnr je,4naxa' oun cy cJf,q[tr.

Axo cy y nnra*y npaBoyrnn ryoymoBx' sa yropfurarc EtrxoBe cJrnq-

Hocrl{ AoBoJr,Ho je 4a ce ua$e no jegau (onrrap) yrao jeguar. 3amo?

Peurnuo caAa HeRe 3aaarxe o cJlllqnocru rpoyrnoBa.

l) llcrrrarr A. nl cy c-.trx

.rur rpolruorn lBC rArBrCv vlxie cy crpaH4e

a: I m, b: 15 dm, c:200 cm

ar:20 dm, 6r: lOQ c6, a, : { 6.O6paeyjuo pa3Mepe o,qro-

rapajyhux crpal ua (vepxr 6Po-

jealr aa ce oaHoce Ha ncry jean-

snqy):

a lb 15 1c 200 Ia,, 2'b, 30 2' ct 40O 2

Ilormo cy re pa3Mepe jeguaxe, rpoyr.noB cy cnntrna' tra trfilueMo:

trABC-[,A1\cu

Cn. 86



a qfiTaMo: 'rpoyrao ABC cnFIaH je rpoytny ATBLC\. Kao Iqro BnAl.rMo, 3Ear3:r cJlr{trHocr cacrojt ce us o6puyror rroJroxeuor Be.luKor cnoBa ,S (-).

2) Aro xpo: FeKy rrqKy crpsHfiqe rpo]'rna troByqeMo trapureJry c lua

xojoM oA Ap)'rnx ABejy crprHtrqa, oBa nap&nen& ogcehn he rpoyrao

cJrllqiH aaToM. AoftaxxTe.

trI: raqre M Aaror rpoyrta ABC (cl. 86) noayueua je npara MNllAB.Tpe6a uora:aru Aa je ,,o.qceqeH" rpoyrao MNC cntlqlar .rpoyl'r,y ABC.

O6a ra rpoyr,ra uuajy no jegau yrao jeanax: ro je saje.qHuqnr yrao C.

Iomro cy orurpn yrnoBn c napaneJrrrr.rM xpa{r. rra je.qxarn, cnery v LCAB:: LCMN,

Iourro lrllajy no aBa y$ra

ru rpoyrroBr.r cy cnrrrrr{,

Id'Kre AMNC- LABC.3) AarE Tpol'rao yMaBtrrn y

pa:uepu 5:3,3agarax saxrena Aa ce Ha-

upTa Tpoyrao cnllliaH AaroM Tpo-

yrrly raxo 4a je pa3Mepa cJrr{rlHocrlr

,r3TOr npeMa xosora 5 :3,Hera je ,qarlr rpoyrao ABC

(c:r. 87).Ha

cfi.rqr{ c€ BuAu KoH-

crpyxqrja no,qere ocnognqe. lBua ner je4rarux gelooa. na je

1tAD:1AB n DB:AAB,

55

tr{s ra.rxe D uyteuo DEll BC, na je,,o,qceqeun" rpoyrao (a}r.an nperxo,qgrr

3a.qarak) cfl.rsaH AaroM LI ,,]MaILeH" y pa3Mepl.r 5 : 3, aar:re LADE- LABC.

4) larx rpopao yre-hartrypatMepr2:3.

O6pnyro uperxognougaaarry, os,qe rpe6a Hatrprarn

Tpoyrao cnrfiaH AaToM 'rpo-

yrny raxo .4a pa3Mepa cnlrq-

HOCTI{ AaTOI npelvla I{OEOM

6yae 2:3.

Hera je garlr rpoyraoABC (cl. 88). IlpeuororuuoocHoBHIIY l4 \a je AD:DB,r jegxy no.norury npeHecuMo

Cn, 87

5?



na trpoAyxerax ocnoBrqe, raxo ,qa je BM:AD, nnv AM:aAB. Cag, uz M2

ryrerr,ro MrYff BC Ao trpec€ra y'f ca npolyxexou cTpilHuqoM AC. lo6njeuytrAMN 1e crrsar jraroM ABC (vwro?), r piBMepa crFftocrtr Aaror q)eMa

EoBoM je 2:3.Tpoytao AMN 6u ce uorao x noce6so ronorpyr{cartr, a trcro raro tr

AADE. a1 nperxonEor 3aAarra.

. 5) Aar & rpoyreo lBC, srje cy crprus a: 12 em, b : 15 .cm, c: 18 cm.Je4au rreuy c,mqrx Tpoyrro, ArBrC' tva, nrjnrany crpouxqr ar:8 cm.

IlrparynrJre ocraJre aae crpruqe, b, t cu oBor rlloynra n Habf,re

PSMePy Crf,ctrocrf, rt ,Et Tlloynn&

Karo cy ogporapajyhe crparmue rD( rpoyrnoBa npolopqgoxaane, o6pa-

:olahevo Hajnpe npouopunjyb, :o,ba

xJrn, rocJre aarueue 6pojuux Bpe,qnocru;

4:1. oou-" ie 6. : ro-t5 12'

a sarnrr nponopuujy:

L = a, 16 is61 3 :1, oou*" je c, : 12 ".. a l8 12

,{ar:re, crpanuqe rpoyrna cnrqgor AaroM cy 8 sm, 10 cm z 12 cm. pae-

Mepa cnntlHocrrr ora pa rpoyraa je

al23/a,82\a, 8 2 \ a 12 3l

3A,EAIM 3A BE]KBA-BE

1. nc[irtirc m rr cy cJrrrm.4lo-yrro[ .[ie ay srpoEnBl

r) a-lm D-1,5 m, c-2m;

ar - 10 co, D, - 15 co, cr - 20 cf,;

O a-lm, b-2 , c-1,5 m,

at- lZ dD',ir -

8

rh,c,

- 16 dtca I(4I( r sy xelty co6ox c.JB.ru tpo-

trnorr qrlr Je Da$lep{ cnr'EosTx lt :lutrTo?

3. As ry ce xo,f,€ Ttprrf, Aa cy

,Er iegarorqnKa rpoyur crnsa-rtao n,rqit o lemr yrao io.ulxl (Brryre cJBrq 89.)



il. CtpaExue rpoynra ABC cy'. AB:12@, BC - 18 cr,,, AC:15 cm. llajBeh crpa-

x6rr BeMy cjrlqllof rpoyEa lr4cr ie

3ll cm, llsDa'IyBrire rpyre aBe crtEDnqe

rof TpoyMa,

5. Arr je ie,lar rpo]Tao. Kof,crlry-

rrnxre E€My cJrllqaf, Tpoyrao roltoty lema-raxr yrnoEll, rt4 onpe.Itrre r(o€lDxllrrienr

c.lxqfl)cm ta ABa tpo]'Dra.

6. Aar je je,uaf, rpoFeo, Ko crDY-

il|lrrr rc€My crfiqaf, Tpoyrao: a) warseE y

pmMeptr 2:1 (ri, cr 2 ryra Rpah{M qrpa.'

f,xq.Ma), O yBehar y parMepx 1:3 (ri. qie

he crparoqe orrrf, 3 ry"rr A are).

. 7. Pacroiarge rp'i e.ra' A' B' C'

na_ repery je AB:l2O m' BC:210m,4C-

- 240 D. Konnr.a ltrrr pacrojarGa oaroMDaiyrr tuaf,y Koilr je f,tpaleu y po3Mepx l: l0 000?

t. JerEoM r4EDoyrnoM Tlxtyrny I(o{-

crpytlruure cruiar rpolrao t(oa t(oiefa lte

6rrx: a) xtrtroreEyla aBa nara rpaha, O ieffaKAreTr TprltryT ayxa.

9. @a6prvrr afiMr6&K 6au cenKy

.4nxrue 3518 m, y icro BIreMe' Bepflrt{a.la

lrn:mr nyrftxxe 1,9 m, 6aq cenxy, .qtrn$trtre

1,62 m. ILDnre EBcxf,y -{ii,r6arcr.

10. Y rpoyrnoBnrra ABC \ APPIT'To ie. LA-LAIA LB- LBP' AKO iE

a:10, b:14 \ at:25' q-20' topaqy{aire

cryosnrc ,r f, c,

11. Eoqf,r crDanua jelror rpoyina

noae.rero je f,a 5 ie.,snaKq Ae.'rcEa f,trroagoEt[ ToqaKa troBy*Ee c / npaBe lrapo-

r€Jrne ct ocso flloM. Ar.o ie ociorEa 20

cdl, KoJllrKrr cy oaceur trapane,rDir trFatEf,

f,lueDy rpoyrnosm crlamia? (Olrorop:

4, a, 12 s lO.

12. ,qra jella$orpara rpotrrra rMaiy

npf, Bpry yrnose je,mar{€. KpaK x ocnoBltla

je.qtron itnoce 17 cm x 10 cm' a ocf,oBtrIla,uDyror ie 8 cm. O,qpeanre rtpax ,{pyror.

13, Koacrpyxurrre rpoyr,roie lBC n

,4,rrcr xar ie naro:

LA-35', LB:80", AB:45 l'''rn rLAr35"' Lcr6" APt-50 mm,

lla lloKat(ile aa cy lo cJrtqEu rlbyfnoDe.

OApearre, larlot, ErdoBy p@Mepy c-rntllocrt

14. Arr ie ieaaE TDoj/rao. Koncrpy-

nurre .qrym rDotrro srie he cryramqe 6BTr:

&) mpaneJrre ca crpatrKtaMa Aaror rpoyrJla,

O EopMa.rne f,a crpasxue ,aror-ue Boaebr

paqlrra o ae,Intllo crl,aBn[a. flor(axl're aa

cy KoEcTltyx@f,f, TlxtyrnoBx cnlllrf,i aaToM

u naltxre Koeoro{rieEre ctFrEocrlr.

15. Afto c y3My y o6lrp @Mo yrnoat'

K ,{ he 6lfr ouqf,a . Ea rpoyria' f, To o8a:

a) jeaf,ar(oxpa(a, O npaBoFna? floxaxrreTO i rlaouqK{.

16. Ilol(rxnre . a xtrlloreEy3f,na Dtcxf,ll

.ue,m trpaBoyr, { Tlroyfao fia asa cJrFma

tpoyrna.17. ,ifi 6x xlMeprc .qyxxlIy,4,B iearor

pn6Ba$a (or. 90), reoMerap ie rpacf,pao

.t Ar,y AM rAB r f,3Mepf,O AM:30 m, a

Tarobc $ npaEy CD L AM, 3Mepf,o

Cr. 90

55



CD:4 m s CM:S m. Karo i€ ilDaeyf,ao

r r(oJtrn y ie AD*xEy .48 A0610?

. C^.9l

18. IIa c;rxqn 91, ae.rax je ra6lo

repfirtrnEo y 3eMrLy r[TaE 9rlia je tryxxtra

Etll 3€r{JE{tM ieAHrr.a EeroBoj Ef,Cf,ffx, Tj,

BC:AB-l,6e'"I rl.fro ie Ea6lrro ra(o Aa

ie y roneheM aonoffijy (AB) Bnqeo Bpx NaDBeTa quiy BtlcrEy xohe Ia f,3Meplr, xao u

TaqKy C y scmi npaBoj nmrlj . Joul ieMepeBeM [aIIIao Pacmiabe-RM:6,5 m. Kal(oje paqyEao l| r(onErry je B{cssy M-^I ApBera

Ear{ao?

19. OApeAure pacroia.rre aBa Mecra,

.4 {,R, a pa3f,uM crlraaaMa p€rte, trpf,MeEoM

cr{{aocrlr rpoymoBa. IIIreMa ortlllll 92, sa

\oioi ie DEllAB, otruurnre f, o6pa3noxrre

nogrynaK.

Gr. 93

ornelaro M fi fiIIao ]Ta3aa qre Aor r i€y orneAarry yflreaao 4rr,4 cry6a,48. [ITa ieMopao Mepsas f, Kar(o ie rqla{yrao rDcxtry,{r?

92n.

20. Ar 6n mMepro DscrEy rer€r.Doo-

cKor cry6. AB (cJt, ), rero ie ra xra6pa-

rlo Megto topt ()ITarior 3oMJD{ltr.a cTaEno



6) CIUTIHOCT MHOTOyTIOBA

Ha cmaqr.r 94, y neroyrury ABCDE nor,yrcue cy gnjaroua:re AC n AD,E je rc neKe ra*e (M) crpanIare l.B noByseno MNllBC, NpllCO t pe|OE.Ila raj rlarru go6njex je roru

rrroyrao - AMNPQ, sojn t:ol;a

rqu o6:ur Kao r{ Aaru neroyrao.Ta asa neroyrra cacroje ce n:rcror 6poja cJrlFrrIrrr rpoyrnoBa(rugeru sag. 2 y rercry q)erxoa- Exor oge.nxa) rojn cy nopefanutrcTtrM penoM, a To cy:

AAMN-AABC,AANP-AACD,

AAPQ-61P9.Onaro 4o6r.rjena ,qDa nero-

ylna cy cnfiqEa.

Vrlloln osrx neroyrnoDa cy jeggaru, trao yrnoBr{ c napaneJHr.rM Kpa-IIr.rMa, Aox sr,t je LA :ajegnrvrn. ,{axae, voxeuo HanucaTn:

LA:LA, LM-LB, LN:LC, LP:LD, LQ:LE.flonno cy o,qroBapajyhe crpannle crnr {ux rpoyrnosa Ea. cJtqu 94

nponopufloHanue, a ro cy fucroapeveno ogronapajyhe crpaxr.rqe y trocMarpaua

,ua MEoroylra tm ucroj c.nrqr.r, MoxeMo xarucarr,r:

AM MN NP PA QA

AB BC CD DE EA

Karo nperxolFo E3Jraralle oqfirJreAuo Ba:Ku r{ 3a MHoroyflroBe c npor.r3-

no.nuuu 6pojeu srpannla, uoxervro peht yonrlre: MtroroyrnoBf, ca f,crf,M6pojenr crpanrm

"n"to:cygr(o cy f,M yr.nosr IleAoM je4raxn r o.{ronpajybe

cTpalxqe trpotrop4trofieIse.

