6 Razred - Borislav Bajic - Zbirka

6VrPHA 3AAATAKA -*BEY YI AOAATRY HACTABY ' P ~ P E A OCHOBHE Y NIKOAE

36lrph.a ]e ti0lillcflHfl y ~KflfldyCfl 113,WetlOMfl 11 d0~ytLfl.Wliupflmtiuxn o nacmnetioM lrnany u ZpoZpa~y~allleualuu~e 3a octioetiy uKOJly,K 0 j l l Ce fipLl~~tby]e fi0'1eL3 0d UlKOflCKe2002/2003 ?0dutie. 361rpk.a ce cacilioj~lu3 dca dena. Y lrpeo~deny je 36~tprca3abaQ a ~ cia a ule~nna,cr y d p y S o ~deny cy yUyi2cEsn 11 peuletba 3adaCiarcn. 3ndayu cy pncliopebenu Uo i i i e ~ a ~y3ayeawasatbe ocnosnlcw b u b a ~ m u r ~ uUputLr(uUa x K ~ K O6u ce sndoeo/bune unduoudyantie ~ ~ ~ f i n o c yretiurca. iiiu 3a cee aabaiii~ebaiiia cy peue%a, a 3a cnonceltuje u deuta~bua06pa3nome&n. Ha ~pajyceaKe iiie~aiiic~e yenrlne damu cy aaba~fu3a yeex6arcoju ce wo?y rcopucEufiu 34 iiposepy ycooje~ociiiu 3 ~ a h a y%eHUKfl. Y3 c e a ~ yE e ~ iloce6tio y cy damu aadaqu 3a dodniiiny nacmaey K O ~ UceroSy rcopucLiiuEu 3a Uprriipeua&eyretiu~a3a Mame~aiiiur~a i%llKM~refbfl. nprr oda6upn~yu caciiiaeha&y 3adaiiia~a,~opuciiiuocaM ce ceojw u c ~ y c ~ i i ye pedoetioj, o~ dodaiii~ojnaciiiaeu u i i p ~ ~ i i p e ~3aa a ~aEenzaluur~a cZa~~ure%a. H a d u ce da he oea 3 6 ~ r p ~iio~ofiu a y iioeeha%y L ~ K ~ ~ ~ u ~ H o c &YreHUKll Ha YflCOeUMflLl ocCoco6s~aea%y3fl cfl~o~Uifl.nfl~:pfld u yretbe y tiaclTiasu waLTi~aiiiu~e. 3axemflti hy 6uiiiu csuMa otirwtn rcoju fie c e o j u ~cy?e~iii~jflMfl doiipu~eiiiuUo6o/buta+t.y K~sflfl~&eiiifl oee 36up~ey cnedehu wda&UMfl.

H a d a ~ce da he osa 36up~a6aure 6aeheae Maiiimar?iu~onr yr~ltiuciluRCIKULUM,nenullcu u 3atiu~~busujw.

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2 TPOYTAO ....................................................................................

.2.1. Yrnosu Tpoyrna ............................................................................ 2.2. ORHOC cTpannqa u yrnosa Tpoyrna ......................................

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.................................................................. 2.4.3a~awsa ysex6asae ............................................................... 2 3 Kouc~py~qaje yrnosa 2.5. 4

................................. . .

~ O A Y U ~ P H OT~ ~T o ~ ~ I I .................... o B ~

2.6. KOHCTPYKUHJ~ TPOYUK3 ......... 2.7. 3 ~ a r r a j TauKe ~ e Tpoyrna ........................ 2.8. 3 a n a q ~3a yaex6aeaae .................................. 2.9. Tpoyrao (3a~aqli3a AOnaTHy H ~ C T ~................ B~)

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3 CKYII PAWOHAJHMX 6POJEBA ....................... 3.1. P a u l i o ~ ~ a n6pojesu. ~u 6poje~Hanpasa. ynopebasa~en UeqnMannu sanuc paquoeanaor 6poja ...................................:. 3.2. Cabnpa~bey Q ............................................................................... 3.3. O ~ y s n ~ a ynQ e ............................................................................ 3.4. Ca6npane u o u y s a ~ a n ye u e q u ~ a n n osanucy ~ ....................... 3.5. J e a ~ a s n y~ sesu e c a 6 a p a ~ na o ~ y 3 u ~yaQ~ a.................... 3.6. Heje~narnney sesn ca6npa~bau o ~ y 3 u ~ a ynQ a ................. 3.7.3anaq~sa ysexbanane ...................... . . . .............................. 3.8. M H O X ~ yEQ~ ..................................................................... 3.9. R e n e ~ ye Q ....................................................................... 3.10. Jennasu~ey sese ~ ~ o x e nn~ae m e n ya Q ........................ 3.11. H e j e n ~ a s s y~ sesn e MnoxeBa n nenena y Q .................... 3.12. I I p o q e ~ a........................... ~ ... ............................................. 3.13. 3 a u a q ~3a y ~ e w b a s a ~............................................................. e 3.14. C ~ y Q n (3anaqkI 3a UORaTHy H ~ C T ................................ ~B~)

1. CKYn UEJIIIX EPOJEBA 1.1.Ancony~neBpeRHOCTU U CylIpOTaU 6poj 1.3anums eneMenTe cryna: 1) A = { + E N,n2 ; 3) 1x1, -1 ; 4) 39. O g p e ~ xanconyTuy BpenHocT Gpojesa, na uspaqyuaj: 1) I- 231 +I+ 51 + 1- 71; 2) 1-281-1+161+[-41; 3) 1-271.1-991; 40.Onpen~BpeRHOCT H3pa3a:

1) 1328-1-72361:l-181;

4)

1-

/XIO;

78. Epoj -56 ysehaj 3a 36ap 6poje~a-18 a 38. 79. Gpoj 48 yManu 3a pasnary 6pojesa -28 a -16. 8 0 . 3 6 ~6pojesa ~ 26 a -16 yMana sa p a s n ~ i ~TEX y 6pojeea.

88. Onpenn BpenHOCT ~lspasala - 31+ la1- 3 aKo je:

1)

a=-3;

2) la1 = 5 ,

a O; 4) - (- (- a)) = -2 . 89. I U Tje~ seiie a sa KonnKo: 36ap npsax 10 Henapnux qenux HeraTuBHux 6poje~aMnM 3 6 IlpBHX ~ ~10IlapHMX r(enHX HeI'aTHBHKx Gpoje~a? 90. kicna~ajga na je 36ap p a qena 6poja yseK s e h on Ma Kor ca6lipKa.

91. Pewu j e n ~ a ~ nn npoeepu ~e Tamom peureaa:

''

I

1) xt7=91; 2) X-516; 3) 6-x=4; 4) xt19=7; 5) x+(-5)=-6; 6) -4+~=1; 7) X-9=-2; 8) X-(-I)=-2; 9) X-(-4)=5; 10) 6-~=8; 11) -4-x=-4; 12) -12-x=8. 92.3a KOnUKO ~ p e 6 ay~efia~ll-12 na ce no6uje 3 6 8?~ ~ 93. K o j 6poj ~ yMaraeH 3a -6 gaje 6poj -4? 94.3a KOnNKO ~ p e 6 aYMaFbHTa -25 fia 6~ ce n06Ei0 6poj -15? 95. On Kor 6poja ~ p e 6 aOnyseTu 36np 6pojesa -27 n -16 TaKo na go6njena pasnura 6yne jenna~a36npy gaTnx Bpojesa? 96. K o j n ~6 p o j e ~ p e 6 YM~BNTW a pa3nu~y6pojesa -41 u -65 TaKo na ~ 0 6 u j e pasnara ~a 6ynejenHa~a36wpy n a ~ n xBpojesa? 97. 3 a KonnKo ~ p e 6 ayseha~upasnn~y6pojesa -18 u -9 na 6n ce no6110 36np ~ n 6pojeea? x 98. Pewu j e n ~ a ~ yu cKyny ~ e 2: 1) (x+7)+5=1;

4) pasnaqu Gpojesa -6 u -5; 6) pamnqn 6pojesa 0 u-l?

101. Koja op, jea~asmta:-3=4+x, x-(-1)=2, -3+x=-5 MMa RCTO peluene Kao N j e ~ ~ a s u ~-1-(x-I)=-3+2? a: 102. O R ~ ~BPeQ, R HHOCTIIpOMeHJbllBe m, aK0: 1) 21 -((8- rn)+7)+ 2 UMa B P ~ A H O C T-6; 2) 6 - {7-[2 + (5 - x)]-9) UMa BpeRHOCT -1. 3) 3 . n - m = 8 u 11-3-1; 4) 4 . n - m = l 8 a 3 . n - 1 2 = 0 . 103. Pe~unj e u ~ a s u ~ e : 2) 1x1+4=8; 3) 6 - I x / = 3 ; 1) lxl=2; 4) lxl+5=2;

5) Ix-lj=2;

6) Ix+21-1=2;

7) -6+1x-51=-1; 8) Ix-21t15-81=17-8-9+61:1-21. 104. 3 a ~ n c n n ~HeKu o 6poj AKOMY RoaaMo pa3nn~y6pojesa -16 e -20, ~ o 6 u j ace 36up 6poje~a28 n -12. Koja CMO 6poj ~ ~ M U C ~ U ~ H ? 105. Oupeau spenHocT sa x a r o je:

106. Hanuurn CKyn qenex 6pojesa ~ o j cy: n 1)s e h OQ,-3 a M ~ OQ,U4; 2) H H C~~ a n OR n -5a an on 1; 3) ~ e 08h-5 N HXCy ~ e ORh4. 4) HUT M a 0 oQ, -2 Ei HHCy ~ e ORh2. 107.3annmn cse qene 6pojese ~ o j cy a pemene ~ e j e a ~ a s u ~ e : 1) - 6 < x < 1 ; 2) 3 2 x 2 - 2 ; 3) 1x1< 4 108. Oupenu 36np qenux 6pojesa ~ o j cy u pemena ~ e j e ~ ~ a s u ~ e : 1) - 3 < x < 7 ; 2) - 5 < x < O ; 3) - 9 < x < - 1 ; 4) - 2 < x < 4 .

110. Oupenu cKyn qenux 6pojesa x 3a ~ o j saxe-~owna e BpenHOCTu uspasa: 1) (2 + x ) 3 ~ ~seha j on e -2; 2 ) 5 + (x - 4 ) ~ u j ~e a n on a 1; 4 ) ( 8 - ~ ) - 5je Mana 08-1

112. O ~ p e cKyn ~ u pemena ~ e j e ~ ~ a s 8n- nX e> 3 y cKyny: 1) N ; 2) N O ; 3 ) z-; 4) z . 113. Koje qene 6poje~eMoweMo RouaTu 6pojy -2 Ra ce uo6nje 6poj sehu on -l? 114. Koje qene 6pojese ~ p e 6 aO A Y ~ ~ TORH 6poja 10 na ~ 0 6 H j e ~pasa naKa 6yne Mana on -3? 115. On Kor 6poja ~ p e 6 ao ~ y 3 e ~- 1u9 , na go6uje~apasnuea He 6yne seha on -5 ? 116. Koju 6poj ~ p e 6 aUoUaTa pasnnqu 6poje~a-99 u -88 TaKo na no6 u j e ~ u36up 6yne sehu on 36upa 6pojesa - 1 1 n 3 ? 117. on pen^ BpeuHOCT IIpOMeHibHBe y Il3pa3y 7 1 - (-1 1- X )3a KO$ je spenHocT u3pa3a Mana OR 36npa 6pojesa -24 u 17. 118. Epoj n E Z uMa CBO~CTBOga je n + 2 > 12 onaocao n - 2 < 1 2 . OnpeBPenHOCT TOP 6poja.

Oupeu~: I)AUB;

4) B\A;

2) A n B ; 5) A\(BnA);

3) A\B; 6) ( A ~ B ) \ B .

121. AKOje a > 0 M b < 0 I4 a = b = 131, OApeAH BPeUHOCT K3pa3a: 1) a - b ; 2) - a + b ; 3) b-a; 4) a+(a-b). 122.3a K O J BpeUHOCI ~ IlpOMeHJbHBe xje: 17+x 7

1)aspas -seha OR, 1;

x-7 2) uspas M ~ B Uon 1; 4

24-x

4) ---j-je seha on 36apa Gpojesa -10 a (+11)?

123. O~peua6poj ~ o jjea 3a: 2) 5 Manu OR Gpoja 2 . 1 ) 9 sehu on 6poja -5 ; 124. g a ~ je cKyn A={-7,-3,0,+2,s). Han~l~lracKyn B sajja cy eneMeaTx Gpojesu:

125. llopebaj Gpojese -3,+2, 0,41,-37,-20,Sl: 1) OR,~ a j ~ a nno e rnajseher Gpoja; 2) 08 ~ a j ~ e h80 e r~ a j ~ aGpoja. ~ e r 126. llpeuc~asaHa 6poJe~0jnpasa Tame ca KoopuaaaTaMa A(--, , B ( 2 ) , C(-3) a 0(+4J ,a s a m o@eua ~ ~ y x a Ryxa: ~y

1)

;

2) lc~l;

3)

IAD~

;

4)

l~cl.

127. Ynopeua la1 ca Ibl aro je: I)==-(-2)+(-s), b=-3-(-9+4); 2)a=5-(7-12), b = - ( - 4 ) + ( 5 - 6 ) .

128.113pasy~ajspennocr aspasa: 1 ) -6+(8-10); 3 ) -(-43 +52)+(58 -74); 5) -7-[-12+(5-1911; 20

2) -6-7+(-13+21); 4 ) 6-(7-9)-(13-15+6); 6 ) 8+{-9-[-5+(8-12+4)].

132. I I I T ~be BHTN ca p a 3 n c r ~ op~a 6poja, aKo yMaraenuK n o s e h a ~ o 3a 60 a ywamnaq: 2) n o s e h a ~ o3a 40? 1)cManEMo 3a 40; 133. Pema je~nasuney Z: 2) 2-( x+S)=-1; 1) -3+( x-4)=-2; 134. Koju 6poj ~ p e 6 aKoKaTu sGapy 6poje~a-6 cr 12 na 6u ce flo6ana nuxoea pasnu~a? 135. KOJIHKO ~ p e 6 aoRy3e~uo~ s6apa 6poje~a-83 u 28 r(a 6a ce no6ao 6poj KOJH je 3a: 2) 8 sehu on 6poja -3? 1)8 Mana on 6poja 3; 136. O n p e ~ nBpeunocT aspasa ( x l x+ 103 aKo je: 137. Pemn ~ e j e n ~ a s u yn cKyny e 2: 2) - 3 + x c l ; 3)6-XL-1; 1) x + 4 ~ - 2 ; 5) x - ( - 2 ) t - 1 ; 6) - 4 - x < 2 . 4) (x-2)+5 O ; 6) 707.x=O

145. Kor seaKa je npon3Bo~6pojesa a.b.c aKo je: l)a>~,bO;

2)a0;

5) -1268.(-a).(-I).(-25).(+875) * -1268.8.25; 6) I%.(-I).(-252).(-376).(+1) * -178.252.(-376). 147.Hannma Aria 6poja raeo Ra HM je 36ap ~ O ~ ~ T Ia Inpoa3~08 B ~ H , -12. 148. I U Tje~ sehe a 3a ~ o m o : l ) n p o m 3 ~ 06pojesa ~ -9 a 4 anu naxosa pasnura; 2) 36up 6pojesa -8 a -5 una nuxos npou3sofi? 149.Ra nu he n p o a 3 ~ 0 ~ 6 wseha, a M ~ m Uu jeunar ~ y n au ~jeoy BeMy: 1) napan 6poj HeraTueHux ssnanaqa; 2) HenapaH 6poj HerarmBHax a napae 6poj IIO~UTUBHHXYananaqa; 3) jeurur-wneram,~p~~ ~ u ~ u o q a n o 3 m ~ a a jjee~ w a nnym? 150. kIcIIUTaj TasHOCT ~ ~ ~ H ~ K O C T H : 1) -3. 8=-8.3; 2) (+25) .(-4)=(-4).(+25); 3) (-6.8) .(-2)=-6.(8.(-2)); 4) (-24+6) .5=-24.5+6.5; 5) (3842).(-10)=38(-1092,(-10); 6) 6. ( - 5 w .(-5)-3 (-5#6+4-3) . (-5); 7) ( 2 n - 5 ~ + 8 ~(2-5+8) ) = ~ 3a F-4; 8 ) (-5x+ 4 ~ - 3 ~ )-4x, = 38 x = 5 . 151.KopIIc~ehPicsojcrna Maoxena mpasyaaj Ha e a j n a ~ u Hasan: a 2) 4.7. (-25); 3) - 7 . 1 9 ~ 9 . 7 ; 1) -5.7.(-20); 6 ) -6x+3x. 4) 8.(-18)-8.2+8 10; 5) -16.2+4(-2)+10.2; 152. AKOje a . b = - 1 0 , uspary~aj: 2) a ( - 36); 3) -a ( - b4); 1) a (b.5); 5) [-al.8).b 6) -(-a)-b1.(-10); 4) ( - a 6 ) .6; 7) 3ab + (-5ab) ; 8)-2ab-3ab+4ab-ab; 9) 5ab-(-3ab+7ab). 153. AKOje a+b= -4, a3parryaaj: 2) -5a-56; 3) -6a-26+4a; 1) 3a+3b; 4) -8a-13b+5b; 5) 5. (a+b)-3(a+b); 6) -2(a+b)+3(a+b)-6(n+b). 154.Y ~ e c r o* crasu ourosapajyha 3 n a TaKo ~ ua mpt)ene 6yue raseo: 1)a ~ o j eaO, omaje -5b * 0 ' 3 ) a ~ o j ea * b, o n ~ a j e - 3 . ( b - a ) > 0 .

a, b H c ,a 3aTMM M3pa'iyHaj BpeRHOCT ki3pa3a:

1) ( 3 a + 2 6 ) . ( 2 c - a + b ) ; 2) ( - 2 b : c + n ) ( 3 0 - 2 ~ ) . 157. n p a ~ e saKoHa ~ o ~ unc~p~6yqxje Mnoxena y o~nocyHa ca6apan e ynpocTli ~3pa3,a ~ ~ T Mn 3Mp a ~ y ~BpeRHocT aj 3a a = -8, 6 = 4 : 3 ) 6 a + 46 + (- 2a)-26 ; 1) 5 a - 2 a + 7 a ;

2) 9 0 - 9 b ; 4) -5a+26+3~-4b. 158. Y n o p e ~ no ~ l senxqma anconyTHe BpenHocTu ~ s p a s aA, B M C aro je: A=)-7-21.(-3)+5;

B=-(-2).16-91-3.5;

c=-19-181.(-4)+(-7+5-12+6).(-1)-(6-8+4).(-2).

159. Mspasy~aj: 1) -36+6.(-8); 4)-12.5+6.(-5);

2) 48-8.(-2); 3 ) 8.(-5)-(-30); 5) (-56+28). (-9));

6 ) (-72+39).(47-54);

7)-6-2-[-24+(-6+8).3];

8)-21.11-(6.8-1-16+(12-16)5].(-2));

9 ) 48.(-2)-{-16+6.[-24.2-(32-48).(-2))). 160. R ~ T cy H 113pa3~:A = 4 . (2a + 3 ) , B = 3 a . (- 2 + a ) . k l 3 p a ~ y ~BpeRHocT aj uspa3a: 1)A-B 3a a=-2; 2 ) A. B-A 3a a=3; 3 ) ~ - AB. saa=-1; 4 ) ( A - B ). (A+B)3a a=-1.

163. Pa3JlJXKy IIIeCTOCTpyKe BpenHOCTH 6poja -17 21 WTBOpOCTpyKe BpeUaocTu 6poja 25 yMana 3a n p o ~ 3 6pojesa ~ o ~ -25 PI -1. Koja ca 6poj R O ~ A O ? 164. Hajseba HeraTElBHK qeo 6poj yMaEM 3a nPOM3BOU 36apa M pasnaKe 6pojesa -10 11 -2.

1) -5OO:(-2); 2) (+1250):250; 4) -7001:(-1); 5) -1836:(-18)+108; 168. n p 0 ~ e TaYHOCT p~ ~~RH~KOCTB: 1) -18:9=-2; 2) 196:(-14)=14; 3) 1874:(-I)=-1874; 4) O:(-256)=0. 169. n o ~ e n aGpojese: 1) -3216:16; 2) 164:(-4); 4) -5436:18; 5) +4125:(-25); 170.3a ~ o j BpenHocTa e npo~enmasexje:

3) -808:l; 6) O:(-5740).

171. Koje s ~ a Mory ~ e aMaTa 6pojesa a a b aKo je: 1) a : b z O ; 2) a : b < O ? 172. I U Tje~ eehe a sa KonaKo: 1)KonarnaK 6pojesa -18 a -2 ans Kaxosa pasnana; 2) 36ap 6pojesa -36 H 9 ana KonucrHaKTUX 6pojesa; 3) rrpoase0~6poje~a -24 a (-3) m ~ o n u l ~ ? ~ 6 p-144 j e a~ -2 a ?

174. Mspasy~ajspenHocT aspasa:

179. AKOje a:b=-6,a3pasy~aj: 1) ( 2 . a):b; 2) a:(2b); 180. AKOje a+b=-24, onpena:

3) (2a):(2b);

4 ) (-6aJ:(3b).