Pa:rvrepa ogrorapajyhnx crpaurqa aDa c.rlrlrrHa MHoroyfira 3oBe c€ upa3Mepa unfi noe{uqnjenr cJrf,qtrocrn ra IBa MrroroyrJra.

Aro je sagaro ga xdncrpylrmeuo Mnoroyrao cJrr{traE rreKoM,qaroM MHoro-yrny, ca AaroM pa3MepoM cJr[rruocrn, qre6a nocrynrru

oraro(npeua c,r.

94):aajnpe noayhl cse Aujaroxare ts jegnor rewena. (AC, AD), sarr*r.qary crpa-Hnry @M) rpe6a npexern xa ogloaapajyhy crparrr,lrqr (AB) lanor MfloroyrJra,a 3aruM peAoM ayfix napa:reae ca ocraJrnM crpanlrrlaMa Aaror Mxoroyrura.Hajsag ce raro 4o6ujeuou lnroroyrny o4Bojexo xbHcrpy[ue rbeMy uo,qy,qapafi

MHOrOyrao.

AMBCn. 94



Aro je pa:uepa crtrrlsocrn reha og,1, rj. aro cy crpanr4e rpaxeHor

Muoroyflra rehe oa crpannqa aaror, oHAa ce qpBa npesece Ha nporyxesy

oarorapajyhy srpaEqy AAror, tra ce pary pal,e Rao rrrro je ouzcano.

Hexa je sag4ro, f,a trptrMep, Aa ce roncrpyume qerBopoyrao cnnvax

jeaEorr,r .qarorraqerBopoyrny,

raro ga je pa3Mepa ,qaror npeua rpaxeuou 2/3.Ha uwn 95a aar je qerBopoyrao ABCD, a rpaxetn je EFGH, xa crnrqn 956.

E

Cn. 95

Hexa yueurryr o6jacHe rrp€Mrr qIIqtr Raxo ce Aorruo Io serDopoyrn EFGH.

V oor:regy cnurmocrr.r npaBulrHax MHoroyrnoBa, oqlrrJrellno je aa cy

.cBn trpramrnf, MtroroyrnoBf, c8 f,crf,M 6pojeu crpanxm uelycoEuo c;rnrrm.

7. CIIPISE 3A IIPTAbE yBEDAHI/X II yMAbEHI,X OI{[I{PA

Ca,(a helro ytro3uartr Eere oA ctrpaBa roje crryxe 3a qponopqr{oEil.Jrxo

AeJseBe Ayxt tr 3a qpraBe yaehanrx r yrvranenrN furypa.

1) Ilpotropqf,omnnn urecrrp cnyxtr 3a

)rlr{anrrBa*e x ylehara*e gyxr no ,qaroj

pa3Mepx. Kao rrrrro ce BnaB ca cnme 96,ou ce cacrojn u3 ,qDa MeraJrra nennpa (apa

napa rpaxoaa). rojn cy ra rpajenm,ra oaun-'nexl. Ora ne6nptr cy no ,ryxtrHa npop$asu

Taxo,4a ce tro Tr{M npope3uMa Moxe lroMe-

parf, norperrrr 3aBpra6, Rojtr ysBpmhije

rpaxoae y Nsa6parou lrecry n1rcpesa. Kpaqu

ce Mory parrupxrn u.nu npu6luxrru jeg4u

Apyr,oM, rao xoa o6HltrIor lrlecrapa. UoAyr('Eff oBux nebtrpa - rpaRoBa mecrapa,

yqprana je cra:ra (Upnqe pa:pajajy jeanare

AenoBe, IyxtrHcKe je4m4e, a 6pojeBtr c€

rarmae yr rorue).

D



Aro .4ary Ayx' 48 rp6a yua*urn, peq Mo, 3 tryra, onqa salllrapyupuhyjerrao y ra'rxlr O ralo',qa je AO =.3. OD. 3arnu rpajean"ra roecrapa

A n B a6yxsatauo Aary y4 'lB, nfa reuy he pacroja*e apyrux rpajera(C r D) 6nrz 3 nyra rrraraq 3j. rpehuxa op, AB. Orlo trpotr3na3tr tr3 cnnrEo-,crlr rpoyrnoEa ABO n ODC (semro cy oxu clrrnu?), o.qanrre cneAx:

',48 _OA _3cD oD l'

Hajgal, aary ayx CD npenert rae rpe6a.I 2. flonpernr paruepmx jd. cupaaa 3a Mep€Ee Ayycx, t 3acufrBa ce rarofe

ACll. 97

jegnaxrax .qe.nola (peqrrrao aa cy ro geufiMerpu). Pacrojarre Mp nogereuo je

xa 10 jeguarux ,qe.rroBa, tra je rpos craxu noAeJbar troB1ntexa IIpaBa napanenHa

c.a MA. Ocuw rora, Ayxr{ MN n PQ troAeJb€He cy cBara Ha 10 jegHarlx

,ueJroBa, ra cy noAeore ra*e cnojeue, Kao rlro noxaryje cnnxa,

Ilparoyr.nu rpoyrao, rnja je rcarera lfP, npnxaoan je yoehano raor. 98. Ty fiMaMo cm.rlHe rpoyrroBe NDE u NPC (sanro cy

c.nnuxu?), Ilourro je ND aecern Aeo oA lVP, ro je r ED gectru"qeo oA CP (a aro je CP: I cm, or.p Je ED: I mm). I4o ucrorpauora je

cr:l.ct (u-nu 2 mm), u'rg.l0

Axo caAa lrrecrapoM 3axBarnMo A,tx Kojy rpe6a ga I,BMepuMo,

IIa xcTr.r oTBop [ecTaPa npeneceMo rra noupeqHrr pa3MepHr,rk o.q xgo

I,xa rbeMy rrtrraMo: o4 I

4o Zw"E 2 aehe jeaunmle (2 dm),

sar vr 3 Aecera Aena re jearcrqe (rj. 3 cm), rrrro c€ qrrra rra

cKa.JM. MN, u rmjra4 5 Aecerru AeJroBa re none je,unnlrqe (rj. 5 mm;,

Jlrro qxraMo Ha cKaJrtr MA, ALra A,x, AaKJre, wta pynmy 2 dm,

3 cm, 5 nm.N

Cn. 98



3) Ilaxrorpa{ je cnpara 3a qp.taBe c;rnrun $urypa. Ilprex ua cmqn 99npmasyje rorcrpyri{trjy lanrorpaQa, Kao mro ce 34ry, ro cy rrernpu nerErue,

o6prxe oxo rasltna M, N, P, Q, xoje cy reMeua trapaJrenolpar\ra. IlaurorpaSc€ nocTaBrr raro .qa raqra O ocrase HetroMnFrm, n rag ra norpeheuo, qerBopo-

y;ao MNPQ cegerfoprrrme,

a;rn ocraje yBeK napdrenorpaM, a ralxe O, P, A

Cn. 99

ocrajy yrer y lparoj ruHnjtr. ItrH;rar (rorrnh) P troMrrqeMo no KoBrypE

r[urype rojy xoheMo Aa yaehauo, a Epx onoBre I ra,aa onucyje c:rnruy

rlnrypy yrehary y garoj pasvepr. flpoveuolr Meira roqxuha P r o.noaxe IMoxeMo qprarn ylrlanexy Olrrypy.

3AAAlpr 3A BE,I(EAEE

l. IlocMarDsjre cJrry 100. Y nelo-

trry ABCDE yrera je r4nr*oEuo ra.rra O,

ED

BCn. 100

r cnor€Ira c reMelrf,Ma ltemyflta. Aa rr je

aerolrao Appppt, qxje cy clpaflfletrapateJte crpagxqaMa aaror fienoyrna, cJllr-

qrs .uaroM ncroyr,ry? Obiacfllle saurrc

KonrKa je oBAe pa3Mepa cJrxqf,ocri

Aaror r eupra{or neroyma? Xaro 6r c€

nocqnflno xrlq 6r ra pasMelta 6wra >1?

2. CrpaErqe iersor ncroyraa rrroc35 cm, t4 cm, 2E cm, 2l cm x 42 cm, a Eaj-

rp6ha crpaErqa I6eMy cjrr.IBor .l€royrna ie

n}Ta 12 cm. O.qlrc.qure ocraae crpanrqeolor lerpyrra.

3. npolloDunollanln\| uatcrapoM: a)

Jliia6Bre a&ry a]'rR 3, 4, 5 r 'Ia' O yBehaire

,qary ,qr'* 3, 4 nFa.



4. IIoMol[r lrpotroDqroxarnor urecralxl

saDrre Ayx: a) ie$aKy 0,2 Aare .qyr(r,

O 1,5 ryra ,4,rNry oa aare ry,.{.5. Y qeMy je rpensocr ionpe.Bor

ttaSMeporrta npeMa o6{qxoM neoxpy a

uelere ,uylreirue?6. Ha{prairc ieaan qErBoDoyrao tr

ttMeplTe w crpaf,ele llolrlrccarM Ita3MeIF

lxKoM.7. IIoMohy nonleqEor pa3MeplEaa

Koscrpymlrxre rpoyrao .sie cy c4taE qe oa

4,35 cm, 2,7t im 3,96 cln.

8. Xollcrpym|lEre noryrc$fi palMep-

Exl{, y3rMajyh[ rehe jearmqe no 4 cm (f,era llx 6yAe cEera 3), [a rxM p&3ndeptrxxoM

x3MeplTe ueKe ay*x.9, ,llaa ie rpatre3 qrje trapaneJrre

cTlrurqc f,lEoce 12 cm r 4,5 cm, r Kpaqr

9 cE f, 6 cm. Kossrpytruxre EeMy c-'rxqatl

4l€ltre3 Tarao.qa ie pa3uepa cJtr.qsocrll 3:2.10. HauprajTe jeAa{ [p6 {naB Dero-

yrao, sarnM Kotrclpyfiurf,re ,upynr, ule he3

crDrm|e 6llT{ - crpaExqa AaTor. Kar(nle4

3aurTo cy ra JEa treroyrJra.an rf,a I l.'tuErit

ie r6f,xoBa Itr3MeIt r cJl{qEocrt

11, Aer ie np€Brnatr tne.croynto

Koqcrpyf,urrrg cruqaf, rrJecroyrao: a) ca ara-ft "r K'laher spa@aMa, O qrie cy srparue1,5 lr]Ta ,ry:me oA capauqa Aaror, a) ypa3MeDx 4:5,

lL Asr Je DpaBruraE ocMoyrao. Kotr-

srp 4rurllre orf,qatr ocMo /Ilo y parMepfi 2:3.

13. Koscrpyuulme nparnraE rgro [ao.frir trorylrpeqfirx oruc1uie Kpyxmue rusocx

4 cm, a 3&rrorl rSeMy cnlfinu Maoroyrao,

]'rraEeE y pa3Mepr 3:2, raKo aa etajy3aje.qEf,.rKf, ueErap f, 3ajeAsxqre ooe c{M9-

rpnie.

14. Axo je pl:'Mepa crrf,.Bocrf, ,uBajy

nDasllrfis ocMoyrnoDa 2:3, y ltoioi patMep{

cy EuoEf, o6xMf,? Aa nu Aofrieu oAmBoP

EAr(r 3A Ma KOja ABa CJrtr*I r4l,lDrlnEa

Mf,OmyrJra r 3r Ma Koiy pdtMepy En(oEe

cnxqf,ocrf,? Ha ocEoBy roxa Aon]'df,re rDa-

BrJro: o6rn r ,lBriy cJtrlwrx Mf,oroyrnoBa

DDOtrOD{rOf,aJmr Cy...



, , ::..:

ryTEPEIICKO MEP. EBE

. I. CHITMAEE MHOTOYIJIOBA

Crmrdrn Hero seMJbtrrrrr€ y o6rf,Ky Mloroyrra 3nairff, rao mro je 6rnpeqeno 3a 3eMJBn[re roje nira o6rm rpoyrna f, qerBopoyrna

- trpeAcraEErtro 3eMJbr{mre rpar$nrru, )&laBeuo y ueroj oryefexoj pa3Meptr.

,{o6njemrp+.

Qux je o''la trna[ Tor 3eMJba1nra.

3a oso curntabe rorpe6no je, ocnir ocranor, r Mepnrtr yrnoBe Ea r€peqt-Beh je parzje (reorraerpuja ea VI pa:pea) 6r:ro o6jaunexo RaKo ce F oHa r€peEy (y xopnsorrarroj panxr) r"repg novohy Maruerue nne, Tj. asrpolF6rjyvov.

IIpe rero uro npefeuo Ea caMo crf,ldane, yf,parro hertao noxoamoApeDnare yraa acrpo.la6ujyuou.

Hexa je saggro .qa oApeAuMo a3mryr npaBqa lB (cl. l0l),cTaB,baMo TaKo Aa My qerrap C n r{t{e Ea rta 4ojy ra{(y nparqa (rj.

Cn. 101

lll, npa veuy he varuerm m;ra (roja oe cro6o,qgo o6phe oxo BeprnraJrreoce y C) 3ay3ern nonoxaj cerep -

jyr (Ha cnrilu S4. Cag lerlrp MN acrpotn-6njyua o6pheruo oro C gor Ee rronotrtr npuy, AB, rj. gor u rasKe M z NEe qptrtraArry nploj AB. To ce ylemara BaflFs€M, .{o6tjeuz yrao SCIY jea3ruyr qpaBqa AB. llpt r .rTa6y oBor yrna yB9( ce nAe o.q rawe ,S y cuepy

o6praEa Ka3aJLke lla qacoBErxy. ffperrla



rloapa3yMeBaMo- rao mTo cMo paHr4je yrBpaxjrE _ yrao rcMeDy Tor rpaBqa

u MarHeTEe u re, MepeE oA ceBepllor noJa MarHeTge trrne y cMepy o6pTaEaka:arre xa qacoBgtrxy. 3aro asrxryr Moxe nMarz BpeIFocrE o,q 0. Ao 360.,

Ano rpe6a trahu yrao usrraefy .pa npaaqa (rj.,qpe npaBe), oEIa c€,3Mepea3xMyrri rux nlmBarla, na ce oa seher a:|nvfyra, o,qy3Me Ma6tr.

Mefynnr, rra cBaNoM uparuy (upaaoj) MoxeMo, no norpe6u, pasnd<orarr,@a cMepa. Ha npruep, 3a trpaBy AB, jeaan orep ce go6uja u4yhr og raueA xa ra,lKtr B, a Apyr|a - oA TairKe I xa rarnx l. 3a csaru o,q rm( cMepoBaRaxeMo ,qa je cynporax Of,oM Apyr.oM. Ipeua ror,le, y EaBeAesoM nptrMepyr]alrlnu cMo a3r vryr npaBqa AB, a ae u a3r.rMyT npaB\a BA. AsraMyr osor,qpyroF npaBrla je nexorrercau yrao ,SCl. Mefyrnu, ca cJrtrxe je orxrnegpoaa pa3Jrr{Ra onHx .4BAjy yDrOEa

- a3ElvfyTa rr3Hoctr olpyxeH yrao, ro jecrHe(oHBeKqrH

ISCA- ISCB:IgO".

Ooy sarlrcuocr a3l.tMyra, MepeHr ( y IFa cylporxa cMepa usre rrpaBe,MOXeMO trqroptrcrrm sa nporepaBarse n3MepeHnx a3nvryTa npu TepeEcxoMclrulv{arsy. Jep, a5o Rao pa:}nnxy ra ABa ix}trMyTa xe go6r.rjeuo 1g0., snavxIa cMo y MeIreBy cBaKaro norperrrrrJrrt. A aro go6ljeuo pa:nnry 1g0., repo_rarxo je g3 uncrvro rpy6o fiorperururl,I, jep je remo lpertrocraBrru ,qa ce yo6a uepena j' racrou curcry x 3a ncry BeJrtrsury norpeErr, Maaa Er{ y roMcrryuajy ra pa3nura oA 180. xe 6r yna3rBara Ha rp€Ery.

2. ABA HAIIITHA CHI,IMABA MHOTO}TJIOBA

.{ra xavxua cuuMaba seM;bnrrrra y o6nmry qerEopoyrra npnvreuuhervroE sa cuuM:tae 3eMJLr{Irrra y o6Jflfty Muoroyrna yonrure.

Cr. 102

1) Hexa Je 3utro ,[a cnf,MtMo 3eMJEmns y o6Jtmy reroyrm ABCDE,



roM.reMesy, tr ro rroqeB or trpaBqa ceBep _ jyr. Ha raj uamr ce go6tjy asnvryrro6a cltepa caaror trpaBqa, urro je norpe6no p4qr [poDepaBaba pe3ynrara.3arn'r ce x3Mepe tr AyxrEe cr-pantrqa uEToyrta y MerpnMa, ra ce cBurnnoaaqtr cral;bajy y c;reaehy ra6eay:

ED84"

EA32r

DC38"

DE264"

CB280"

BA235"

BC100"

AB55'

AB:44 BC:52 CD-42 DE- 56 EA:40

Vs ra6ete oAr\,rax [poBepaBaMo a3uM]rre: pa3Jrr{m a3r,rMJrra npaBrla Blu lB je 180"; ncro raxo, no l80o lranoce pa:alfte asuwyra npasr\a CBaBC,CD n DC, DE u ED, EA.u AE.

Ilouro cy ayxxHe crpaH[qa MepeHe Mcrp[Ma. 3a uprarbe y rauoj cnecqr.r

troroAHa je pa3Mepa I :2 000, ro jecr yuecro rraerapa rpe6a y3err4 Ayxr.rHe y

Mnnr.rMerpar\4a, ua rrux jour npe-

rroJroBr,rrr.t. (3aruro?) 3a uan her'lo,

.uaKjre, r.rl"{irTx: AB:ZZmm, BC::26 mm, CD:2|mn, DE:28mm u El:200 mm.

fJourro cuo np[npeMrrnr{ creD re no.qarre, flpgcry[:rMo rrprary

caMor ruraHa :ev.rrmra. V xeroj

Ta,tKa (A) qpraMo rrpaBarl Marrer-He r{r.Jre ceBep

-

jyr (ua cn. 102

npaBarl npeAcraBJBeH uprrrrlaMa).

V roj rarru oAi\.{epaBaMo a3[Myre

nparraya ABU ld sarrttr,r opyepa-

r€Mo AB-22 mm, a3lrlrj/Te rrpaBa-

\a BA BC, saruu ayxnuy BC-26 mm rrrA., gor ne go6njeuo cBa reMerax crpaHuue Mrroroyrra. Taro je wauprau r{ Mnoroyrao ABCDE Ha cnturr 102,

2) Hauprarn trlarr seMr'r,r rra y o6n{Ky treToyrna /BCDE {cL l03) no-IapIloM MeToAoM.

Kao npu cHr.rMarby 3eMJr,ltlrra o6nnra rpoyma r,r qerBopoyrna, u oBAe

ce y3Me npor.r3Bo,rto jeaua raqKa y yHyrpaurbocr[ M]toroyrna: ta raura je

no.n (P). IIol caAa cfiajaMo ca cBr.rMa reMeHuMa E Ha rrera ce craBJba IIeHTap

acrpoaa6njyr'aa, na ce oHAa n3Mepe a3lrMyrx npa*a\a PA, pB, pC, pD, pE vAyxr{He rr.rx ,qyxu, a pe3yJrrarr.r c€ ynoce y ra6eay.

AE142"

I

Cn. 103

CD2t 8"



Ilpagaq PA PB PC PD PE

A3uMyr 2s3" 298" 39' 103" 158.

AyxrEa (y m) 31 2t l8 23

fl.nax ror seMJbtrrura y pa3MepB l:1000 npeacrarJbeH je Ea cll4n 103.

IIIra 6u ce oBAe jon MopaJro Meptrrtr aro 6vcrvro xrerr Aa trpoBepffMo

,4o6ujene asrwryre npanaria rao y rrpnrrepy l?

3A-4AI[.T 3A BEXE,{BE

1. I4oreprre amryr ie,Ercr lrpalqa

r [poB€pf,Te m.2. TDacrlajre Ea repetry aB MeDy-

coff,o Koca 4nrqa xoia oe oe*y, oa rrM€-

|,lrre yrao rueby 16rr: a) troMohy [oJbcrcor

yrnoMepa, 6) nouohy acrpojrrdrjyM..

3. CMMme rlrRoncrco AropElrre irf,Eero

.qtyroxoltrrorranso 3eMJbrtrne y 06-

nnxr Msom n'r , Erpajyhf, DoroAxy p$Mapy

trpei{r DeJtlc[m 10r o6Jexra"

4. JeIEoM 3eMJrltrr'ry y o6.,mq/ rc-royl|nr ABCDE Mepese ctr/ crpaqe B ynt(F

Ba - a3trMyr , na cy aooriefi troailE c:ra .J €E y c.nerehy ra6ejry.

IlaqDTajrc dnas ror €nJf,rorr y

noroAf,ojpa3Mqrf,.

Teue A B c D E

II aBa( AB BC CD DE EA

A3m,ryr 90' 165" tm' ffi 160"

Ayr$rrr8 (y n) 55 7 30 t7

5. lh ropr3omsrEoM TeIE y o6ene-

xtre reMed& jeAaor trgroynt& a 3trf,M tro-

Julpnou Meno,(oM csimre rsj trcmyrro ytrorotr{ol D63MeDlr. (Ibjme r3MeDffre ffroie mrpefoo, ua lo6riese noAarKe yxecrr€

y a6eny.)

6, fipexs maaqrira y ra6eJE 3a-

Agfi( 4_ Tlacrpqjre Ea reMe :y leToyreo

,4BCDE,^

3arrM Tpocf,prlTe r r3Mqlxrc

Inm je iont trorpe6so r f,lpa{y{ahe-rrygoBDuury y rEa.qlrorf,x MerDx a"

7. TDacrpajr€ i mMeprre urro ietrorpeSso f,

'3paryrajteo6nr e uorprrry

tlrKotctaor' ItopErrm.

t, JeJ[o rE(oJrc{o x aBe Dls o6JE(

ruocroyr,Jrr ABCDEF (cn, 1/M), rn{e ie LA -

-LB-LC-LE-90",Aor( cy i{y oc.rJ

ereMerrf, o6enersem f,a cJrilr. nogra lG

tarla rnegrcy:lo El Eexo topxSoorulto

3€l|J rrlre, 3afElrr rpacrpair€ | rro ie jom

trorpe6Eo, da Bpatyf,ajre r|o4llllEy Y m:.



Cr. 104

3. BPCTE 3EMJbI4IIITA I,I PE,'bEO HA NNAIIY

y ntraHoBuMa repexa Mory ce rpaQntlru rlpuxa3arrl H Btrrre 3eMJrbuIrITa

Ea rnM repelrtrMa (Ha trPnuep- rrryMe' IlrlBe' Dohlsa4{, 6ape rra')' Ilcro

raro, y rulaxolrrMa ce]ua:rase yupraux cao6pahajnn o6jerrx (npyre, uyrerv,

MocroB[ HT,q.), 3amM py,aEnqt, 6a*e, :ualajue BIlcr{HcI(e ravxe (rore) u apynr

o6jerrn. ,{a 6r ce cgaxn oa oB} ( eJleMenara npnra3ao y cBnMa ruarioBnMa

Ha [crr{ 8a9fi, trpontrcaBe cy HaPoqrre o3lmxe' raro3BaHH Totrorp&Dcr(n 3llrqrl'

Hex4 najra*nnju, o,q rfix sHlroBa nplrKa3aHrr cy Ha cn. 105'

BO:h15AK

F..al"'."% '.

I

NYMA I{ETTH.

Txr:--i-x;-lI*rtxllx r x tl

mm;;;;1l, v t r tl

lr r r r rl

EUHOTPA'q

m

NPYfA HOPM. KOJI.

flPvrA v3AH. KOn.

: A{tom/r

NI'RA'TA IIAIITILAK

MOqBAPA

r_-_=_ _ _ -tl---lCn.

Ocnr,r oBora, rla nnaHy ce nptra:yje n ltenerfi repeua, rj ua'alropcx'r

Escxxa pa3arx raqaxa Ha repexy. Ilpegcrar.nenN penet Ha unany rpe6a'ua

trpyxn Erro aepxujy c.nmy pacnopeAa Brrcrrll rr Htl,3:rirra r.a' repexy' Oro ce naj-

F &e( MOCT

:|]ffi HActrn<::- XE,'683. CTAHI{IIA

\=7 EAISA

6AIIITA

nvMA nncron.

G-;=-llo'"o."o'91loO oo o ol

05



6ore nocmxe r3oxmcaMs, rrrr xopl3o[raJrsM& a ro cy nnunje xoje uoreryly

arre je.uEaxnx Ha.q{opcrnx BLIclHa.

Ha trpuMep, aro ce 4mlnjou noBe)Ky cDe ravxe roje EMajy Ha.u{opqy

sfictrfly 50 m, go6uja ce u:|oxfinca Ha4\4opcx€ sxc[He 50 m. Axo ce cai nonexy

unnjor'.r cae raq(e c HaaMopcroM El.tctrxoM oa 60 m, aoSxje ce u3oxx[ca uaa-

Mopcne Bucr{He 60 m, urg.tr{aoxnnce ce nocrar:rajy xa je4xan.rw BrrclIHcxItM

pa3MaqtrMa. Ha aera.msnw nJIaHoBt{Ma xpynmje pa3Mepe BnclIHcKa pa3J}rra

r,r:ruefy cmre ,Be y3acro[He I43oxr{[ce 6fie 0,25 m ao 5 m, Aor ce 3a on[rre

npuEpe.qHe Kapre, pa3Mepe l:25000 Ao l:200000, ra BuctrEcKa pa3nnxa Kpe'

he o.u 10 m .uo 100 m.

Aa 6 cMo 6o,ne cxgarruu rBoxllnce, 3aMucJlrrMo je.uuo ocrpro y uopy

(c:r. l06a). Jlrurr,ija np€cera noBplxx Mopa ca noBprnfi 6pera na ocrpry

npeAcraBJba }l3oxuncy Ha,qlr,fopcKe sucutc'ie o. Aro ce cag HIrBo Mopa rt3,llHrHe

sa 5 m, y rpec€Ky rberoBe noBpruI.r ca norpurlr 6pera Ha ocrpBy, ao6nhe'uo

al b)

Cn. 106

rpyry rr3oxuncy, vrlja je naAuopcra s cuHa 5 m. Aro saMxc,lnMo na ce HrlBo

Mopcx€ Boae It gaJLe u3al4xe, npecequ onltx norpuu he Ha cBaxllx 5 m gucuse

narr.r HoBy u3oxfircKy; cre cy oue [peacraBJbeHe .nxxljaua sa cllull 106 a.

Ha cnlnln 106 6 npe,acran:seHe cy re lr:oxllflce Ha[praHe Ha rlnany.