2) -a:2-b:2; 3) a:(-3)-b:3; 4 ) -a:4+b:(-4). 181. AKOje a-b=-36, onpena: 1) a : 2 - b : 2 ; 2) -a:3+b:(+3); 3) a : 4 + b : ( - 4 ) .

1)a :6+b :6;

182. Pemu jeusareay a nposeps pemene:

1 ) 5.x=-35;

2) - 7 . x = - 4 2 ;

3) x:(-6)=8;

14) ( 2 . x + 1 ) . 3 - 7 = - 1 0 ; 15)-24=-16-(h+6).4; 16) 6 - ( 4 - h ) : 2 = - 2 ; 17)-lo=[-6+(2x-4)]:2-8. 184. Koja 6poj ~ p e 6 IaI O M H O ~ T ca F ~ -7 Ra 6a no6ujeea npoasaon 6x0 84? 185. Koju 6poj ~ p e 6 na o u e n a ~ca ~ l- 12 Ua 614~06vrje~a KonllrHuK 6x0 -2? 186. Ca K O ~ H M6 p o j e ~ p e 6 anonenaTa 6poj -36 na 6a ce ~ 0 6 Konas~ 0 HUK 6poje~a -12 a 3? 187. Kana KOJIUYHllKy 6pojesa -63 U 9 UORaMO lIeTOCrpyKy BpeRHOCT Heno3sa~orBpoja, uo6ahe~opa3na~y6pojesa -1s a -13. Koja je TO Bpoj?

190. O ~ p e n uspenHocT 3a rn y j e n ~ a s ~ a2,n+~=4, u aKo je x peuene j e n a a ~ ~ n2 ~. xe- 4 = -8 . 191. Peuu jenaasaay 2a + 3b - c = -8 aKo je -2a = -6, 46 - 3 = -7. 192. AKOje x = -3 a y = -9 - X , onpena 3a KonaKo je npoe3son X . y seha on: 1) x + y ; 2) x-Y; 3) y : ~ . 194. 36sp T ~ y3amonaa H qena 6poja je -18. Onpena Te 6pojese. 195. Onpella y3amonaa TPH qena HenapHa 6poja a ~ jeonexos 36up -21. 1 9 6 . 3 6 ~n e y~3 a c r o m n a p x ~ e n m 6pojesaje -10.Onpew ~e 6poje~e. 197.A~ojea.b-c=24,0flpenU2 a + b + 3 c a ~ o s a x a . b = d a a.c=12. 1 9 8 . 3 6 ~nna ~ qena 6poja je -18.Onpena Te 6pojese aKo je j e n a ~on &ax nea nyTa seha on npyrora. 199. Kojy BpeRHOCT ~ p e 6 fla a MMa DpOMeHJbMBax, TaKO na A3pa3:

201. Ha 6pojes~ojnpasa pac~ojaneas~etjyTaqaKa A(9) a B(-4) n3HOCM 391nm.K o n a ~ o asaoca p a m o j a ~ ea s ~ e t j TaqaKa y C(-37)a D(-62)?

202. Onpena cse npaponHe 6p0je~e~ o j cy u peureEe ~ e j e g ~ a s n ~ e : 1) 2 . x - 5 < 7 ; 2 ) 11-3.x>2; 3) - 2 . x + 7 2 3 .

204. O ~ p e cKyn ~ a pemena u e j e ~ ~ a ~y ~Zl- a: e 1) -16.xS64; 2) - 9 9 : x > l l ; 205. Penra ~ e j e ~ n a ~ a n e : 1) 2,x - 1 2 ; 4) x:(-2)2; 7 )(x :2)+4>1; 8 ) (-2.x-2):3-16 ;

1 1 ) s - ( 1 2 - 2 ~ ) . 2 2 8 ; 12)(-2+6):4-45-4.

206. Y cKyny qenux Gpojena pemu ~ e j e n ~ a s u ~ e :

1 ) 21x1 < 2 ; 4 ) 2/x1-2-2;

3 ) 2ix1+1 x ? 209. AKOce EsocTpyKa BpeRHocT H ~ K Oqenor ~ Gpoja yseha 3a -4, ~ o 6 n j ace 6poj He seha OK 2. Onpenn cee Tame Gpojese.

210. Ha@ cse qene 6poje~e~ o j au ~ a j ycsojc~nor(a neTocrpylta BpeRnocT THX 6pojesa yMa%eHa 3a: 1)p a s n u ~ yABocTpyltor Tor 6poja u 6poja 12 ~ u j seiia e OR -6. 2) 36up TpoCTpyKor Tor 6poja u 6poja -16 ~ u j Mana e on 8. 211.3a ~ o j BpeRHocTu e npoMeemwBe X, x t z je TasaH u3pa3: 1) -7-(3x-1) - 4 ?

218. Elspasy~aj6 p o j e ~ ~ BpenuocT y nspasa - 3(a + 2b)- c :2 aKo je: 219. Onpenu BpennocT aspasa ( 3 .x - 2y). (X + y ) aKo je: 2 ) ~ = -aI lyl=2 1) 1x1 = 2 a y=-I; 220. Peuu jennawi~e: 1 ) -5 . x=-40; 2) x:(-5)=+4; 3 ) 2r+7=1; 4 ) -2x-I=-7; 5) (-6-2x):4=-3; 6 ) (9-3x).2=-6. 221. Onpena qe'rnpa y 3 a c ~ o n ~uena a 6poja s ~ j jen 36np: 1 ) -26; 2) -6. 222. Onpe~n4 ysacroma qena nenapna 6poja wjn je 36p: 1)-48 ; 2) -8. 223. O~pefln 4 ysacronna qena napna 6poja ~ a j je a 36q: 1) -60 ; 2) -4. 224. P e u a j e g ~ a w i ~ e : 1 ) 2x-7=2+5x; 2) -3x-5=1-5x. 225. O ~ a qm a 33 ronaae, a can 6 rouaea. IIocne KonaKo ronuea he 2)2 ny~a? o ~ a ~q U T Ucrapnjn ofl cnna: 1 ) 4 n y ~ a ; 226. Onpena cKyn penrena nejenuasane: 1) 3.x+155-3; 2) 4x-122-24; 3) (2x-4)-2)-1 ABC aKo je AB = 7cn1,a = 75' n p = 60°, a 3aTHM KoucTpyams cuMeTpane csa T ~ HyHyTpama yrna. LUTa npnMehyjem? Y n i m ~ p p m ~ u y . 4 7 2 . K o ~ c ~ p y ~ lKpywnaqy rra ~ o j nouupyje a cse cTpaeaqe: 1) oru~poyrnor; 2) ~ynoyrnor; 3) npanoymor; 4) j e , q ~ a ~ o c ~ p a n u s ~ o r ; 5) j e ~ ~ a ~ o ~Tpoyrna. pa~or 473. OnIIIIIII H YnHIIIU KPYXHMqY Y ~ ~ ~ H ~ K O C T P ~ H H TPOYI' THOM nY CTpam q e 4cm. I I I T npn~ehyjelu? ~ .

~

475. K ~ H C T ~ ~ jHe InU~ H a ~ o c ~ p a n Tpoyrao u s u ~ aKo je nonynpesum: 2) yntica~or~ p y r a1.5cnz. 1) o n u c a ~ o ~r p y r a3cm; 476. KOHCTPYKT~BHO onpenn TewaruTe: 1) om~poyrnor; 2) ~ynoyrnor; 3) npanoymor; 4) j e u ~ a ~ o c ~ p a ~ c r s u o r ; 5) j e n n a ~ o ~ p a Tpoyrna. ~or 477. K ~ H C T ~ ~ I ~I I~I I~UH ~ K O C T ~ TpOyraO ~ H H Y RyXIIHe ~ H TeXHIIIHe nyX U 3cm.

478. K~HCTPYHIUII j e n ~ a ~ o ~ npasoyrnH pa~o Tpoyrao aKo je nywnHa TewHmHe gywa ~ o j onrosapa a xano~euy3a2,5cnz. 479. K~HCTPYMIIIM npanoyrm Tpoyrao KaTeTe 4cm E TewKmHe ~ y w a ~ o j joj a onronapa 2,6cm.

3) A o ~ a x na n T e r n m a n y x ~ o j onrosapa a x m n o ~ e ~ npasoyrnor y3~ Tpoyrna, nena ra Ha asa j e n ~ a ~ o ~Tpoyrna. pa~a 481. KOHCT~~HIIIII Tpoyrao ABC ~ K Oje ~ABI = Scm, ~ACI = 4 c m , a Tewamua nyw nOByYeHa 113TeMeHa C je 3cm. 482. Texamna AYX,~ o j aonrosapa KpaKy j e n n a ~ o ~ p a ~Tpoyrna, or sUju je 0 6 28cm, ~ nenn ~ 0 6 Tpoyrna ~ ~ Ha gna nena, jenan je 21cri1 a npyrn 19cm. llspasyeaj RywuHe cTpaHxqa Tor Tpoyrna.

485. K o ~ c ~ p y nCBe m seTHpPI 3HasajPIe TaqKe y ~ ~ Q H ~ K O C T ! J ~ H H V H O M Tpoyrny cTpaaxqe 6cm. I U T Moxelu ~ ~~KI~~YIITH? 486. KoHcT~~PIIIIH Tpoyrao ABC aK0 je naT0: c=5cm, BIlCHHa h, = 3cm II

h,=4cm PI hb=4,5cm. 489. O Q ~ ~OR~ PHI MTpOyma, aK0 Cy MePHH 6pojesn IbeI'OBEiX CpenEbHX n a ~ a j 33,cm. a 2cm n 4cm. 490. h e ce ~ a n a sqeHTap ~l onHcaHor Kpyra a opToqenTap KOR npaBoyrnor ~ p o y r n a ?

491. Y KpYXHPIqPI k(0,2cm) nOBy4eHH Cy Ilpe'iHllqli MN I4 PQ,TaKO Aa je W Q . f l o ~ a x nna cy PIM j e n ~ a TeTaee. ~e 1) I M P I = : 2 ) JMQI = I N PI . 492. RaTe Cy nBe KpyXHPIQe k,(O,, 2cm) Ei k , ( 0 2 , 3cm) TaKO Ra k, nk, = {M, N). , Q o ~ a x nna je: 2) D 4 ~ 0_= ,D 0,MO,. I ) LO,MO~= LO,NO, ;

I N Q~

IAM~

He cy TasKe M PI N TaKo Ra je = ~BNI, na cy y THM TasKaMa KoHcTpyPIcaHe HopMane QO npeceKa ca Kpaqma y TasKaMa E a F. f l o ~ a w nna je: 1 ) LAEM = LBFN ; 2) I M E I =INFI; 3) P ECP j e n n a ~ o ~ p Pa ~= ;ME nBC ; 4) 0 FCQ j e , q H a ~ o ~ pQa ~ ,NF n AC.