CBaxe oa n r.rx je uarlprana, y pa3Mepx n:raua, je4Ha nopea apyre raHKoM

rrurujorvr, a cBal(a reTa x3ly{eHa je ne6re, Taro .ua ce naxo yoqe Br'rcuHc(e

pa3irme oA uo 25 m.

4. qr{TAI}E Cr{TyAqI{OHI4X KAPATA

Axo je Ha ruraxy (rapru), oct{M apyr}Ix noraraKa, npeacral:beH He caMo

xopr{3oHTaJruu pacuopea repena neh r.r neron pen'etf, onga je ro cnryaqltortx

raH, nJrr cmytqnoHo naprr. Cnryaqiona rapra trpyxa troAarre o roMe y

rojoj je pagruept ona pafera; KoJIEKa cy uefyco6ua pacrojama noje,{uxux



Mecra; Raro je qeo repen (n nojegxrn aetoBtr Ea EeMy) troqmBnes nperra

crpaHa"Lra cBera (crpenEa x3EaA cnora S noxasyje ceaep); g3 nf, repeE rpeil'

craEJEa ruroaEo 3eMjr'urtrTe n roje ce ryJrrype ua frteMy raje; Aa Jm Ea TepeEy

r.rMa py.4ulrra, 6aw., ustu gpyrux snarajnux Mecra, natraB je neror peneQ;

raro je parrpaxara cao6pafiajxa Mpexa, xra' Hahu one u cJlrme troAarre o

Tep€Hy npeMa rreroaoj cnrya{uoaoj rapru 3EaqE 3narlr qf,ntrf, clrryaryonyKapTy.

fiporrrahelr.ro caAa crryaqsoxy rapty [peAcraBJEeEy Ira cnqn 107. npe

ciera, ucno,q rapre rpn"rehyje ce pa3Mspa y xojoj je rapra pafexa' 1:2500'

To esasn Aa jeasoM qenrlrMeTpy Ea xaprs

olroBapa 2 500 cm:25 m na repery xojr

ona trpeAcraBJba. Taro je, Ea nPtrMeP,

uxo;ra I yn?lre'Ea OE npyr€ y3aEof rono'

cexa 100 m (Ea Kaprn 4 cm), a ol py,uma.8, rryhu xoncKEM tryreM rojn n re:Yje,

237 m.

lpyra yrauor xo:rocera trpeceqp repex-

rnyhu oa jyroncrora Ka ceBepo3atra,ry.

OHa jeaxw AenoM npotastr rpo3 Moqla'

pan Tepex (mpqsre o6e;rexeue 6pojeanra

(8 n 20), s6or rerq je ca o6e BeEe crpase noamayr nacun. Tepenje

6peronn:r'ca oHe crpaHe niyre Ha rojoj ce uanasa mrota A, ca uajrruolt Ko:roM oa

32 m uaguopcre nucme. Ty ce HaJli{ttr tr tr3BoP M, st B:aczns o,q 15 n'Ilapueae ornalene 6pojernaa 10, ll, lZ, 16 u 18 jecy Isaee opauue'

13 n 14 cy 6rrnr:, 17 je rohnrx, 9 je namrar, 15 ruBa,qa, a 19 je rnlcro

laasa EyMa. flpyra u rolcxu nyr yxpurajy ce loa yuou oA 77o.

Ipernocranrvo jour 4a je norpe6no .qa uspaqyuaMo E€ntrquHy rstrBe

o3uaqeue 6pojer'l 16. flomto -ona rnua o6nur trPaBoyrtor rpalre3a, MepnMo

joj napanerure crpilgtlle u go6ujar'ro 2cm n 2,4 cm, ruro Ea rep€Fy mnocu50 m, ognocuo 60 m, u pacrojane u:ltefy *nx l'2 cm, uer"ry o,uporapa 30 m.

Ilpeua. o6pacuy 3a noBplrrlruy rpanesa 6nhe:

r:50+60.10-1650, ro jecr 1650mr:16a 50 m2.2

3AAAUII 3A BEXEABE

1. Aro a Jucxjcl(oM palMapdf,(Y 5mm

trpeacraliLa rqnxf,ry qa 100 d, y ttoioi ieDq3mcpf, [praE nJrr ?

2. Aro je trltrE paDeE y puMepr

1:2 500, (ojorl 6tr ce 8tseou ra JrnsjcxoM

3. IfDe*€ cnhuf, lot f,lPscYf,siTe

pacrojere aueby Mecrr I x Megra-B ta

clrDormt o6a.'raraa DeKe.

4. nDoEaDrre Ea reDef,Y AEN uogrt

lr250OCn. l0?



rlMeplna, tla rM To pscrojaEe f,3loqyEajre

npxMebyjyhr cjr |rtrocT TlroyuoBa.

5. 3allrm cT Ea t|na[[y s3otxme

ryDhe rAe je repen crpnrrjr, { lxrrte (ry

D@lMr$f,lrre rae je r4rs6 :epre 6nrr?

Cr,

6. lla trJraqy (cr. 109) lDo{rrjr€cneaete:

a) trpe r nf,f,rjcKoM porvepnncy o,a-

pe.Irre pa3Mepy trJraEli

O iouolly ,rxEdcKor F{rxepmkr fi3-

MeDnre aBMef,liie uxone M f, ExoJtcKof

:

N@ $ti ABCD r 8p4{yf,aire lolnrylr(oJrcltor ampluft;

B) noluco le yAalDena uroJrr ./l,[ oA

6as€ S tllybr lyreM rojn nx Eelyiei

IE:F-E::=t0 0 ?0 40 60 80 t00n

Cn. 109

108

r) urr ce xox{e Dehx o peJleoy

SeMJllrrITa roje ral rurar [D€acr8lJrri

a) Koie ce sDcre leMJrf,ulra tla roM

nrpef,y f,anale f, nore ce Kyntype o6paDyry;

lt) m Koioi lc ayxrsx ryr o5€r6efreE

4Cnrou oa noqrrDe:e) s roJol oe crDsss oA tfKone (M)sajt8f, 6sEa (S), a Ea KoJoi c ctttaf,f, o,q

6rEe f,aJraas uroJra?

7. Oalearre troEpury 3erJltrurta

o:arceror qDJeu l8 Ea trJrgry ittEAcmDJr€-

rorl xa cJBqf, 107.

lo ollo i-

\:".:1I ""

l^ &m



vIII4TATOPI{HA TEOPEMA T{ IbEHA IIPI,IMEHA

1. II,ITAIOPUEA TEOPEMA

Crpauxqe npaBoyrJror rpoyrna cy y Hapoqr4ToM rrrefyco6aor"r ogxocy,xojra y reollerpr{r{ nMa BpJro Daxly ynory. Bnueheuo caga xoju je ro o,qnoc.

Ha c:ruqu 110 xoxcrpyncan jenparoyr:ru rpoyrao rnje xarere usxoce 3u 4 u3a6pzld.e ,qyxHHcr(e jegr. rrue. Xulorelly:a ror rpoyrra usuoctr 5 r.rcrm

.qyxuHors{x jeauuruta, y rura ce MoxeMo }tsepxrr.r vepenelr.*

Ha4 crarou crpanl roM Tor rpoyrna rorrcrpyncau je roagpar, a craKuoA rI.Ix JoaApara nogenen je ua o.qroaapajyhr 6poj rnagp4rnnx jeAmrqa.

. .qa je rpoyrao ca ratGnM crpasruraMa (3, 4 E 5 ayx. jea.) trpaBoyrnt, 6rdro jeno3Aaro joltl crapltM Erx'IiaBI{Ma, ra ce 3aro raxaB Tpoyrao rofekaA 3oBe n,,ermarcxd.rpoyTao.

Cn. I10



Mepnn 6pojean Klagpara Haa rareraMa cy 9 u 16, a MgpHn 6poj xBa,f,para

HaA x[rroreHy3oM je 25, na ce o,4uax npnuehyje ,na je

25 :9 + 16.

3a osaxne npaBoyme rpoyrnose uoxeivlo aaue pehn:

troapunua xBtaparo na.q xnnoreryrov jegnaxr je rftrpy norpmuna rragpara

HAA rgTeTOMA.

Caga rpe6a ,qa cnnraMo ,ua nn oaHoc crpaHnua rojx cvo.{o6rur saxu

u sa r,la xojr apyrn npaBoyrnr.r rpoyrao.

Cn. ll I

llcequuo ol xaprnje 4 noAynaplra ryxmoyrna rpoyr,ra, rnje cy Karere

a u b, a rurnoreyy3a c, rra ( o3uaq Mo 6pojerlrlyra l,2,3, 4, Axo nx c:ro-

xr.rMo Rao urro noxasyje cauxa 111-I, .qo6uheMo xaaapar rr{x je je.nau ,ueo

HerrcrryBeH (na c:rruu oceHqe.Hn 4eo). A aro rx cJroxnMo rao r:a.c;ruun lll- ,

oxn oapebyjy rBa4par noAyAapan npBoM, y xove ocrajy Hel{cnylbeua .qBa aena

(oc.emena).

OcenqeHr.r .qeo npBor xBaApara, je oner xmapar. vnja je crpaauua xnno-

Teny3a c, ro jecr raaapar xaa xrrnoreHy3oM c, Ocen{erm aBa Aena apyror

rB:rApara cy rarofe- raa4par4 uuje cy crpaHlrqe Karere d ll D,ugrxx rpoyr:rora,

ro jecr xaaaparN Haa KareraMa.

Karo cy orn ocenqeHu aeirosu jeaHor tr apyror l(Barpara jeaHald (3auro?),

oraj orae4 noxasyje ,ua je rraapar Haa x[noreHy3oM npaBoyrnor rpoyrna

jeanar r6upy KEaapara rraa r(areraMa.

Karo cuo qpaBoyr { Tpoyrao y oBoM orneay fiegau og qerfipll npaBoyrta

Tpoyua) y3en[ npo]BBblbno, :amyryjer'lo aa ntrxuuno xoje cuo ]BBcnI{ B:lx[

ygnurre. To nparu.no ucxasyjeuo, rpahe, omro:

KBrAIrsr rs.q x xlmHy3oM llpaBoyrJron rpoyrno ieAtlaK je rbrpy Kargpera

Il8A XATeTAMa.



Hera yuexqr n3Be,ry rroKa3aHu oDreA c H€trIrM nporr3BoJbHI{M trpaBoyr:IrrM

rpoyrnoM.

OBo trpaBnro no:uaro je no,q uMeEoM flmaropf,Ha reolrcua (no c.nanxolra

rptrKoM lrareMarrrapy Iuraropr, rojn je xnreo y vI BeKy rrpe name epe).

llr.,ra mroro 3aHlIMJbrrBIt( E,aqr{l a, @noraxe flr.rraropr.rHa reopeMa. lI3BeEheMo

caaa jom jeaan orner uotuohy c.nme I12.

Ilou',rarpajruo [poln3BoJbaH trpilBoyrnt

rpoyrao ABC, uag rujur*l cy crpaHnqaMa

KoucrpyricaHlr (BaApartr. Taure O y KBaApary

Haa ayxoM rarero:r,r ECje npeoex .uxjarotara

Tor (Baapara. Aro rpo:.ry ratlry noByqeMo

HOpManyHa xnlloTelry3y,

a3aTn4

rnpaBy

napanemry c x[norexy3oM, uo.{e.rnhevo raj

KBa,4par Ha ser pa AeJra, o6e.nexena 6poje-

nav,a l, 2, 3, 4. Aro ra qeruptr 4e.na:ajea-uo

c roaaparoM naa rpahor"r xarerou (o6e:rexe-

rnrr 6pojeu 5) trceqeMo, EtrMa MoxeMo raqso

noKpnrr{ KBa,4par saa xstroreEy3oMl,{C, rao

nro uora:yje c.nma, Ogu aenoBu Mory ce

AoBecrrry xore no;roxaje naPateffiIrM no-

Mepa* eM (rpaxcmu4iolu).

Ilonno ogaxag oulea, tr3ae,qex Ma KoJIM npaBoylmM TpoyrnoM' aa1e

trcrrr pe3yJrrar, alrglrvo jonr jegnou ga fluraropriua reopeMa Biunrl yoDlure.

rleuy cnyxn lluraroprroa rmpeva? V ,ua,'rer*l u3naraby ErtreheMo aa

oHa r,rMa MHorocTpyKy rpn{eHy y reor'aer;iujr.r (augehelrao To tr y H:urIeM paly

y paanourrqu). Amr y cruv rIrM npl,rM€puMa ocaorno je ro aa ce y flpaBoyrnoM

:fpoyrJryJ aKo cy tro3HaTe .qB€ crpa-

trurle, Moxe noMohy ntrTaroprure re-opeve nahn rPeha. To hevo caAa

Hayqrrru Ha je,urov npra'repy..

Hera jeano 3eMJbumre nr,ra o6-U

nrrK npaBoyrnor rpoyrna ABC, c npa- -

Br{M yrnoM y C (na cnrarr.r 1 13

np€,4claBJEeHO yr{aEer{o y pa3Meplt

I : 1000). Ilpernocraanr'ro jour ga nauje norpe6x4 .Ayxrna xr.rnoreuy:e, i:tt.i

gaje 16or nerrx nperpera xe MoxeMo

Helocp€AHo n:lrepurr. 3aro hel,ro r{3Meplrtr xarere n uahr gpje a: 60 m, b:25m,KBaApar rojr 6u ce l(oucrpynca:rlt Eaa oBuM KareraMa rar'tatn 6u [oBpIuKIre:

Crr. lt 3

C[. ll2

60'? : 3 600, ro jecr 3 600 m2, u 252:625, ro jecr 625 n2.



Ilpeua Ilnraropruroj reopeun, 36np oBr.rx xBa,qpara jepar je tc4lpqfHa.{ x}rnoreHy3oM. Aarne,

3600+625:4225,

ro jecr rm,apar Eaa xr{rlorexy3oM nsaocu 4225 m2. Mebyrml, ualvra je norpe6ra

caMo xrrrroreHy3a, a ny heuo Ao6nrr ra,q Ila noBplrxHe, ro jec,r ua 6poja4225, uzpauyuawo KBaaparxn ropex. Onaj KBaaparslt ropu uoxeuo uahn y

m6luuaua, na je

., iurorerryge: 1T+nS : eS, ro jecr 65 m.

I4s osor npr{Mepa BIra}rMo gp nau je n[TaropxHa r€opeMa xoptrcHo

nocJryxltra aa Ha repeHy uafelto je4ny .qyxlrxy, rojy uxaqe HncMo MonIIr

Ao6rlru xenocpeannM MepelreM.

flperuocraauuo ca,ua ,qa ffa repeHy HtrcMo Mor.rtlr InMepnrn jeany xarery'rpoyr;ra, il aa cMo xr.rnoreuy3y tr npyry (arery n3Meprntr n ao6rurn Ayxune

65 m n 60 m. fuyry xarery Ao6uhelvro oBaxBr{Ir.{ paswm.ir,aner*l: axo je

ftBaapar Haa x[noreny3oM jeanar a6upy xBagpara Haa Karerauir, xBaapar f,lJegnou xarerou jeasax je pg3rf,q KBa.qprrr f,rA xtrfloreEy3oM r APyroM

Kareror, rroruro c€ jegax ca6npax HaJIa3Ir raA ce o.a:6r{n o.4FMe,qpyrn.

Tpe6a, ggrure, FahI.I rBaapare Eaa xtruo:rery3oM tr ,qpyroM (areroM tra lt(oay3srn, a fi3 ocrafia Bpaqyl{ar{ KB:rAparEu ropel. AeKre, 6 he:

652 : 4 22s, 60': 3 600, 4 225-3 600 : 625,

oAaKne ,4PYra *rn u :y'6 -25, ro jecr 25 m.

Vsvrnao jom jeaau upn*lep. Ilpernocrarur"ro ,{a rpe6a z.cn;rtarl Pa Jtra

je je.qax rpoyrao npaBoyrnx aro cy My crpamqe 9 cm, 12 cm, l5 cm' Aa

6ucrrlo ao6lr.rtn noBprlrr4r{y KBa.qpara Ha,q crlxrnnrlarvg rpoyr.na, noalhz heuo

oae 6pojeae Ha xBaApar:

92:81, 122 :144, 162 :256'Karo je xuuorenysa xajgyxa crpaHlrua npaBoyrnor rpoyua, ca6pafier'lo

(Baapare .aBejy Kpatfix crparuqa:

8t + 144:225.

nourro HucMo rcinsg 256, 3aKrby{yjeMo ga raj rpoyrao nlje nparo-

yr:ru, jep i,My 36trp rnaapara Ha,q .EBeMa xpahur'r crpanuEaua ruje je,axar

rgaApary uaa Hajryxolvr, nrroje cnyvaj (o.4 cBaxor trpaBoyrnor rpoyrna'

Ipernocraarano caaa aa cMo y oBnM 3aAaIInMa r"repre 6pojere xarera

Ttrx npaBoyDIID( TpoyrnoBa o6srexunu onnrrnu 6pojenn'ra (uroruua), peuoao

ca a u b, a xlnorexy3y ca c. flogttaro nau je ga rloBprtrrrle rBanpara Koucrpy-

ucaH'x HaA oBIrM CTpaEqal\ra o6elexaralrao ca a2 , b2 , c2. Kaxo Crtlo naUlrn



KBaAflar Ea"q xrtrorery3oM, ro jecr.c2? Ilpeua lltraropnxoj reopervr4 ca6pa.nn

cMo rBaapare Eaa rareraMa. Aro to uannueuo a:rre6apcxn, n'lahevo:

c2.: a2 + b2,

n ra jeAHarogr mpaxara Iln:raroptuy TeopeMy aJrre6apcRur,,r jesnrolrl. A rarocrrao 4o6rut KDaapar Ha,( jegxou xarerolr, a2 v,J t 6r? Og xratpara Ha.q

xxflorexy3oM oAy3enn cMo KBaapar HaA oHoM apyroM rareroM. Aro r,r roH4[firueMo ame6apcrn, mr,taheMo:

a2: c2_b2 u b2: c2_d,

Kao uno nrgnruo, alre6pa npyxa uajxpahr.r sarruH .qa rcpa3uMo HeKlr

ogpelen ognoc uely reauwnraua, .{a Hanu reMo HeKo npaB Jro.

Axo y:uevro y o6rup aa ce r{3 trare noBptrrune(BaApara

rleroBa crpa-unla ao6 ja kao rBa,qparHx ropeH Te rroBprrrrHe, crpaHn{e llpa.Boyrnor rpo-yr.na, o6enexexe xao Mano c, 6she:

t - l/sz-d4br. o l/F lr, b-l'i2-a:,.

I,IcKaxnre ore o6pacqe pe'rr,ora.

Iouohy oarx o6pa3aqa, .qax;re, MoxeMo rtpasyHarr.t jeary crpanr.rqy

npaEoyrnor Tpoyrna m,4 cy no3HaTe ocr:ure 4Be,

Ha npmrrep, aro je xunorery:a je4nor npaaoyrnor rpoyrna c:25 cm,a jegna rarera b:24 cm, ipyra xarera (a) 6nhe:

a:/e:F =,/19=14i:/624--5i6:/4s:7, ro jecr a'-7 cm.

Crparuqe trpitBoyrnor rpoyrJa, pa3yr{e ce, ne vopajy 6nru yoer o6e;re-

xete cll,oYul,la a, b, c.

3AAAIIIT 3A BEXEAISE1. tla{prqiTe rlruoyrJu rpoyrao, na

rry o6ensrarre ra&Tqr€ cn n x ,r q xf,ttot€-

Eyly ca p. 3aTlcl, rDrMeEyjytr IlxraroprnytcoDeMy, I IIIrrTe qerqr Je ie,qnrl(s cBara

on fEr crpaf,rqa.

a lllMepr:re y MxJrBMerpxMa crDa-

Eqe nllaaoyrJrcr rlrotrnl cBor nD|Boyrnoa

n€6qrl, [a loMohy llnramDtrse reop€Me

yrlpatre raje

ro raxqr trDaloyrlrr Tpoyrao.3. .qe ri cr@Eqe lpaaoyrnor rpo-yrna Mory 6xrx ,qDrfl qf,ir q/ MqrB 6lloieD{(eexr llcrrM A ,rxf,locel,l JeAErllrlllMa):

r) 15, 20, 25; O t3, 14, 15;

.) lO, U, Z6i r) 33, 56, 65;

-ltlD) 0.7, 2,4, 2,5t ll 2 - , 3 -, 4 -?.364. IfupaqyEajr€ xxnoreny3y fcl npaao-

yrnor rpofna'rde cy Kerere fa Il 6):a) a- 12 cm, ,:5cm;6) a-65 mh, ,:156mrniB) a:33 mm, ,:5,6 mm.

5. Habxre ie,ury x'tery (a".t

b)npaloymof Tpoyrna xaA cy aate *rflroTe-

njra (c) l apym r€Tem:

a) c:25 cm, 6-15 cn;

O c *3,5 m, a:2,1 m;

o) c-9cm, 6:5cm.



. 6. I{rpaqyDaire ooxM f, lroEDrxrury

nFtBo}fnor rpoyFrr rsra ie lrollraro:

a) xfinoreny3a c-50cm t Karera

,:,|8cm, O r(arsre. a:30qn i 6:40cnr

' 7. Xf,noreryla le.lf,or trpdBo]Tnor

1TDoyrna je c:

-ctt, a ierf,{ r€Tgra mf,ocx

2ipfl nel|llrc ixttorery3e. I'ITa{yBajre Apyry

Karery. (O.uromp: 2/5 cm)

8. Jenra Karera [eror rrPoloyrnor

rDoFna je lrsrxyr aID*. oI .{pyfe- I{3 a-

ry{ajTe lsenoBy xtnorety3y aKo te ooBp'

ulllxa 16 cm'. (O,aroEop: tP4 ,..)

9. Iecrse' ryre l0 m, npf,cJronefe

cy Ea 3f,A r(yhe rsl(o aa ilM ropnn xplnocTf,nae BucxEy oa E m. Konmo ie MeTepay'uarbes .Ilosf, rylaj tecraEa oa srlr?

1O Aro uecrap ctoirrr rPamrr' ry-,r.rf,e 130 nm, o6ytrara .ry/f, or 32 mm,

ro,nrro Je neror Bpr yasJtaE o.I're srol?(OrroBopr 112 mn)

', 11. V ;IEopnurry r?e6a r(atrtnoM orpa-

.ulx rtpaBo]TJrt rJtyrao ctia ie noBplrrf,Ea

60 m'z, r ie,q[a KaTcra 15 m. KonxKo lexroatra norD€6ro? (OJroDop: 40 m)

12. Vlt iealror q'|lcra[f,ufir lllr€ry

f,groBpeMello .qEe naDe, ll To jea{r Ea ceBep,

trFena3€hfl l5 ktrl Ea sac' a rpyra Ea aaoal

Ilpenrtehx 20 km ra qtG Kofioco he o',e

5f,Tr yaa,r €He je.stra o,[ apyre nocne 3 qaco-

r. nyroEarba? (OrroBop: 125 km)

13, KalaJLKe ielsor qacoBuxKa tMaiy

al'lrxEy 5,6 cm fl 3.3 cm. Konixo be uemx-Merapa apxoaf, oBrlr( LdlaJbKll 6rm yaa.te.

Er jeAa[ oa ,qpyror y 9 ,racoEa? (Oarosop:

6,5 cltr)

14, Ton maa ,loMer (Ti. Mo?lce aa

aofaur rpasrry [a xlBecry .urJLtury) 6 kD, a

ynarbeE ie 4 ko oA jerse trpalofioxjcKe

xeJresE ru(e nDyae. KoJm(x ,qeo trpltrc Moxe

r&i ron na ryqe? (OI|'oEoD: t 9,14 kn)

15. EnerarprqEx cryq Drcox 8 m'

yllDplIrhea ie Daa{ ocEypaEa @ tpl stTer-

ryre xxue, Koje cy leluxM rapileu te3ase

3r qry6 Ea Brcrrllf, oa 6m. ADyrr rpqjesx

oBEx xiqa yqBplnh€sr cy a l(oq|rhe rc6o-

aeBe y xoptl3omardo 3eMJ6truTe, f,l Docm-jaoy 4m o,u rorsor(ja cry6a. Koarxa ie

,l Dt$Hr Te ,r(rqe? (OnroEoP: 21,63 m)

16, Epo[ trroB{ y rqlarry ceBelpso-

nlAar 6p3[uoM oa24

kDHa qlcr

aoK loile-ra crpyjr llus r4nlaq oeBepoxcroK, a 6p r-

ry lokD sa qac. Ko,rxr(o he Rf,roMerala

6 oa ornjroBxrx 3s jenaf, qac? (O.smDop:

26 kn)

17. Asnod nerl y npaBrry ceDepa, 6p-

3[doM oA 24okDl xa qac,,{ox xcloquB aerap

,uyEa dptrlxoM o.q 100 km ua {4c. l(onlrohe os 6rrr yralb€E otr jea[or yoeeDor Mecra

Docre 6 i{ryra? (OaroBoP: 26 km)

l& no,quoxja aEejy kyna gloie y

xsroi xopx3ot€&oi p&BEr' ia pe ,[ajfifix

oA l5m. JeaBr ie Excor.a 35m,4tyre 43m.

Korrr.a je .4'lr'sr ,ftf,ue. Koia cmia (rrpa'

Bontltrldcrai) rrrooe rpiore? (OaroaoP:

17 m)

t9. Jlecrae, ayr*xtre 15 m, lacrroEele

cy Ba 3';l Kyhe ra*o.qp aogrlr(y ao tpo-

lopr Da E{clr{Ir 12 m oA 3eMJEe. Arao re

necrBe okDe{eiuo btro no.qaoxge raqre na

npyry cTpa,.y yrnne. olre,{ocTxr(y ,uo iealrot'

npo3opa xoj4 .ie ctr ru"'re y.la,'6en 9 m.

KorxNa je ullrpnl|a yxlqe; (OaroBop: 2l m)

Cr. ll4

20, Ha c {rqfl ll4 iet AB-z' BC-4rr cD-l. filDaqyrajre ayrirny AD. (O',.o'

loD t

75



21. Hs ctn|r ll5 te, AC-CB-E,CD-12 t LACD -90', Ilallrre m:ryrper-

/ 2\Ef,r rprflcrrqe.

IOaroeol t

;]12. Hat crDaxmlaMa rDaroFJroa

\W$JA ABC Korrclxl 'lrurf,Te na,qprre (cJr.

116), .rrflc| nporyxrre rBc crpoBlrua rEs,{-psrr taa xf,foref,ytoM TaKo .[r rodf,jer€ql,tt AE t BD, t, rt raqKe D Dolyqrre

DFllAB. Illr^ ca,q rpe6a aa DaAxre Aa

6xqre f, oBf,nr orneAoM aoKalantr nrt4io-Dnry reoleMy?

,/\F/)\D

\r lz

Cn. 116n. 116 C,t. ll1

a3. H. qnlx 117 alr Aeqa(s ce BpasyEajTe raiEehy Brcssy Ao ftoje .:€ cta(r

,,xJraltrCri}*'. Ilp€ua @Aaurxa rs roj c.rw oa Ert Mor(e norths. (O.qroBop: 3 m)

2. TIPI{MEHA III4TATOPIIHE TEOPEME

Y I(OHCI?YKIIIIJCKI{M 3MAIII,IMA

IIptrMeEoM lfuraropmor npaBfi.lra MoxeMo peurltrn HeKe xoucrpyKuxjcre

3aAarre, trlto her'ro szgertr ug caeaehnx trptrMepa.