494. nujaroeane npasoyraouaKa ce ceey, AC n BD = (01. Kpos TaqKy o noBysena je npasa p , Taeo ua p n BC = { M )a p n AD = { N } . n o ~ a w uUa je: 2) D CMO z D AON ; 1) P BMO z D DNO ; 4 ) LDNO = LBMO. 3 ) LCMO = L A N 0 ; 495. Hag CTpaHIlqaMa AB PI BC KBaupaTa ABCD KOHCTPYHCaHM CY j e n ~ a e o c ~ p a ~ rpoyrnosll ur~u ABM a BCN. ~ O K ~ XuaHje: 2) D ADN j e g n a ~ o e p a ~ . 1) r> CDM j e n u a ~ o ~ p a ~ ; 496. Hag c ~ p a u a q a ~KsaupaTa a ABCD, KoncTpyacanu cy jeunaeoCTpaHHYHH TpOyI'JIOBll ABM, BCN, CDP PI ADQ. ~ O K B X I IJJajet 1)D MNQ j e ~ n a e o ~ p npasoymu; a~o 2) r> MQP J ~ R H ~ K O K ~llpaBOymH. ~ K O 497. , Q ~je T j e ~ u a ~ o ~ npasoyrna p a ~ o Tpoyrao ABC (LC = 90').

498. n a je~ npaBoyrnn Tpoyrao ca O I L I T ~ M MyrnoM on 30" a 36up xuno~eny3eM pah he ~ a ~je e18~17~. ~ eOgpegu gyxuny: 1) Tewuntne uyxu ~ o j ourosapa a xuno~e~y3u; 2) nonynpewma onncane ~ p y x a u q eTpoyrna. 499. n a ~ jea TemIlma ~ y - CC,=7,8cm. x Oupenu pamojarae TexuntTa T on: 1) ~ e ~ c ;e ~ a 2) TeMeHa C,.

2) BC

= jcm,

LABC = 60"

LACB = 45'

, a 3aTHM OnHILIPi My KpyI'.

S O l . , ~ ~ c ~ p y unpaBoyrna urs Tpoyrao u onpegu o p ~ o q e n ~ aKo a p cy: .~,

502. K O H C T ~ ~ M j e IuLuIaM ~ o ~ p a eDaABCU K O H C T ~ ~ K T M B Houpeuu O TewuluTe Tor Tpoyrna aKo je UaTo: 1) ocHosuUa 6cm u yrao ua ~ o52'30'; j 2) KpaK 5cm M yrao npu spxy 67'30'.

1)cy nyxaHe c p e n m x n a a ~ j Tpoyrna a 4.8cn; 3cm n 2,6cm; 2) je pyxaaa o 6 a ~ aTpoyrna saje cy cTpaanue cpenne nuaaje Tor Tpoyrna 7 . 2 ~ 1 ~ ~ .

505. Mspa~~yaaj yrnose Tpoyrna aKo je j e p a ~yrao 50' a apyra nsa ce pa3na~yjy3a 20'. 506. M 3 p a ~ y ~yrnose aj Tpoyrna KOA Kora Baxn pa je 507. O p p e p ~yrnose Tpoyrna aKo je

a

+ p = 104' a

508. M s p a ~ y ~ yayTpamne aj yrnose Tpoyrna

L

I

5

4

a =- p =-y

,D+ y = 130'.

KOA

Kora je

2

a=-p H

7

509. O ~ Tyrao I ~ on 54%o~enaHa ~ p j ne n a a ~ apena, K O H C T ~ ~ K T I I B H O . 510. Y Tpoyrny ABC, a = 75" M3pasy~ajocrane yayTpamKe yrnose aKo nonynpasa A, n BC = {D}, penu Tpoyrao ABC Ha nsa j e n ~ a ~ o ~ Tpoyrna. pa~a 511. AKO je y TpOyrny ABC, a -)' = 3.,L?, OHW B W H J ~ A H ~ K O C T 512. n o ~ a x ana je Tpoyrao npasoyrnu aKo je 36ap nsa cnomamm yrna 270'. 513. B o ~ a x ana je Tpoyrao npaaoyrnr? aKo cuMeTpane Usa yeyTpam n a yrna rpane yrao op 135'. 514. ~ O K ~ XnaHcnMeTpana Ma Kor yeyTpamner yrna ca HacnpaMHoM cTpaHaqoM o6pasyje nsa yrna s ~ j jea p a s n a ~ aj e a n a ~ apasnnqr? ppyra gsa yrna.

51s. Y T~OY~JIYABC, cuMeTpana yrna BAC, AM(ME BC) ca BC rpanM 2 yrna. AKOje LAMC = SO', aa$a p a s n a ~ yyrnosa y H p . 516. , Q o ~ a x ana yrao ~ o j nrpane cnMeTpane yrna p a nonosuau yrna a.

y,

j e n a a ~je

517. Cnomanma yrao npa spxy j e n ~ a ~ o ~ p Tpoyrna a ~ o r asaoca

3 2

-

KeMy cycenHoM ynyTpamBeM yrny. H3pariyaa~ayrao ~ s ~ e b y cHMeTpane H s n c m e ~ o j nonase e n3 TeMeHa O C H O B H U ~ . 518. kispasyaaj yrnose j e ~ ~ a ~ o ~ Tpoyrna p a ~ o rPI ynopeux KpaK ca OCHOB~~OM aKo HopMane nosysene a3 TeMeHa ocHoeuqa Ha KpaKe o6pasyjy yrao on 50". 519. Hspasyaai yrnose ABC H ynopenn c ~ p a a ~ aKo q e je y = 75', a caMeTpana cTpamqe AC ceqe cTpaHnqy A B y TaYKn D TaKo na je LACD = 2 . LBCD. 520. M3paclyaaj YHyTpaIIIlae yrnoBe je~HaKoKpaK0rTpoyma, aK0 CHMeTpana cnomarulaer yrna Ha O C H ~ B M ~ Mn caMeTpana cnomaruKer yrna npu spxy rpaue yrao on 70". 521. n a je~ j e n ~ a ~ o Tpoyrao ~ p a ~ K~ O Kora ~ je yrao npn spxy, nsa nyTa Mann on yrna Ha O C H O B M ~ U . C n ~ e ~ p a nyrna a A Ha O C H O B H ~ Hcecre KpaK BC y TarrKH D. n o ~ a wna ~ je l AD=AB=DC. 522. f l a je ~ j e n ~ a ~ o ~ Tpoyrao p a ~ ~ lABC K O Kora ~ je KpaK nsa nyTa RYXFi Ojl OCHOBHf(C. Kp03 CpenHHy KpaKa AC, TaYKy M IlOByqH napanenay npasy p ca AB a p n BC = { N ) ~ O K B X H ua je BM cnMeTpana LAW. 523. f l a je ~ j e n ~ a ~ o ~ Tpoyrao p a ~ a ABCa npasap ~ o j jea HopManaa Ha O C H O B H ~ YAB a cege KpaK AC y TawH D, a npoAyweTaK KpaKa BC y TaYKli E. f l o ~ a x ~a a j Pe CDE ~ ~ ~ H ~ K O K P ~ K . 524. n a je~ D ABC. kis TeMena B KoacTpyacaaa je HopMana Ha cmeTpany yrna c ~ o j cTpaHaqy a AC cecre y TasKa D.n o ~ a x na a je D BCD J ~ ~ H ~ K O K P ~ K . 525. n a je~ Tynoyrnu Tpoyrao ABC ca LC =50°. C a ~ e ~ p a yeyTpane rulaer H CnonaruBer yrna y TeMeay c y npeceKy ca AB onpebyjy ~~,I(H~KoK Tpoyrao ~ ~ K H CDE. O ~ p e ~ l z Tyrnose ll D ABC.

526. X ~ n o ~ e n y sAB a npasoyrnor

T p o y r n a ABc

nponyxeHa je npeKo

TeMeHa A 3a nYWMHY AC, a IIpeKO TeMeHa B 3a UyWHHy BC H TaKO no6~jeaeTacIKe M H N. M s p a s y ~ a Mepy j yrna MCN.

527. A

~ jeT T y n o y r n H

ABC (LACB = 1440). C ~ ~ e ~ p cTpaHxqe ana AC CeYe CTPaHHqY AB Y TaYKEl D TaKO Ki je LACD 8 n y T a M a b H 08 r>

LBCD. Hspasyaaj YrXOBe D ABC. 528. Kpos npeceK S cHMeTpana yrnona a H y D ABC, nonyseHa je npasa p, napanenHa cTpaHnqH AC, K O J ~npeceqa npyre use c ~ p a ~ ~ qpe n: AB = { D } u p n BC = {E}. f l o ~ a x ~ua je

I A DI + I c E I

=

I D S I +I s E I .

529. A O K ~ XHaHy J ~ ~ H ~ K O K ~T p~ OKy rOn yMYHjH je YraO H a OCHOBllqH 7 9 , nyxaHa K p a K a je 2 n y T a ~ e h a 08 UyxaHe nHcme noByreHe 113 TeMeHa OCHOBHqe H a K p a K . D t BC TaKo na cTpaH H ~ Ynena H a nyxane on 8cn1 H 4on. H 3 T a s K e D nosyreHe cy HOpMaJIe H a &lyre R B e CTpaHHQe. H3pasyHaj yRaJbeH0CT Ta'iKe A O R non~oxja T U X HopMana.

530. f l a ~je j e n ~ a ~ o c ~ p aD ~ABC ~ r Ha Taws ~

531. f l a ~je

TpOyraO ABC

A D 1 AC PI

I A D I=I

AC

H

y

TasKH A KOHCTPYHCaHe CY HOpMaJIe

1 , OHHOCHO A E l A B H I AEI =IABI.

AoK~WH

naje IBDI=IcEI. 532. @T ]e KBanPaT ABCD U Ha8 CTpaHHqaMa A B El A D KOHCTPYUCaHH cy nouynapHn KsanpaTa s ~ j ce e ~ n j a r o ~ a nceKy e y TasKaMa O, H 0,. ~ ~ O K JJa ~ X~eHTpOyraO CO,O, ~ ~ R H ~ K O K ~ ~ K . 533. A ~ jeT npaBOyraOHHKABCD & Han I CTPaHMqOM AB je KOHCTPYMCaH j e n ~ a ~ o - c ~ p a n D~ ABM s a ~ y cnofiam~oj O ~ ~ ~ C Ta M Han , Y Y H Y T ~ ~ U I0 B 6 nOa Jc ~ ~~ 1~ ~ H ~ K O C T ~ CTPaHUllOM BC T p o y r a o BCN. A o K a w u na je L B A C = I B M N . 534.

A~TH Cy cTpaHe nMmTe WH.

~ ~ ~ ( H ~ K O C T ~ ? ~ HTPOYrJIOBM IIYHH ABC H BDE

~ H

C a MCTe

npase TaKo na AB H BD E p , a IAB~ = 2 . J B D ~AKO . je M c p e c T p a m q e AB, oHRa je D CEM j e n ~ a ~ o c ~ p a A~o~~ sa a- ~ .

535. A a ~ jea n p a s a p a A e p, B e. p , a ca a c T e cTpaHe npase p. O ~ p e n ~ H a npasoj p T a r K y C T a K o Ha je AC+CB ~ a j ~ a % ~ oer y i i e .

536. AKOje ADcaMeTpana LBACTpoyrna ABCnpn seMy je D t BC . oHna saxa: na je I A B( > a IAC~ > I c D~. f l o ~ a x a . 537. f l o K a 3 a ~na~CBaKM H ~ ~ ~ ~ H ~ K O C TTpOyraO ~ ~ B HHMa: ~ ~ H H 2) 6 a p j e n a ~ yrao sehu on 60". 1) 6 a p j e . p ~yrao Masu og 60'; 538. f l a je~ OIUTap Lxoy PI y y ~ ~ ~ p a l u o6nac~x &oj TacIKaA. O ~ ~ ~ U H T M Ta'lKe B H C Ha KpaqHMa yrna TaKO @i TpOyraO 6 y ~ eMUHHManHor 0 6 a ~ a . 539. E ~jeTTpoyrao ABC Fi TaYKa M y y~y~paUIEb0j o 6 n a c TaKO ~ ~ na je AM=AB. ROKBXIIna je AC > AB. 540. Y Kpyry k(0,r)gaTe cy use jenna~eTeTase = npase AB a CD ceKy ce y TasKa M. a o ~ a x na u je npasa MO caMeTpana yrna AMC.

~BDI

JABI IcD~.

542. Y KsanpaTy ABCD noBysene cy gse HopManHe nyxa ( ~ o j eHHCY napanenHe ca cTpaHxqaMa ~ s a ~ p a ~sa~) j, a~pajesunpunagajy HacnpaMnaM mpaHnqaMa KsanpaTa. f l o ~ a x jae ~ a a ~ THX o c uyxa. ~ 543. ,@T je j e ~ a a ~ o ~ Tynoyrna p a ~ o Tpoyrao ca TynnM yrnoM y TeMeny A. B a c n ~ enoByseHe x3 npyra nsa TeMena o6pa3yjy yrao on 50" H~spasy~aj yrnose Tor Tpoyrna. 544. H3pasyaaj LACB,P ABC aKo je I A BJ = TasKa H je opToqeHTap b ABC. 545. f l o ~ a x ga ~ l je n y x n ~ acxMeTpane yrna Ha ocaosn~~a jenaanoKpaKOr TpOyrJIa nBa nyTa seha On BElCHHe IIoByseHe El3 TeMeHa npH

IcHI,

L

~pxy,aKo cno&amIsuyrao npa npxy ns~oca; yayTpamner. 3

546. Hspasy~ajyrnose b A B C ~ K Oje BD BMcaHa, BM cnMeTpana yrna a AYX, MN je BHCMHa b BMC, LMBD = 30' l3 LBMN = 40'. 547.Ogpena yrnose Tpoyrna aKo TexamHa n y x a sncnna nosy1eHe a3 TeMeHa C nene yrao Ha 3 j e n ~ a nena. ~a

549. Mspatry~ajyrnose npaBoyrnor Tpoyrna aKo x a n o ~ e ~ y 3 aBUCK~a Ha UenH XHnOTeHy3y y OjJHOCy 311. 550. Hspasy~aj omTpe yrnoBe npaBoyrnor Tpoyrna aKo BHcHHa K T C ) K L I I U Hn~ y x nosyleHe us TeMeHa npaBor yrra rpane yrao on 9'9' . 551. &IT je IIpaBOyrJIM b ABC, TaYKe M U N C y CQeflHHeKaTeTa, a D je ~ O ~ H O X JXHnOTeHy3UHe K ~ ~ BUCUHe. Kojoj BpCTU T p O y m O B a npnnana Tpoyrao MND? 552. klspasy~ajyrnose j e n n a ~ o ~ p a ~Tpoyrna or aKo BnccrHa noByueHa H a K p a K A BUCUHa IlOByYeHa H a OCHOBUL(y CeKy Ce y Ta'iKM H T a K O naje 553. Y ItpaBOyrJIOM TpOyrJIy ABC K O H C T p y U C a H a je HOpMana CD H a XUnOTeHy3y AB. T a s ~ aE je CpeRMHa Te HOpMane, a TaYKa F J C cpenUmTe ~ ~ ' K HQA D O . KRa npana ~ BE je HopMarrHa H a npaBy CF. 554. kispasy~ajyeyTparune yrnose Tpoyrna K O ~Kora cy qeHTplr onacaHor A ynncanor Kpyra cnMeTpHwH y O ~ H O C Y ~a jen~y CTpaHllqy TpOylTa. 555. Q o ~ a w ana je Tpoyrao npaBoyrnB aKo je jenna TewHmna nyw jenaaea nonoBaaa onro~apajyhecTpanKqe. 556. AKOy D ABC je a J = go0, Tana cy ouceYqE cUMeTpana yHyTpaurner u cnomamner yrna KOA TeMeHa C , 80 npeceKa ca npaBoM A B M ~ $ Y C O ~ H jennaKU. O flo~awn. 557. f l a ~je Tpoyrao ABC ( A B 2 A C ) . H e ~ aje D n o n ~ o x j eBncnHe ( ~ o p ~ a n113e )TeMeHa B H a cmeTpany yrna a , a E cpenaruTe CTpaHUqe BC. k I 3 p a 3 l l ~ UDE npeK0 CTpaHHqa b K c . 558. T e w ~ l m nyw ~ a Tpoyrna Mass je on n0ny36~pacTpaHnqa ~ o j noe na3e A3 ACTOr TeMeHa TPOYrna K a O B TeXMmHa nYW ~ ~ O K ~ X B 559. AKOcy t o , th u t, UyxnHe TexHmHxx ~ y x nnpaBoyrnor Tpoyrna ABC(LC = 90'1, A O K ~ ~ na ~ TjeA t, + t , > 3 . t , .

~cHI=IABI.

-

560. f l o ~ a s a ~na u npaBap ~ o j acanpwu cpennme M cTpaHuqe AC n ~ o j jea napanenHa cTpaHtiqa AB ~poyrnaABC nonosn cTpaHuqy BCTOr TpOyrjra. 561. Q o ~ a s na a ~je~cpenna n a ~ ~Tpoyrna ja napanenHa ~pehojmpaH M ~ Mn ua je neHa u p m a p a nyTa MaKa on nyxuHe Te mpaHnqe.

.

562. f l o ~ ana~ je~ TewumHa l nyx Tpoyrna Mass on n o n y o 6 a ~ a~ p o yrna. 563. AKO je TasKa D y YHYTPaIIIILOCTU DABC OH@ BaXB: a+b+c < DA + DB + DC. flo~axkl. 2

--

564. n o ~ a x na u je qeaTap ynucaHor Kpyra Tpoyrna ~ a j 6 n ~ TehfeHy xa ~ a j s e h eyrna. r 565. Ha8 CTpaHEIqaMa AB H BC TpOyrna ABC KOHCTpyHCaHll Cy KBaRpaTll ABDE a BCFG. H o ~ a x ana je DG = 2 . BH (BH je TeXHlIIHa nyw 4 ABC). 566. n a je~ D ABC,saH Tpoyrna cy Han cTpaHauaMa AB a AC KOHCTPYBCaHH KBaRpaTll ABDE ll ACMN. ~ O K B X H je CE = / BN 1 II CE IBN.

I I

1

567. J e ~ a nonmap yrao npaBoyrnor Tpoyrna je _npyror oruTpor L

yrna. T a s ~ M a je cpenHHa x ~ n o ~ e ~ ya3Ne ,je Ha CTpaHMqM AC TaKo na je MNI A B . ~ O K na ~ Bje A C =~3 . 1 ~ ~ 1 .

IAEI IEBI

568. Aam cy rrpmoyraom ABCD B KsqpaT DECF ( E E AB, = ). n o ~ a x na n je D ABFjenBa~oKpaK. 569. AKOynacaHa KpywHuua b ABC ~onapyjeKerose cTpaHHqe AB a

570. KoHcTpyama b ABC o 6 ~ 8cm ~ aH yrnosa ,B = 75', y = 45' 571. KoHcrpquu~TpoyraoABC a ~ jeoc=6cn1,BllcBHe h,,=cm B h,=3,5c1n. 572. KoHcT~~RIIIB D ABC aK0 Cy AYXEZHC CpCuEIIX J ~ M H Tpoyrna M ~ ~ Zcm, 2,5cm a 3cnz. 573. K O H C T ~ ~ HbUABC, I H 061iMa locm, yrJIa a = 60' ll BllCHHe h,=2c1n. 574. KOHCT~~BIIIB Tpoyrao aKo je c=6cm, BllCMHa h,,=5cm H TexuurHa nyx t,,=4,5cnt. 575. K O H C T ~ ~ HTpoyrao I I I B aKo je c=5c1n B TewHmHe ~ y w tD=4,5cm a a t,l=4cm.

576. Ko~crpyacamTpoyrao aKo cy RymHe TtXCiIIIHKX nymf 3cn1,4cn1,5cn1. 577. K O H C T ~ ~ MDCABC ~ T BaK0 Cy AaTa TeMeHa A, B B OpTOqeHTap H.

579. K o ~ q y m D ABC a ~ o j e c = 5cm, b = 4cm HTemmHaw to = 3cm. 580. KoHcT~~MIIIPI ~ ~ ) J H ~ K o c T ~ ~ H PTpOyraO I ~ H ~ ! aK0 je 3 6 CTpaHIlue ~ ~ PI BHCHHC 7Cm.