1) Korcrpyncrrr xra.qnr rrja je norpm jegnaxr :6rpy noll,ltrn.EBa Aara

r(Bqalr8fl.

Hera .qarr Kmapartr n\aajy crparlqe a t b (ctt. 118). Tpe6a l(oHcrpyu-cartr trpaBoyrfl{ rpoyrao vrje cy Rarere a n b. Kwlpa:r xoHcrPyucau HaA

xtrtrorery3oM c ror rpoyum 6nhe rpa,xeur rrarpar, jep je uo lluraropnnoj

reop}{lr C:&+02, (Pasyrue ce, rBaApar wrja je crpaH[ua c tuoxe ce n gace6-

EO KOHCTpyUCaTtr, a Ee XaO Ha Cn4tr.)

CBCn. l15



2. Koncrpyrcarn rraAllar .[is ie noBpm icAf,lttr prrJtu [o.p[r ,raItTa Kraatr)tr&

tla cfiry 119 rea4larr nrajy crpaqe nr n n.

cJI. r18

Axo roncrpyuuer'ro npaBoyrntr L?oyrao tlnja je xruorexysa m n jegta

,N, oug4 he rragpar xoxorpyucan EaA ApyroM rareroru (p) 6a:rn, rpelra

reopeMu:

f: t?-nz

1?axeEE rBanpar.

3) Koxcrpyxcaru nraglrar strjr je norpm paryr reha oA troBprnf, Aaror

rBa,4llars.

Hera je crpautqa naror KBaAIlara a (cn. L2O). Aro r"ry loayrevo .urja-

u r'ap, rr oM, Kao BaA crltaHs-

roncrpytrme do rragPar, 6nhe:

&:e+&:2e.

I

ln

r--llllm



Ipeua ror'le, rcaApar EaA

Behy noEptu oA Aaror.Ha ocHoBy rof pemeg,a pernr,rre onaj 3aaaraK: Kotrcrpytrcarf, Kla.[par

aBtlrrr Molbe troBpmf, oA troBpnlf, AaTor xBaApaTr.

4) Koucrpyrcaru xrupar rrja je troBprtr jelnrrca s6upy troBpurl Brme

AATf,X XOAAp T&

Ha crrqu 121 aare cy crpaurrqe xBa.qpara a, b, c, d. Hajnpe xoxcrpy-rrrueMo crpaHlrqy m rlaApara .ruja je uorpur jeanara s6npy nonpur,r gnajy

aa'rru, ro jecr

m2:a2+b2.

3arrau roucrpyxutr.rMo xr.trroreuy3y rt npaEoyrnor rpoyua vuje cy rarerem c, ta ao6xjaMo:

n2:m2+c2:a2+b2+c2.

d

c

h

ap(.r. 121

Haj:ag xcncrpyuurer,ao xnnorenysy p npaBoyrnor rpoyr:m rnje cy Karere

nud: pz : nz + d2 : a2 + bz a p2 I d2,

tlttMe je ao6r.rjeH TpaxeHlr KEaApar, I{eo onaj pa'q npaxrr.rqHo cq r.r3BoAr.r Kao luroje nprra:axo na cnuqlr 121.

3AAAqrt 3A BEI(EAIbE

gnjarouaror,r ,4aror xBa.qtara uMa aBanyr

1. Ha{prrjre ars jeAsarr KBraDNTr,

na Kolcrpy{unre rtaagDar qrja ie norpur:a) jefllaKa 36rpy Ef,xoBxx roBDurr,

O ietr-llaNa D&3JUIU | lgltxoEx)( trODpurll.

2. Koffcrpyf,[xre raBglprrllolpu e[r o,q nospurf, Aat(ra) Ira ry.a, O qerrpr tryr&

78

3, I(orcrptnurre KB{.[prl fiia je

norptn j€.UoKa nturoElltfi noDptllx Aaror

(aLIIATS.

qf,J. je {' I'I pmJre rpr reje.warta RBraIta-

xrqlprllri 1A' rn Ko{ClDylltrfrrc Xra'[PaT sia ie rcaDul

je,qrrKe r6xpy Brrorllx DoDputr.



5. HaurrsiTe rDs reierEaKa l('r,Iln'T& tcoEsryrurle r(laaltsr {f,ia Je trorDul

,Drl'yf re[4 oA prlrXKe bllro f,x llolpltrl.

6. Kor.crwrrlfire rlsAllaa cxir ie

trouxltrna 13 cE'. (n85trel 13-9+4,)

7. Koqcrpyxurrre4rrr

ryxtxe y'2 cm.

(l/r-ltr+t)t. Korcrlyrrmre ryx avncme y'5cn"

(/{-ttr+1)9. Tpn rcaqpan xof,crpYxcals (Y xao

uro norcaryle cJEfo ltl Axo Je s.Dsstm

f,.iMrF€r 3 cttr; Epaqysths cr?s@y rri-rahar.

10, Aro ie' nlleMa crfil la3, norP'

trmr nileh€r xBa,qrart 16 cD'z, BDacyEaiae

,qrxq crponue rojuarer.

11. II .q Ar€Ml crprmard& trP{Bo-

yraof,fia noxcTDyscanf, cy Kta&rrTll, rsro

xa cJlrqr 12l. Ar.o Betx rta'qtar nlr trG'

rDlltlrEy il ryra €iy oA MaEer' lura gc Moxne

lrrsby{rlx o ruresltj$lr nrt{rr qraro-

yraourra? Kucrpy[rrre rlrxat qllloyrao-

ur ero je trotDums r&r xrg.4rrrr 9 oD"

r MlE€r -4 ryTa MrEo.

C\. 123

aelmraaa. Mebi"rrnr Mor(e ce o,|pentlrr ayxqrJf, je Melra $poi 16 Koncrpyruure ry

Cn. 124

at"x( nllnrcf,ou llrraroDrne rooIFMe' { Tlt Lpa rsqirr, rpeng crea€irlM ieararocrnua:

"rE-y'ET-r,,qlE-/T-A

3. fIPIIMEHA IIIITATOPIHE TEOPEME

HA TEOMETPIIJCKE (DITI. r'PE

nnraropl4Ha reopeMr Moxe @ nptrMeHltrlr Ha coe reoMerplrjcre Q{rype

lojc ce, nolJraqereM u3Becxn ayxlr, Mory flo.qernrlr raxo 'qa Meby,qeroBi'tMa

llMa fi TIPaBLTHID( TpoyrnoEa. v serotlKo c.neAehnx trpuMepa Bq[eheMo raxo

Cn. 122



l) UpaBoyreonff. Ilparoyraomn je gEjaronalor\tr noaerex sa lFa troAy-

Aapra trpaBoyura rpoyrm (cn. 125). flprueaov firraropnxe reolrve na je,qa

oA Erl MoxeMo mpaqyEarz ,q jaroEany ra,qa qr no3xare o6e gnuen:nje, xao

r je,rry oa gnraerrrja xar je noauara grjarorara n apyra ant{er3[ja.

Hera ce, Ea trpm"rep, rparu o6nr.r tr troBpuurla nparoyraorrra urja je

Ntffia, a:12cm, z pgjgr.orla:rra- d:13cn.

a

Cjd. 125 cn. 126

Vselqtr he, rpnueuorrr Ifrraropuue reopeMe, ao6urr cneaehe pe3yJrrare

O:34mm s P-6Oamz.O6jacnrrre trocrynar oBor rcFr.rygaBaba.

Oxo npaaoyraoxma, Rao rIlTo je noenaro, Moxe ,qa ce ontrme rpyxuln1a.

IIIra je nonyupewrr r r,qe je qeardp re xpyxarqe?

2) Krrgprr. AujaroEara aem rEa,q)ar Ha ,qBa troayAapEa jeaxa*orpara

nparoyr:ra rpoyua (cn. 126). Ilpnueaou ffrrraropme reopeMe Ea jegar oa

rrn, urremherr.ro o6pa:aq sa usparynaaane .ryjarouane raa je no3xara crp:ruurla

(nperua o:naxar'ra xa cnnqu):

a:YE-ad:lW:ay'I,ro jecr 4njrrona.ny Klcapsra ,qodf,irMo Kra MepEf, 6poj neroae crpa@e troMno-

rrlrro xroAprrf,f,M r(opeHoM 6pois 2.

rrorro ce /2 ne uoxe Earrucarrr f,ao ronaqaE ,qe@ ,raJrrr pa3JroMar

(ao6xja ce Eeorpasnqea 6poj geqnuana), sa .ryjarotrany ce.qo6uja yBer caMo

npn6mxxa Bpe,Focr. gecro je AoBoJbEo M u"o

Vlllsr"tu npx6:nrxoro rpen-

socr ca 2 Aeqrovrara, To jefl 1,41.

Taro je, xa qpnMep, &jaroEaJra KBa,rpara vrja je crpau4a a: 10 cm,

a:ay'I:t0.1,41:14,1, ro,jecr del4,l cm (rae 3sa( Ai 3gat r,,npa6;ruxxojeasaro").

3) Jegf,akoxpsKtr rpoyrao Brcma trol5r,reEa u3 Epxa AeJrx je4narorqraxr

q)oyrao crrMerpruo, rtrro 3Earm Aa troJroln tr ocxoBqy xa xojy uaga. .{ratroAy,qaptra npaBoyflra rpoyrna, xoja raro Ao6njanro, nr,rajy rarere npeMa



o3HaKaMa ra cnrqr{ 1,27 u l, a xruorery:y b. Axo gnauo ,@a oa rnr1

eJreMeHara, rpeht woxeuo aa rnpaqyHaMo npxMesoM llrraropme reopoMe.

Hera yrenn{n peme onaj 3aaaraK Aara je ocuon44 d: 10 cm fr BvcEE

h:l2cm; E3paqyuar[ Ayxnxy beroBor o6rnra. (Omoiop: 36 cm)

4) JeAru(ocrprElsun rpoyF:to. Br.rAenn cuo aa ce jemaxocrpaErFrH[ Tpoy-

rao Moxe Br{c{FoM ctrMet1)FrEo troAertrTn Ea IFa npaBoyrna rpoyr.na. Aroo6e;rexrMo eneMenle Kao Ea cJlr.rqfi 128, xunorenysa rarBor je.qaor rpoyrna

al

Je 4. a KaTcTe cy h t4-

.

z

azan. 127

ll3Be,quMo uajnpe o6pacqe 3a lrpa{yHaBane EucI{Ee:

t=1;g:tl;4:l .,lT:+aI,I: oror o6pacqa BH,qtrMo Aa y jeAEaxocrpallfiErou rpoyr.rry rncnry go6r

jauo xa4 tronoaruy rseroBe crporue troMrrorf,Mo xBaAtrlaTmM nopeuou 6pja 3.

3arru rrsearft{o o6pa:aq sa noBprrrtrEy jeggarocrpagngor rpoyr.ra. Aroy o6pacqy 3a BeroBy troBprnnuy lsaMesruo srlcrry h xaferou rpeaxonrhy,rarraheuo:

p -L . ah- r--o ." vrt :ty't

2224liJrr{ pgquMa: troBpmf,Ey jeguanocrpanxunor rpoyrna 4oSujonro ltoA qerBprf,Hy

KBaApara rreroBe crptHnqe troMtrorntrMo r$aAportrf,M nopeuou 6poja 3.

(V ga,qaqnMa MoxeMo y3l4Marr y'T: l,ll.\Beh crrao rlayqsJlr. Aa. IIeHTap o lrcaHe oaEocxo yrtrciute xpyxuuqe jea-

Harocrpantlqqor rpoyrJra aenu BeroBe BucEHe y pa3Meptr 2:1. Ha ocxory

Tora, a npuMeuoM fluraropune reopeMe, u roprcrehn ce ooxaraMa rra cln{u 129,

4a2 - a2

c.r. 128



My (aro ca a o3EaqxMo. crpaEtrly A,{BC) xunoreEy3a

av3E .ryXa KaTeTa 2 .

Ilexa yrreHnq[ x3De.[y o6pacue 3a noJryllp€qHtrKe OI tcaHe r ynllcaEe rpyxHl.llle

f,3paxef,e floMohy cTpaEnqe rpoyrm 4, ro jecr:

R: 6. ,: tE.36

Ilocvarpajuo je,Farocrpaxnwa rpoyrao ABC (cn 130-I). EeroEa Bu-cma (CD).qentr ra cnMer?rlrmo Ea ADa noAyAapHa np:rBoyrna 4royrna. Tararjegan rpoyrao (cn. l30JI) trMa orurpe JnnoEe oa 60' u 30', a crpaxglre cy

a, xpa.ha"unu I

c,n, 129 I Cn. 130 Il

Axo, gaure, y .qajbtrM reolr,rerpajcrrau ricntrrnanrnaa nabevo la jelax

lpaboyuu rpoyrao xMa xltrorery3y rlanyr Ayxy o4 xpahe rarerg sxaheuo

oAMax aa cy My crnrprr yurorr 60' N 30'; r:rr.r, axo nalelto 1a jegSu nlmlo:

yrnn rpoyrao uwa jeggx yrao og 60' (o4nocxo 30"), :nahevo gg je neromrpaha rarer:a je,4nara uonorunu xuuo-

TeHy3e,

a lyx(a -npoKlBo,qy xnnoTe-

-.\try3e r{ opoJa2 , *4.