581. KOHCT~~HLUK ~ ~ A H ~ K OD KABC ~ ~aK0 K je 3 6 UyXHHa ~ ~ OCHOBHue a KpaKa 8cm, a OAHOC TPIXnyxma je 1:2. 582. K O H C T P ~~M~ ~~ HH ~ K o KTpOyraO ~ ~ K P 36apa I OCHOBli~eA KpaKa 7orr a yrna Ha ocHosaqa 75'. 583. K o ~ c ~ p y a u rj ne n ~ a ~ o ~ Tpoyrao p a ~ ~ l ocaosaqe Gem PI BucnHe ~ o j onrosapa a KpaKy h,8=3,5cm. 584. Koac~pyamnjeflHaK0KpaKH TpOyraO aK0 je BHCHHa IIOByxIeHaHa KpaK k,,=3cm a yrao npa ~ p x y30". 585. KoHc~pyHura~ ~ ) J H ~ K o K TpOyraO ~ ~ K E ~ aK0 Cy RyXKHe IberOBMx HaCMHa h, = 5cm Pi hb = 6cm. 586. Konc~pynmanpasoyrnH TpOyraOABC, aKo je:

587. K o ~ c ~ p y ~npaBoyrnH run Tpoyrao a ~ jeo no3aa~o: 1)nonynperIaae onacaHor ~ p y r 2,5nn a a m n o ~ e ~ y s amcma aa 2cm; 588. KOHCT~~HLUH npaBoyrna Tpoyrao KaTeTe b=3cnl a yrna a 3 ~ e ) y x a n o ~ e ~ y 3BHcmie a ~ e H cuMeTpane npaBor yrna 15'. 589. K o ~ c ~ p y a m npasoyrna s Tpoyrao aKo je: 1) 36ap KaTeTa 7cn1H xlmOTeHy3a 5cm; 2) p a s n ~ KaTeTa ~a lcm a x a n o ~ e ~ y 5cm; 3a 3) nonynpesHHK onacanor Kpyra 3cm a ynacanor Kpyra lcm. 590. , Q ~ je T yrao L x@, ~ a j jea spx san nanapa, a TasKa A y yeyTpaurnoj o6nac~ayrna. KOHCT~~PIIIIH n p a ~ yAO. 591. KOHCTPYMIIIM cmeTpany yrnaxoy, KoMe je spx HenocTHwaH.

3. CKYn PAUHOHAJIHHX GPOJEBA 3.1. Paqno~~annn Gpojeen, 6pojensa npana, ynopebunaee s ReqnMansn 3annc p a q n o n a m 6poja

595. Hanamn Gpojese -2, -3, -6 Ha HeKOnHKO HalIHHa y BHny HeraTllB-

Hor p a s n o ~ ~ a . 596. Hs csyna

(

2

A = - 6,4-,-23

1

2

5 1 ,--,0,-,3 ; 6 4

598. Koje cy on cnenehnx ~ ~ J Z H ~ K O CTasHe: TM

1) Z n N = 8 ; 2) 599. Onpena eneMeHTe csyna:

QUQ-=Q;

a3aTHMOnpeRH P u R ; P n R H P \ R . 600. 3aoKpy~HIlCTHHIlTe H3Ka3e: 1) Z - u Z c Q ; 2) Z ' u Q - = Q ; 4) N c Q ;

5) Z c Q ;

113~soj noncsyn:

602. Ha 6poje~Hojnpasu RaTa je

k!

Taws A 3-

. O ~ p e KOOppLHaTY ~n Ta-

n

604. Hanlrmrf cKyn paqnouanullx 6 p o j e ~ ao 6 n u ~ a-a ~ je:o 2.n+l

606. nope$aj no s e n u r m n op, u a j ~ a n e rp,o uajseher 6pojese: 3 1 5- 1- 7- 5 4 ' 8 ' 12' 2 4 ' 6 '

2 1 3 614. Koja p w o n a n m 6pojesa majy anconwe Bpemom:-, 1- ; 2 - ? 3

615. n o n y ~ ~a6eny: a

616. klspa~rynajspeuHocT nspasa:

2

4

9)

1x1-lyl+2.(-x),

3a x=-2,y

1

=-.

2

617. K o j u 08 6pojesa U M a Many anC0JIyTHy BpeUHOCT:

632. AKOje a = - I - ,1 b = 2-,1 2

3

1 ) a+b=b+a; 3) (a+b)+c=a+(b+c);

633. I I o n y ~~~al 6 n ~ q y :

3 4

c =-

nOKaXH Ra s a x e j e u ~ a ~ o c ~ n : 2) b+c=c+b;

4) (a+c)+b=a+(b+c).

+a;

1

6)

5) (-a)+3-+(-b); 2

643. kispasyaaj BpenHocT uspasa:

3) [-7++d]+k;+(-3)];

4) [-5:+4,2)+[-4,4+2;];

5) -64+[-5+k+[-1;])]; 3

6) [-9+(5+4f)ji[8+(-7f

2 1

1

2

4

10) la/+(-b+(-a))+\-bl, sa a=-2-, b = 3 - .

-311;

a+b a-b

( A :I[: 6)

3) 3--4-

-

--+I-

;

4) [ - 2 , 4 + 3 i ) + G i - t ) .

657. K a ~ tie o ce npoMeHHTu 3 6 q nsa 6poja a ~ jegau o ca6upaa yaetia~osa

659. Onpe~apa3~1a~y aspasa a - b + c a a - b - c aro je:

660. Cabepa UeqaManHe Gpojese: 1) 3+2,708; 2) 4,9+5,368; 3) 0,99+1,909+0,9; 4) -6,8+(-3,956); 3 -4,75+(-2,368)+(-1,909);6) -6,9+4,75; 7) 7,2+(-0,976); 8) -0,756+(-1,244)+4,3+(-3,857);

661. O ~ y s UequManHe ~a Gpojese: 1)2,75-1,96; 2) 0,978-0,75; 3) 3-1,99; 4) 3-4,99; 5) -4-5,76;

4) B-(c-A); 5 ) c+(A-B); 6) - ( A - B ) + c . 665. HCIIHT~J HCTHHETOCT ~ ~ ~ H ~ K O a+(b+c)=(a+b)+c CTEI 38: 1)a=-2, b=0,75,

1 c = 1- ;

4

1

2)a=-1,5, b = 2 - , 4

c=-0,s.

666. AKOje a+b=-3,25,~ l s p a q y ~spe,qHocT aj aspasa: 1) (0,25+b)+a;

667. AKOje

a

1b

2-

I

2) ( a - l + ) + b ;

b = -2,5 oupenu BpenHocT uspasa:

o+b

I a-b I -a+b /

3) b - ( l f - a ] .

671. Epoj -3 yMa%n 3a p a v n a ~ y6poja

5 a s6npa Bpojeea - 2 5 a -1,G. 8

- 4-

~ p e h n3a 3,625 MaeK OR npeor, a npyra je

I

- 4-.

8

673. t i s p a q y ~ aBpenHOCT j a3pa3a: 3 8

1) a - [ b ( a + b ) ] , 3a a=-5,5a b=-2-;

3

2) (b-a)-(a-b)+[a-(a-b)],

3a a=-3,25n b=-1-.

4

674. K a ~ fie o ce npoMeHaTu 36np nea 6poja aKo j e n a ~c a 6 n p a ~yseha1

3a 3 - , a npyrM CMaEbHMO 38 4,625? 4 675. K a ~ fie o ce npoMeHaTa p a s n x ~ ansa Gpoja, aKo ce: MO

1

1

1)YMaIbeHEKY AORa -3- + (- 1,25) a YMaIbaOqy- 4- ; 4 2

AKO x+5=1

nposepa

oaga je x

I

-4

I

-4+5=1

4)

6)

5) 1x1= -0,5 ;

1x1= 3,2 ;

- I X I =1 ~ ~ . 1

678. n p o ~ e pna ~ inu je 6poj -3 pemene jennasx~e(x - 2,25) + I- = 4 . 4

679. Perur?jeg~aranea nposepr? peuena:

1) x + 3 - 1= I - ; 1 4

2)

2

3

x

1

6)

5) (x-0,75)--=-I-. 4 2' 681. Pema jenfiasuae:

4)

1x + 031 = 1,2 ;

+ 2,5 = -0,7

5)

;

3) -x

1 (6,375-x)-I-=--. 4

16,375-xl=l-'

3

8'

+ l,2 = 0,s ;

3

8

6) /x - 04= -1,4 ;

683. Pema j e n ~ a s n ~ e : 1)-3,8+(4,5-y)+2,25-1,6=-8;

684. Kopac~ehace j e u n a s n ~ a ~pemn a , c n e ~ e h es a n a ~ ~ e : 1)K o j e ~6pojy ~ p e 6 auouaTn 6poj 3,25 na 6u ce no6no -7? 2) Kojn 6poj ~ p e 6 ajqojJaTn6pojy 0,6 ua 611 ce ~ 0 6 6poj ~ 0 -1,2? 3) On Kor 6poja ~ p e 6 oj~ysem a 6poj 252 na 6u ce ~ 0 6 6poj ~ 0-25,2? 4) Kojn 6poj ~ p e 6 ayMaranTn 3a -2,75 fla 6a ce ~ 0 6 6poj ~ 0 -1?

685. Koju 6poj ~ p e 6 ac a 6 p a ~ uca 0,4 Ra 611 ce no6ao 6poj peqnnposan 6pojy 0,4?

690. Kana on pasnnee 6pojesa -3- 1 n 8

1 4

- 1-

3 5

o n y s ~ e m36ap - 6- a

no6ajam HeKa 6poj yMaraeH 3a -0.75. Onpena ~ a6poj. j

4-

1

2

695. Ha 6pojeeuoj npaBe ogpeua: 1 1 n e qenex ~ 6pojeaa ~ o j cy u sehn OR -2; 2 ) YeTupe paquouanua 6poja ~ o j cy a Mana ou

3) T ~ I paquoeanaa I 6poja seha OR

1 2

-;

1 2

696. n p e g c ~ a s uHa ~ ~ o J ~ B npasu H o J peurelae cne~ehax aeje~nasuna: 3

1) x > 2 -1 , .

2) x s - - .

2

4'

3) 1x1 < 2 ; I

6) / x J > l z . 5) 1x1 < 4 ; 697. Ha 6 p o j e ~ ~npaaoj oj npuKaxH cKyn peme&a nejeuua~uae: 1 1 2) - 5 -1< x 5 - 1 - ; I 1) - 2 - < x < 3 - ;

4) 1x1 2 3 ;

2

3)

1 /xl41-; 2

4

2

3

1

4) 1x1 > 2-.

4

698.3anuma pa pawonanHa 6poja ~ o j sanosomasajy a ~ejenna~oc~: 1) o < x < 1 ;

3) - 1 < x < 1 ;

2) - 1 < x s o ; 1 5) - - 1 x < 0 ; 2

4) - 2 < x < - l ;

1

6) 1-; -2-.

11) lxl+0,25 $ 2 - ;

1 4

700. Koje OJJ naTnx n e j e n ~ a s a ~u a~ a j y~ C T HcKyn pemeffia:

1) x + 0 , 7 5 t l ;

2) x - 1 < -0,75 ;

3) -x+1x:0,5; 2

2

I 2

3) - x + 2 < - 4 ;

1 2

4) 2-0,Sx>1-.

816. Ha Bpojeanoj npasa npaeaxu cKyn pemeaa n e j e ~ ~ a s n n e : 1

1)-2- a > c ; 2) c > b > a. 409.1) b < a ; 2) b > a. 410.1) c > a > b ; 2) a > c > b. 411. 1) n0ByIp.i npany p ~ p o BD 3 H H e K a je p nAC = {MI* LADM > LABD n LCDM >LCBD K a o cnomauma yrnom T p o y r n a ABD H BCD a ADC > LABC. 2) C J I ~ H On p n o M A o K a s y . 412.1) 24cm; 2) 12cm. 413.1) 90' + 45' ;2) 60'. 5 + 15' 3) 75': 2 ;4 ) (3.600+19):2.414. IcC( = I A ;~ 415. K O H C T ~ Y OCHOCBMCTPHTIHX K~E~~~ T p O y M O B a , D AIBIClZ D ABC 416. K O H C T P Y K ~ C IOCa ~ ~ ~ K M e l p k I j e ; 417. KOPIlCTIl CTaB CYC; 418. KOPElCTIl CYC; 419. KOpIlCTIl CTaB CYC; 420. CTaB C Y C ; 421. C Y C ; 422. CYC; 423. Y C Y ; 424. HCKOpIICTIl CBe CTaBOBe IIOjJYKiPHOCTII; 425. CBe CTaBOBe; 426. CYC; 427. KOPIICTEI CTaB CCC; 428.1,2,3; 429. 1,2,3; 430. C Y C , 431. C C Y , 432. C C Y ; 433. CCY; 434. D ABC G D ABD; D ACD SD BCD ; D AOB Z D COD ; D BOC ED AOD ; 435. CCY; 436. D DBE ED CEF =D ADE 3 = =D DEFje j e n ~ a ~ o c ~ p a n a437. s a ~ CYC; ; 438. CYC; 439. Y C Y ; 440. CYC; 441. CYC; 442. CYC; 443. Y C Y ; 444. C Y C ; 445. 1 ) U o K a x n D ACE Z D BCE; 446. KOPHCTH c T a B CCC n cpenrby nannjy 448. K O H C T ~ Y K Q HH~a~ OCHOBY T p O p I a ; 447. K O H C T ~ Y CTaB ~ ~CYC; ~ Y C Y , 449. K O H C ~ ~ ~ K QH Ha OCHOBY ]~ CTaBa CCC; 450. K O H C T ~ Y K ~H H a ~~ ocnony CCY. 451. KOPHCTB CCY; 452. 1) y = 30' K O ~ B C T HC Y C ; 2) ~ o p n c ~CYC; n 453. K O H C T ~ Y K ~HI aI ~OCHOBY ~ Y C Y . 454. C ~ ~ e ~ p na ynma AB y npecexy ca m p a u o M p onpefiyje T e h l e C. 455. KOHCTPYKT~BHO O&EnHTH CHMeTpMrIHy TaYKy B TaYKll A Y OAHOCY H a p II TO je CTPaHHUa jejJ~a~oc~pannsT n po or y r n a . 456. ~ o n c q y w ~Hj aa OCHOBY CCY. 457.1) KOPECTH CYC; 2) KOPUCTI~ Y C Y , c=4cnz n yrnoea H a % o j 45'. 458. KOPMCTU

IEF~ ID^ I D FI

YCY; 459. KoncTpyme ce

LM

= 4s0,a TAM 113 B

D

AMB rue je

IAM(

= a + b = 7cm, LA = 60'11

HopMana Ha AM. (BWHcnnKy 1).

y.Kopuc~uYCY 3a(BWU

~ O M O ~ HD HABM CJIUKY 2);

bh '"'cn.

C

461. Hpouymu ce (EM(= (U4( u ( BN( = (NC(;

462.

Ko~crpynrne ce

rne je AD=a+b=6cm, c=3,5cm LD = 15'. T a ¶ ~ aC je s , nAD = {c}; 463. Koncrpyuure ce no~ofinn D BCM rue je ~ B M=( AB - AC , 3 a m caMeTpana c~panuqeCM y npeceKy ca npaaoM BM aaje TasKy A; 464. Ko~cTpyame ce n o ~ o h u DBMC jep je BM = a + b , LB = 75' n LBMC = 37'30' 3 a m ,~,s n BM = { A ); 465. Ao6nja ce q e n ~ a ponncne Kpymsiqe Tpoyma; 466. V e n ~ a ponucane Kpyxmqe ce nanasn y npeceKy cuMeTpana crpannqa. Koncrpymuija. 461. K o ~ q y u m yce cmeTpane ,qym AB, AC u BC ; 468. Honynpesnu~ owcanor Kpyra je nonoauna xnno~enyse na je xnno~e~y3a 5cm, KaTeTa 3cm u 3a K O H C T P Y ~ W ~ KOPHCTU ~ CCY. 469. I I p e w n ~AB je xnno~eny3aTpoyrna, a TasKa C je npeceK cnMeTpane npeunuKa u Kpymnqe. 470. H e ~ a je C E k, k(0,3cm). 1.13 C ce Konqyuure K p y q a k,(C,Scm) n k = {A, B ) =, uo6uja ce TpaxeHn D ABC. 471. flobuja ce qeHTap )lIUCaHOrKpyra y TPOYrJIY; 472. IJen~apynncanor Kpyra y Tpoyrny ce nanasn y npeceKy currrepana ymosa. KoncTpyKqnja. 473. fla je C , = C , 474. ,Qa C , u C, npunauajy ocu cu~eTpnje.475. Kopucm caojcrao ua je nonynpezlrrur: 1) onucanor I I O M O ~ HPUABD

Kpyra

2 1 TemmHe uym; 2) ynucanor Kpyra - Temluue u y m . 476. 3 3

-

IIonynpesna~onncaHor Kpyra je

2

-

3

TeHulmHe n y m ,

TO

j e c ~2cm. 478.

n p e s ~ onacasor ~l~ Kpyra je 2'2,5cm=5cm a TO je m o ~ e ~ y npaBoyrnor 3a rpoyrna. CH~erpanaxnno~esy3eca K P ~ X H H ~ O M onpetjyje TeMe npaBor yma. 479. K o n c ~ p y u mn ~o ~ o b n ub CMB me je CM = 2cm, TemmHa nyx 480. 1) HCKOPHCTH 2 TCIKIIIIIHe nYXM 2,6cm = M B , K O ~ H C T ~CCY; ~E Tpoyma; 2) AA, = 6cm;BB, = 9cm ; 3) KaKo je nognomje Temmne g y m noeyueHe Ha x ~ n o ~ e qenTap ~ y 3 ~onxcawor Kpyra TO je CC, = C,A = C,B ; 482. H e ~ a 481. Koncrpymne Ce I I O M O ~ HTpOyraO H ACCl K O P H C T ~ ~ CCC. H je M I Texcllmna uym. K a ~ je o o 6 n ~D AA,C sa 2cm ~ e TOhje KpaK ~ 3a 2cm ~ y OR mO C H O B H I I~p~e.~ ToMe a a + 2a + 4 = 28 3 a=8cm, b=lOcm. 484. O ~ T O I ~je~ TarIKa H T ~ m~e Ce CeKy BnCEHe 483. K o p ~ c ~CYC; n rpoyma. K o n q y ~ m j a .485. Cse Tawe ce nolcnanajy.

"A

PI lip, A. nPI = {c}; (cn. 3)

-:;I-

,= 3 c m

-

C=

cr

M P-.

a.3 L

487. n o n y n p e u onHcanor ~~~ Kpyra je -h = 2cm. 3

490. C, je y cpegmTy xmo~eny3e,a H je y TeMeHy npanor yma. 491. Kopncm CTaB CYC; 492. KOPHCTE CTaB CCC; 493.1) KOPHCTU CTaB YCX 2) K O ~ E C T UcTaB YCY; 3) 4) KopncTn L ca napanennm KpaQBMa. 494. KOPHClX CTaB YCY, 495. 1) KOpHCTH CTaB C y C 3a DADMP BCM; 2)D ABNP CDN 496.1) LMNQ = LMQN = 4 5 ° , 1 ~ = ~ 1IMQ~, LNMQ = 90' ; 2) cnHsHo Kao n nog 1. 497.1) 16cm; 2) 9,6cm; 498. 1) 6cm 2) 6cm; 499. 1) 2,6cm; 2) 5,2cm; 500. 1) K O ~ H C T Umas CYC 3a 501. 1) KOPHCTH KOHCTP~KW~Y 2) KOPECTll CTaB YCY 3a KOHCTPYKUU~~. CTaB CYC; 2) KOpHCTPl CTaB CCY; 502.1) KOpHCTH CTaB YCY; 2) KOPHCTH CTaB m C . 503.1) xmoTeny3a je 4cm=2r ; 2) xano~eny3aje 6cm=2r, Tpefie

on 36' a

TO

cmeTpana LADC

=

swaTaK

1 3

je - yrna

OR

LBAC, TaRa

je

P +5 2

a

54'. 510. 1) a=75', p=35',y=70°.

cnoJbamHa

LADB = y

a 2

+-

2)

OAHOCHO

na je LADE - LADC = y - P. 515. ~ o p a c mnpeTxouHa y-p=2o0.

516.

n-ff ---+2

2

P 2

a

a+,D+y=18o0

a 0 a 0 =.P+y+-+x=18O ax=-. 517. y = 7 2 ,a=,L?=54', LSAD= 2 2 990 -90 0 = 90 . 518. a=/3=65°,y=500,b>a; 519. a=50°,fl=550, ( A B ( > ( A ~ > 52Q. ( B ~a=/3=4@,y=10@. 521. a = ~ = 7 2 ~ , ~ = b3 ABD 6~,

a a ACD cy j w q a m l l > ( ~ D ( = I A =(cC( q ; SZ2 ( A ~ ( = I A ~ ~ L ~ B M = L Kao L j e ~ n a ~ o ~ p a D~ AMB. o r KaKo je LNMB = LABM Kao L ca ) KpannMa; =.LAMB = L B W => ME je merpana LAMN. 523. LADF = LCDE KaO YHaKpCHH L, LADF = LACM KaO L ca )I KpawMa; LCDE = L l C M = LBCM = LCED =D CED j e u n a ~ o ~ p a524. ~ . Tame B a = W BCD D Cy OCHO CHMeTPBYHe Y ORHOCY Ha S jep IBC( =lo4 j e m i a ~ o ~ p a525. ~ ; K a ~ oje s,h, s C D E j e j e u n a ~ o ~ p anpaBoymna ~o a L C D A =45O a a =180° - ( 4 5 ° + 2 5 0 ) = 1 1 ~ 0~, = 2 6 ' ;526.LMCN=

Ic~

(CDI

f f P 1 90~+-+-=90~ +-.9@ =139; 527. D t SAC =, 2 2 2 K O K ~ ~ K DC H A~ = LDAC= ~ ~ ' , L A B C 20'. = 5%.