5) ffperuaeu mecroyreo. Aro y

. rrpaBxnEoM uecroyrny noryrevo ,uuja-

rouane xoje ra c[MerpuqHo ae;re, on he

6urx noae:res xa urecr,je4uarocrpaun-

Hrr( rl troayAaputrx rpoyrnora (ca. 131)'

ffpeMa roMe, nouprtrI{Ey npaBrr;Ixor luec-

royua uoritevo x3paqylarn aro je no:-

HaTa lberoBa cTpaHtrqa a:

P:6.t6-Yv7.42

a

2Cn. 129

Cn. 131



Ko.nrma je, trpeMa roMe, rrogpru{ra npaBu;rHor ruecroyrra sqia je crpa-a:Z cm?

6) Pou6. flourro cc r(o4 poM6a alljarogarre floJroae x croje jegxa ra 4py-uopuanno (rj. jegaa Apyroj je cuMcrpara), ore ,qene poM6 Ha {err{p[ no-

npaBoyrna rpoyr,ra (c.r. 132). ItrTa je xunorery:a, a rrrra cy Karere

cBaKoM oa oBr.rx rpoyr:rona? Aro guauo aBa oA oBrrx eneuexara, rpehufi3paqyHarr.r [pnMerroM lluraropzue reopeMe.

Taro, xa [pI.IMep, crpaHllqa poM6a, npeMa o3HaK:rMa Ha c.nrqu i32,6nhe:.

td,), rd 1' l|--F, pf+$ I /_':VU) -FliJ :V i* i:'l'z-:7'lai+a1'

7. JeAnanoftpaxr rpaner Y je4naroxparoM rpane:y IBCD (ctr. 133) roay-je us Bpxa D Ayx DEIIBC ao upecera .E ca oclroBluoM. THMe je oH

Ha napaJrenorpaM BCDE u jeauarcoxpaxu rpoyrao AED (:auro je raj

jeanaroxpar?). Bucma GE rpanea je tcroapeueno ,I Btrcrua jeAna-

rpoyua. Ha jegau oa raro ,qo6ujeur.rx npaBoy rux rpoyrnoBa MoxeMo

flnraropnny reopeMy.Hexa ce, Ha npuMep, Tpaxi{ noBprunna jeasaroxparor rpane3a raA cy

rleroBe ocHos qe AB:a-28cm, CD:b: 12cm u rpax AD:BC::c:17 cm. Pelrrre raj saaarax. (Oarosop 300 cm'?)

8) Kpyxffiqa' Aro n: qesrpa KpyxHnqe troByqeMo HopMzrny Ea rernBy,

HopMa-ra'. Kao r ro cMo Bl.r.rr,eJrr{, l.rcroBpeMeno nor6ar.r rernry, ro jecr

c.n. 132

CJl. 133



oEa joj je cr.Merpara. Taro je (npeva o3Ha(aMa rra cnrflI{ 134). OCIAB n

AC: BC. Iloryaer'ro rn jolrl nony-

npeq{Er xpyxrrmle ao (pajtre rasxe

rersBe (Ot), Ao6ujaMo np:rBoyrnu

Cn. 134 C.n. 135

rpoyrzo OBC, IlTo 3lraqtr Aa ce n Ea KpprGEy uoxe npnrrexrrn flrra-DOptrHa TeopeMut.

9) Kosr f, rEqApsr. Ha crnrF 135 npe.ucran:r,exa je xoqxa nrtrIe d.

Bx6n ce ga je gujarouala xolxe (D) EcroBpeMero n ArjaroxaJra grjarororxor

trpeceKa, To JecT npaBoyrao-

EtrKa qnja je gyxma aujaro--F.aj1a (d) ocEoEe, a BIrcnIm -nBqa {oqre (a). flpnuexor'.rflrraropme reopeMe trMaMo:

,.D2:&+d2.a xaxo jel

d2:q2 +d:2 & (:amro?),

6utle Dz :3az uri*t D :15& :: alry, Ta&o ji sa roury,uBu"Ie 4:10 cm, 4lrjarOna:ra:

D:t}y'3:10. 1,73:17,3, ro jecr D:17,3 cw.

Ilocuarpajr'lo c:r.qa RBaaap wrje cy Aur,remtrje a, b, c,

C.nnxa 136 noxasyje gg .unjarona:ry (D) xra,apa MoxeMo $patrysarl

trpEMeEoM Ilt'raropnre reopeMe Ha npaBoyrntr Tpoyrao EHja je xrnoreny:a

a

CrL 136



ra A[jaroEana, a Karere cy joj BucEHa KBaApa (c) s ,qljaroHana ocEoBe (d).Ilourro je d?: a2 + b2, rnaaheuo:

D2: d2 + cz: az + b2 + d, D:/A+F+V

Taro, ,ryjaroua;ra rsalpa qrje cy gn"rexsxje a:3cm, b:4cm, c.:lzcm,r.rsuocu l3 cm.

3AAAIM 3A BE'I(EABE

1. Iftrrc(o jeltror rMatra y -odrtrLT

trpaBo 1r&ouxlta, l[tftrDe 156 m ll rusprdg 65 m,

Tpefa racrmr ctasy y rpauy ieaue adaro-f,ane. KoJm(o he 6rrt .4rra{(a ra' cra3a?

2. KoJtnxa je AniaroEara Ksallrara:a) qnia ie crpaEqa 100 rxm, O qf,if, je

o6uM 4t2 crll, E) eEia je troBprltrlsa 784 cm,?

3. Kaaat)ar oa rcaproBa, cT|rafiuqe

4 cm, [pe[onoEJL€n ie nparon llapanenf,ott

c aBeMa E€mBrM crDlf,uFtnta' tra ie o,q Trx

aeroEa cacraBJlef, npaaoJ[aof,{r. 3a KonxKo

je MrnrMerElra Af,laro[ana olor npaBoyrao-

'EtrKa A 4aa OA Af,iaronane xra.qpara? €a-allre plcynclt|r i MepelseM, na ynolteatre

pe3ynrare.)

4. AKo cnojxMo ayx[Ma, tpeamey3acronEd crlraff,ua KBa,qpara, no5{iaMo

ieAafi qerBolroy.ao. flbraxxre .qa ie rajqerBolroplo r$aApaa f, sqi:rq}'IraiTe Mj/ o6rn|t

axo ie cqtasiqa aaror tranpara 4 cm.

5. V rparolraonnx je ynicsf, rx'oy-

rao, Itab llrro soKasyje o1f,r(a t37. ll3pa{y-xajre o6rrr mr rpoyrna, [peMa troaaqrMa

m cnsqi, (Oarorop: 34,7.)

b5

6. Aar je nloDofrao{m( ,{.8C,(cn. 138): lB:24 cm, .BC-10 ch,I{3paqy-DajTe o6Ell f, noBpurf,Fy r ,ryrJt AMC, rn,e

ie M c f,.Atlra ocsoBrqe ,4r.

M

Cn. 138

7. Cpe,qrory je,qse crparxqe (Ea.qpaTa,

'r[jr ie o6f,M 16 cm, cllojBTe Ayn Ma c ae-

MesEMa EacrrpaMue crpaauue. hrpa*ynaire,3arsM, o6nM ao6xje[or je,qEaKoKDaKor rlo-ynra.

8. Aro ce

trojueMepneBnne rrocrate

raro ga je pacrojarse srueiry pa3MaruyEr

RpajeBa 3 m, of,e Aocrf,xy srcnry o.q 3,5 m.

I(ontKo cy stcoKe l(aA ce qtrote?

9. ABe crpaxf,qe jeAfiaxof,?a(or rpoy-r.Ia cy 8 cm f, 20 cm. I{3lraq}Tajre My

B{CUrly.

10. On ABa lloAyAaplxl je,Uraxoxpaxa

Tpofna, exje qy Are crlrnEE{e 5 cm x 2 cm:

a) cacraBxrc poM6 n Ep4qysaiTe My

,nar|(y arjarotrany, O c{craBrre poM6osA Iuqa.rtr'uaire My noBpuulry.

11. HaA ielsoM crpatxqou nleEoy-

moflrK€ ltotrcrtryicall ie ie.marocrpaf,rsu



rpoymo, (cJr. 139). flpeMe roaalf,ua Ea crx-qr, BparyDajTe lmlmry troBprnrsy re orfy-pe. (O.uroBop: 14,92 cm'1 qH6Jrxxoo)

12. I{3pa{FaiTe ayxuEy ie,qrc .qrda-

mxaJle llpaBllrEor uecpyTna, tcoia ra f,e

Ie,r'l cnMerptrqEo (AB $ cn, 140)' aKo My

ie srpamua 3 cm. (ltaxa.B ie ryofrt ABCI3alxro? OAroEop: 5,19 cm nptr6mo*ao).

CJI: I40

13, Al{o ie cryraxf,qa nltaBf,rsot urcc-

To]'rnr 10mm, KorxKr ie crpaflxqa tpaBt,r.

EOr rpolrna Roif, le y t6era ymlcan rut

naqxB Kao uToie

rDllr(a3aBo Ha cn[Un 141?

14, IlpaBnnf,{ urecrota-ao aar je Ay-

r{lrEoM crpau{e: 6cm. I43paqy[4iTe noBD-

lIIlRy ,qpyfor fipaBsrBor rrlecroyrna l(ottr n c-Taie ctraiars€M cpeax[a y5acTonu[x cTllarlf,Ua

Aaror (cn 142).

15. AxiaroE re jeanor pou6a ct16 (D r 30 cm. I{4orpaire My o6e{rxclry, (O,{roDop: O :68 cm, , - 1t,1 cm)

16. Kontlra je ltyxa,ryiaro{ana I}oM6otKo je Kpaha nrjarouala ienrat(r c4raf,nu a?

17. I{3paqyraiTe y rBalparllnM neun-

MeTpriM:r floBplrlErly je,qxaRo(paKor rDanesaqnje cy crpasxrre: napanenle 28cm r 12cm,

a KIraK 17 cm. (OAroBop: 3 dm')18. flapane,'rle crDalrrie jearor npa-

BOI/mor TDale3a cy 35 cm l' 20 cm, a EncnEa

8 cm. Korsrcx My je o6rM? (OAroBop: 80 cm)

19. Ilceulre aBa noaynapxa rEanpara

oa xapTftie, rpori3DoJ fie cTpatlmle

a,na

ie,uaE oA r6nx I{rcnoJroBBTe, ceKyhr ra no

Aliarosann. OA orllx AenoDa lr [pBor KEa-

npata cacraB{Te ie,anarorparr rpaner. Ko-nxra ie ,qliaro{ana oBoa Utarc3a? (OAro-

"op-oy'1,

139r. Ca. 141

c.tt. 142



20. KoBcrpy{ulre ll rceqtlTe 3 [oAy-

Aaplra jeAHaxocrpaH {Da Tpoyana, oA r(ojr8cBaRfl {Mlr y o6fiMy 24 cmr lra rx flopebaire

Ta{o Ia qxue Tpane3. ToM Tpane3y l|3paqy-

BajTe obrM { noBpurmry (OInoBop: O - 40 cm,

P- 83,04 cm')

21. TerdBa. Ayraqr€ 30 cm, yAar6erE

i€ 8 cm or ue[rpa Kpyr(H{ue. Ko,IHKrt ienonyflpeqHlR Te xpyxufi r€? {a ,rt ie raTeT[Ba A]rKa OA CTpJnrI te ynlrcauor KBaApATa?

22. Ha:praire ABe Koljre8Tp[rne(pyxHriue norrylpe4uKa 35 mm II 28 mln,

na RoHcrplrllrllure je {y rerfiBy Behe Kpyr|({ft :e

noia qoqrpyje Mal6y. []paqyHaire, 3arfiM,

.qyrsr y re rerrue. (otroBop: 42 mm)

26, A{MeBrde ie,qqor Rraapara cy:

11:0,3 dm, b:0,4 dlfit, c:1,2dm. KoHcrp}'|r-

ulrre remry upexy. I,I3paqyraiTe noBputxuy,

lafip€vrtHy. arri lfolra,[y t noBprxnEy .ll,ljaro-sanrlor npeceka.

27. Ax m wr 3a rrnererse, Ayraql.]all cm. vor*e ta crane y ryrtiy odlrma

KsaApa qtie cy AnMefi3xje: 9 cm, 6 cm, 2 cm?

23. HauDraire r T}rr@y, mnytr +oftr 34 mm, TaKo ,Ia [pona3tl Kpo KlraieieA s B Nnxs AB-3|6E.3arn,l f,tlraqjiEaiTe

olcrojarLe rleu4ra xlryh.f,IlUe oA ter',ae AB.(OrroBoD 30 mm)

U, Ira c]'fiit,, 143, npaDxnlllt uecro-yrao ie Ea ABa rrnqua no.UeJ'hea sr noAyAaDse

AenoBe. UsDa.ry[aiTe y o6a cny,raia 06l|rr inoBpulry ieaFor oA rrra lenosa aKo crpa-glua uecroyrna t3uoc{ 2 cm.

25. I(oura uMa rrg l{t' a:2,5 ctt\.

Ko{fip}fllx re rr'erly Mper{y. llrpaqytaiTeIroaprnrEy sanpeMrlEy, Iflirrof,a,rjr I noBp-

rrrury atiafosanEor opeoerca.

28. JeAas rcEaApat rrMa'nirMerl3xie:

2cm, 3cm. 6cm. Ofpelf, re EeroBy AHiafo-ualy (oEcrpy({xjcKr, a 3arfiM rrpoleplrepa{lftcKx. Crtrtra 1{4 xoMohu he BaM .qa

j'BqDfIe Kojr llpaBoyrr{ rlroynao Tlreba f,aj-rlpe aa xouctpy& ieTe,

' 29. PeIIrI|re sraAar&K 28 3a c.nyqaj Ran

ie Aara xou{a, rBfiqe 3 cm. Brlallre cn&Ry 145.

Cn. 143

(J-\- --:--

( ----:.i)\'

.\\

Cn. 144 Cr. 145



MEItrOBITTI{ 3A.IIAIIII

1. Tasra i}I geax crpaxqr AB'tpoy a ABC raxo Aa je AMiMB:2:3,orpalllqa AC no4elbena je ua 10 jeAHaxrlx AenoBa (bn. 146). Koja o.q orux

c.n. 147

Cr. 146

Aeour.rx raqaxa ,qenu c{paffuqy lC Ha

geaoae gilonopuroHaJrrre aenoB{Ma.lMn MB' Axo ry raqxy osxarure ca 1f,

y xoM cy uefyco6xorr.r noaoxajy npane

MN n BC2 O6jacrmre :auro.2. Ilper',ra noAa[nMa Ha cnnqa 147 ,

urpauynajre AyxtrHy Jryra l,SB lr norp-rur,rrry rpyxnor oaceqxa l8S,{.

3. Axo je npeua cn.rqlr 148,

AB:BC:CD:z cn, lB- lC- 120',DE : EA : 4 cm, Koucrpyulrure 6lzl.ypy ABCDE u r.rrpavynajre Ber{y uoBprrnuy.

BcCn. 148

o

,ec<l1g>._



4. ,{re ranrenre &yxsrlqe cery ce troA yrnoM oa 60'. Aro je .qyxfHa

tronynpeqHtrtra rppriEnle 4 cm, xo.ruxo je yg3.rex o,4 q,enTpa flpeceK TEx raure-

xara? (Omoaop: 8 cm)

5. ,{ra 6para cy no,qenll;Ia uMabe y o6.nuxy qran*a ABCD, (cn. 149)

no nparoj MN,tAe je M cpeplrra AB a N je cpeguua CD.,{a nu cy go6ulu

jeAnaxe ae:rore? O6pa:.roxure oaoje rap$ene.

M

Cn. 149 Cn. t50

6, Kojrr,qeo rpoyrra oaceqaMo upaBoM KoJa nonoltl Je.qEy crpanl'Iuy,

a trapanenna je ca apyrou? Bururre c: fty 150, rle je MA:MC a MNI AB.

(Vua tn ry cnlFmux rpoyr:roaa? ,{a ;ru je nparou MN rpeuoJroBJbeHa I{

B cnHa rpoyma?)7. Ko.rnro crpaHlrua ultr npa-

BnJraH Mnoroyrao uujn oaaxn Yrao.usHocl50"? Koxcrpynnrnre raxaa

lrHoroyrao, orpaxnue 2,5 crn.

8. Axo je,qaH yrao npaBoyrnor

Tpoyrna IBHocIt 30', noxaxrre Aa

xuIIoreHy3UHa bnCI{Ha AeJrfi x}Inore-

Ey3y y o,4Hocy 3:1, ro jeor nPernla

cnuqs l5l, AD:DC:3i l' (Kaxru

cy rpoyrnoBr ABC n BCD? Roju cY

nM yrrroBl4 je,u,Haxtr? IIIra 3HaMo o

crpaHluaMa npanoymor rpoyrna vnjn

1eje,aau yrao 30"? Aro (arery .BC o3l{aqllre ca 2a' ievy he 6nrs jeauaro

AC, Cm)g. Eyxn*a jeane rpyxrlqe uanocl 4rrcm, a jegua nexa rerl'rBa y.qalberra

je oA qenrpa 3a nonoBllsy concrBeHe ayxl{He:a) nsparynajre AyxnHY rersBe,

6) ncnurajre roju ylucarn npaBltnHu MHoroy€o xMa ry rerlrBy Kao

cTpa*{nqy,

c

Crr. 151

89



B) KoHcTpyr. mrre ry ,rpyxHn{y r.r Terr.rBy.

Ko,rnro peurena xtlta osaj saAarar?

10. y npaBoynroM rpoymy ABC (ctr. 152) ra,txa M je cpepuua xarereAB. Axo je AC:33 mm, BC:65 mm, Korl.rra je gyx MC?

11. Aro je, npeMa cnrutrr 153, AB:BC:CD:4 cm, lB: /C:120.,AE: ED - 4,4 cm r(oHcrpyr{tur{re $wypy ABCDE u urpavynajre rberry rroBpurlrHy.

12. flpelra cnllr 154, u:pa-

EyHajre 06 M rpr.rlollruujcre r[z-

rpe ABC, a 3arr4M r{ rbeHy rro-

Bprxr.rHy.

AMBCn. 152,

BCCn. 153

13. ,{re nogyAapue Kpyxgr{qe, rronyrtpe { xa r: l0 mm, cery ce raKo aacBara o.{ceua urecru .{eo oHe ,[pyre. Kolmo je pacrojane rrnxoaux ueirapa?

(Oarorop: l0l .14. Ikparynajre noBpq[Hy .anjaronarnor npecerfa Koqre, nBrue a.

15. Iloraxlfre aa _ie y cBaKoM npaBuJrHoM Mrrotoyfny, qeHTpanHn yrao

rojN oaroaapa crpaHflq{ cynreMeHTaH jeaHov yryrpalrreM yrny,

16. trI:uepure ruro je norpe6no u n:parynajre. norplur.uuy $wrype AOBCDE(cr. 155). rgie je ABCDE rlpaBwl:rrr. neroyrao, a ra'r*;a O rrerol rleurap.

3 c,gt-- -



17. V xpyxrmly, nonyflpeqHl{I(a .R, ylll{mrrre npaBfiJrlrfi: a) neroyrao,

6) veraopoyrao, 9) mecroyrao - tr y cBaRoM c:Ilnajy crpaur4r ylmc:rxor

qpaBrrrror MHoroyrla tr3pa3xTe noMohy .R.

18. V rpyxnuuy, r:3 cm, yflrunlrre qerBopoJnao ABCD. TaYo M iei

AB crpaswlaynllcauor npaBrurlror rpo]'ma'

BC arpallfina ynucaHor npaBlrnHor qerBopoyrna,

CD qpavw]a ynncaxor TIpaBUII{or mecroyrna.

IIIra npe,qcrama crPaIwa DA|

19. Aro rryr rpyxHuqe' noJrynpelmsra r, It3nocn f ,.r, *ot" t'puu"*ot

ynu€HoM Muoroyrny npunaaa (ao crpanuqa o,4ronapajyha rernaa?

20. V rpyxuway r:2,5 cm, ynEIIIftre AeJftoIIA cfiia AyKa crpaHIEIa wuo'

cn 4cm. (flpexxur re l<pyxxtrqe je anjarosana - cfiMerpaJla .qenror.rga.) I'Irpa-

'ryHajTe noBpmngy ror ,{enro4(a xe uepehN xrmra.

21. na nt oy cm{qHr rpo}'rnoBlr OAB u OCD (cr. 156) dro je OA::7,5cm, AB:l0cm, OB:17,5cm, OD:3cm, DC:4cm, OC:7 im? 3anno2

22. Kpare yrna oA 60" npeceutrTe ,4BeMa npaBaMa raKo Aa HacraHy

.Fa c.nnrtna rpo]'ura oa pa3MepoM cJrurllJocrr'r 3: 1. (Ka(aB he uefyco6alr no-

.roxaj rcuaru re JFe ceqnle?)

23. V rpyxuuau, sireErpoM O n notyrypeunuroM r:2cm, rroByrll{Te rerx-Dy AB, jea*aKy crpa:f.u\tt rpaBntHor rpoyrna yflxcaEor y ry rpyxnrzuy. Kon-

crpyr rnre 3aruM apyry rpyxHlrqy, ca IIeHrpoM y ot (o u or. ua pa:uHu

crpauaMa reruw AB), Taxo .4a rerrlBtr AB 6yae crpauuaa rpilBulrror ruecro-

yrna, ynucaHor y xpyxn[qy c qeurpou Or.

Cn. 156 cr. 157

9l



u. {4r je jeggaRoxlrar lpaBoyrntr rpoyrao ABC, c Yr.roIrr Y I,

Ha raj nawnr uacrao je,,Meceq" tr3Meby

AEa r(pyx6a :ryxa. (c.rr. 157).

Iloraxure ga je norpmma ,,Meceqa.. jegnaxa uorpnnmn rpoyrna.(yflfcrro: noBprnmry ,,Mece[a.3 ao6nhere xaa oa 364pa uoBprrrrrHa rpoyr-

ta rr nonyKpyra oay3Mere.., Cge re troBprrnffe r,Bpa3trTe uovohy rarere c.)' 25. Ilperr,a o3HaKaMa tr tro.qaqnvia Ha cJrmltr 158, r4e je AM:ME t

BD:CD, rrparyuajre o6nr gztype ABCDE.

rntr Tpoyrao A6aJ, c rJpzBvtut yTnol(ld y A,

(scrBprtrHa xpyxEtrqe) 3C, a oro cpeperarere a. Oro I onscax je raa,upar

O xurorery:e .8C oooy*py*rn q, ft.

l'CCrr. 158

26. Axo rpyxra n''K rra cnqn 159 ugnoct 9,42 cm (r.tv3,l4) rornrrje neror nolynpewm?

27. fiujtonatrra upecer jegnor lo" lta je raa4$ar, crpan4e l0 cm. Aroje gyro.rra ror rragpa 8 cm, ro;ruxo n:uoca a6up AyxlrHa cBrr( BeroBHxm'wa?

Cn. 159 Cn. 160

28. OcxoBrqa jegxor rpoyr:la je 12 cm, sncnua 8 cm. I,Ispavyuajre noap-rrlnHy rpoyrna rojr je, nparou trapaneJmoM c ocnoriqoru, a na pacrojarry 5 cm

oa Bpxa, o.4ceqeH oa ,qaTor.



cy rpoyflroBr ABC tt' MNP c;rlvxz (cn. 16l)? Axo yrBp,alrre r(a cy cm {Hf,'Hanr{mnre xoe$nunjenr (pa:rraepy) crltsnocrlr.

31. ,[a .nu cy crrsHr rpoyrtoBlr ABC tt MNP (cl' 16l) aro je:

1A:74", lB:s2", 1N:s2", lP-s4'?

PCa. 161

32. pkMepure uno je norpe6ro u urpa'ryxajre

csnrytt 162,

33. Tpoyrao IBC je jeaxaroxqan, a CD

x lE cy nerore rnclrxe (c.n. 163). Iloxaxnre

.qa cy rpoymoB ABE u BCD cnn'mu.

noBpmr{sy rbxlype Ha

D



34. Ha cruqs 164 npegcrar.iren je naau jegue cale, paleH y pa3Mqpn

I : 200:

a) o,upelure gur.aenelrje ca:re,

6) aa ;rn ce y ygy6r6erbe AB vroxe cMecurr rnarnp, 4yxrure 2,5 m?

(EpojesnHa cnHAI{ o:xararajy ueurrMerpe.)

CrL 164

35, V rpyxuzqy, noryrperrHuKa r, yflvcaH je neroyrao ABCDE, 3a ,vjecTpaHftue r.rMaMo noIaTRe:

.AB je crpaumg yn[caHor rrpaBr{nHor rpoyrna,.BC je crpantltla y[[c:lHor npaBr{mror qerBopoyrna,

CD je srpasnaa ynucaror npaBr{JrHor neroyrlra,

Dt je crparrrqa ynucaHor rrpalunsor recroyrna,

llcur.rrajre troMe ynucaHoM npaDunHoM MHoroyrny

'Jpunaaa

crpasu\a EA.

I --L 7.5 -l



cAAPlr(AJg)royTAo

L floieM l'IoroyrraZ YrnoBtr ri.{Eoro}Trra

.rt +. ra rex6ase,,.......L llpaasax lflroroyraoa XoscrpyRrfsja fpaErinEnx M(oroymoDa

t Grcrprurocr trpaBr4tsax Mrroroyurola

:ha.qr 3a B€x6arie ......,..a- (X-x tcoroyua. Ayxnra o6lrMa

t- IlqDrfira M{oroy0ra' L trqrorya npaBt rgor Mrroroyrta

L.Er 3a Eex6aEe ....,....

KPT/XHUUE. NOBPMI4HA KPYTAL EcrEB€ o rpyx$ IIr-n rpyry ... .. . . . . ,. . . ,

ll@ :a B€x6ar6e . . .. .. . . .

L.frm_ rpyxunreIru3aDex6aEe.........

I|.t* xpyxgor ny{a

h 3 Bex6a6e ......-..f Qm rpyra ......,..........i

h3aEqSaEe..

Lhw xpyrelor uceqxa..3 [.-* xPYAnor trPcrega

3

6

77

9

l2l418

m

24

27

28

31

32

34

35

39

40

4t4t

44

45

46

47

50

50

54

57

58

60

62

63

65

66

67

68

95

AY]I{I{. CJII,IqHOCT MHOTOYTJIOBA

r{tls E:I JeAtIaKe AerIOBe .,,,........

trlryF flpotropqro{anlle ayxlr ..... ..'...

-

.' E6a&€ ..............'...

&lcr c.uEocr rpoyrnoBa .......'

i 4rraee JasehaEux r yvanenls dsrypalGOrOy Ola

EE .{rnl- 3ajeAqt'Ixa Mepa . . .

tF cf,tt&lE l lvlEorolrJloBa

ryrr E p€rE€o rra rrrasy ....o4EIDSE rapaTa

Ely ,{at(tr

3 E6aEeEPEBE

lFroyftoBa

rtr€

r *r .........



V. IIIITAIOPI4HA TBOPT*TI U I}EHA IIPI'MEHA

70

74

76

78

7980

80

EO

8l82

83

83

83

84

85

88

7 razred - obradovic - geometrija - udzbenik

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