=IAD( a P ACD

jeuaa-

LCAS = LASD = LDAS

IAD(

Kao CarnacHH L 31.ASD j e n ~ a ~ o ~3 pa~

ICEI IESI

=

ISDI

; LACS = LCSE =

L ! & ' C E ~C S E ~ ~ J J H ~ K O K=~ ~ K; D n p e ~ a TOMe

AD + CE =

-

AE

BDF 3 BF = 4cm * AF = 8cm ; 531. b A E C a A B D H ~ = C Y C 3 ICEI = ; 532. D BCO, g b CDO, (CYC) 3

= lOcm; 0 3 D

OCHOBH

I C O, ~

IBO, I

IBDI

j e j q ~ a ~ o ~ p533. a ~ . b ABC ZD BMN (CYC) * LBAC = L B M N ; 534. b BMC zb BCE ( Y C Y ) a b CEM j e ~ a o ~ p caa ~ LMCE = 60' * LCME = LCEM = 60' =ID CEM jepmK@c~pa~Hsan:535. H e ~ jea A, o c ~ o c ~ ~ e ~ Taws p a b Ta ~ ~ K yHo A~ ~ o Hap, c y A,B np = {c}. M 3 D ABD 3 LADE > LBAD

""

21. CO,O,

A

A

B

cn. 4

113 D

=, IPBI > IB4 ACD a LADC > LCAD 3 > ;

I A CI

ICDI

537.1) n p e ~ o n o c ~ a s mnao n ~ j e j q ayrao ~ m j e MaBn OR 60°, oHna n o c ~ o j e nse ~ o r y h n o c ~*a @an : yrao je 60 a oc'rana nsa seha on 60°, na je 36up yrnosa seha on 180, m a ** csa Tpa yrna seha on 60' na je H 36up yrnosa n e h ~on 180'. R a ~ n ej e n a ~ yrao Mopa 611m Ma&H OR 60'. 2) WIHYHO npeTxonnoM ( n p e ~ n o c ~ a s na a ~~n ~ l j e yrao n a ~m j e s e i i ~on 6od; 538. H e ~ a Cy OCHOCKMeTpHYHe TasKe TaYKU A y OnHOCy Ha KpaKe A , H A,. Tana A,A, nOX = {c), A, A, n~y = {B}.Tpaace~Hb je b ABC. 539. H e ~ npasa a AM nBC = {N}. Tana je AN > AM(AB) =, 3a D ABN a;)< LANB < ABN H 3a b ACN, LANB je cnoJbaurBa yrao =, LANB > LACB A B C >A C B 3 > 540. D AOB zb COD (CCC) H j e ~ ~ a ~ o ~ p a m Cy 3BHCHHe ON H OP Ha OCHOBHqe A B H CD Cy j e ~ I i a ~ e =IP MOP ZD MON (CCY) 3 LPMO = LNMO 3 MO je me'rpana LAMC. 511 K p T a w A : p l l ~ ~ , p n ~ ~LAMC=LBDE =(~); Kao L c ~

*

IAC~ IABI.

.

3 [ A M \=

IAC~ = I B DI , L ME

= LDBE

Kao L call3 (YCY) a D AME

ED

542. H e ~ aje MLEF ( M EA& N ECD,E E BC F E AD), a NN,L4Ba FF,IBC Kao L Ca HOpManHHM KpaUllMa 544. D ABA, ED CHA, 3 =

IAA,I

A C = 130' 3 LABC => LACB = 25' ; a r , AA,C j e n ~ a ~ o ~ p npaBoyrna a~o

3D

IcA,~

AB 3 ACA, = A

C B = 45'. 545. y = 10XO,a= ,f?= 36' ;s e ~ jea CC,BacnHa a AA, caMeTpana yrna H C C , n AA, = {0},

L OCA

c~

A

UI.

5

, = LCA,O = 54

O

2 D

OA,C j e ~ ~ a ~ o ~ p a

a IcO( = 1041 neKa je D c p e p m x AA,, a K ~ K Oje C, c p e W T e AB rn je C,D, cpeR&a m j a D ABA, =, C~DI I B A, I C A~ 2 LOC,D = LODC =I loc,~ = loo1 ~ j C. C ,

=

DA,

3

Buns C n H K y 5.

1 C C , = - AA, 2

3

AA, = 2 . CC

546. PI3 D B M N ~ L I . I N B = ~ ~OL' A B C = S.2=10@; O~ a3 D B M ~ L B M B ~ ~ a LniMC = 20' ;113 D MNC a LMCN = 70' = LACB; LBAC = 10'. 547. C

H e ~ jea CD nncxaa, CE TemrnHa nym 11 EF HopMana Ha BC,Tana je i.ADC Z D CDE E D CEF AD= DE = EF=a, oHua je BE=2a na 03 D EBF =. ,f?=30°; DEFC=LECF=~$ a A C B = ~ . ~ O =90° D

cn. 6 548. H e ~ a je ABD a D ABE J

BAC = 60'. BURHcnuKy 6.

E cpeunaa x a n o ~ e ~ y 3 eBD npanoyrnor Tpoyrna ~ ~ H ~ K O K D~ AEC ~ K ,

I4

D

AED

Cy

jenHaK0KpaK~

XHnOTeHy3e 3 D ACE je)JHaKoKpaK (CE = A E ) . K a ~ oje CE = EB 3 D C E B ~ ~ ~ ~ H ~ a KKaKO O KBHC0Ha ~ ~ K ,CD UenU BE y OnHocy 1:1 TO je b BCE je,q~aKocTpa~asaH 2 A B C = 60' U LBAC = 30'. 550. H e ~ je a CD BHCHHa, a CE T e m m a Rym a LCED = XO'SI'4 ACE je~~a~o- pa^ L CAE=80"51,:2=40~25~30,* a LABC = 49'34'30". 551. D ACDje npaBoyrna 3 DM je TemmHa uym 3 DM = MC = MA Z D DMC jemaKOKpaK11 3LMDC= LMCD; D BCD npaBoyrnn a DN= NC = NB =x> CDN ~ ~ R H ~ K O K3 ~ LNCD ~ K U = LCDN; LMDN= LMDC+ LCDN= M C D + LDCN

= 90' ~b

MDN

je

npaBoyrnzf.

552.

b

ABA,

SD

CHA,

(YCY)

a = p = 67'30'. 553. FE je cpen-&a n x ~ n j aD ACD : , E F ~ ~ AuCEFILIC, a KaKo je n CDlBF =, E je opToqezfTap b BCF a BE je BxcuHa D BCF =, B E E F . 554.H e ~ jea 0 - q e ~ ~ yrIncaHor ap a S qeHTap onucaHor Kpyra. D AOS je j e n t r a ~ o ~ p a ~=&AB=LBAO=X, vl a " h a ~ jeo A 0 caMeTpana B A G To je d A e X &KO j e [ ~ A=[SC/ ( = r, 3~ SAC j e ~ ~ a ~ o ~ p a m aLACS=3x=\LACB=6x A

1

\ 4

/'

liii

,'% s E s ~ ~ = > O A =f2Of B2P= ~ - = - ~ ~ = P ~

ABC j e n z f a ~ o ~ pz fa ~C E OS ; I I p e ~ aToMe 6 ~ + 2 x + 2 x = 1 8 0 ~3 ~ = 1 8 0 a a = ~ = 2 . 1 8 0 = 3 8 ,

D

y = 108'. (BPIUH CXEIKY~.)

cn. 7.

555. AKOje

(c ,I

=IAGI a b ACC,

j e u ~ a ~ o ~ p a a~ =t lL C , A C = L A C C ; ;

I I p e ~ aToMe a + p = LACC, + LBCC, = y = 90' =D ABC npaaoymzf. 556. H e ~ aje s, cIuleTpana CnoJbaIIIIber a s, CHMeTpana yey'rpamber yma, s , ~ A B = { M )a s Z n A B = { N } , ~ a K oje a - j = 9 0 ~ 3 a = 9 0 ~ + pn

3 (CM(=

IcNI.

Ha T a w y D 3

IAB ,(=[ A B [~

557. H e ~ jea B, OCHOCAMeTpEcIHaTa'iKa T a q K H B Y OAHOCY =[DB,[a b ABB, j e n ~ a ~ o ~ p D a ~jea cpenmrre , BB, u

IBD[

1 1 DEje cpensa n ~ l ~ v r jDa BCB, a DE =-. B,c=-(A4 2 2

- AC)=

D

ABA, Eb A, A2C (CYC) 3 JAB/= JcA,~,KaKO H3

b

A 4 C a A 4 < A C + C A 2 a 2 . t , 2tc CJI.

3

3 2 /a + tb > 3tc ; 560. H e ~ jea p n BC = {P)u ~ p o P s npoayqu npany m TaKo

562.113 a ACC, a C C , ABF

Tpoyrnosa

nohlotim D MNC caMerpane ,s nMN

b AOP

Z",

A

zb AOR 3

(MN = lOcm, L M

& 0

~ ~ ~ H ~ K o K569. ~ ~ ki3 K H nouynapHocTa ;

= {A} a s,

I A PI

=

I A RI .

= 30°, MX np, =

AKO

{c)),

je

3aTHM

n

= { B }. 574. Ko~crpqwne Ce Kpyr k H a IIpe'moM AB=6cm, 3 a m k,!B,hd y TeceKy ca k onpebyje Tamy B, H B2 ; ~ p o 3c p e m T e S ~ y x nAB ~ o ~ c ~ p yce ~ nPu, ~e~ A ;B~I p y rk2(AIi,)nP, = {AIA2). ITpaae AB, nBA, = {c}o w m o AB, nBA, = {c,}. B B ~ a m y 1 0 .

cn. 1 . 575. Koncqqxrne ce n o ~ o b uD ABT ( c = 5cm, AT = 3cm, BT = 4cm), a SITEM ce nponym AT a BT 3a csoje nonosme a noGlljy ce T e m H e n y m AA, u BB,. I I p e c e ~npaBax BA, nAB, = {c}.

h+,, 2 BD = - t , , saTaM ce nponym DT sa csojp 3

A

cn. 11.

B

uymny a no6uje A, a BA, aa c ~ o j y nymay a n06Hje C (A, je cpenmTe TD). Bunu CnKKy 11.

1

AT nponymvulo 3a - A T , Ra cc ~ o 6 u j e 2 TemmaauymAA,; ~ p y k(S,SA) r u KPO~A, Ce KOHCTPYHrUe n U A , Zk n n = {B,C ] . Bwu cnnKy 12.

&

579. H e ~ jea t , = AD = 3cm a DEje cpeulLa nunaja P ABC . K o ~ c ~ p y u u r a npBo n o ~ o f i ~I> aADE ....... 580. K o H c T ~ ~ HCe U~I ~O M O ~ ~ H H P ACID ( C , D = ~ + ~ = ~ ~ ~ L C ~ D A = ~ S 3aTHM SAil n C I D = {c],R(CAC)AAC, = { B ] ; B u ~ u c n a ~13. y

C,

A

B

cn. 13. 581. Hyx OR 8cm nonenn Ha TPPI j e , q ~ a ~gena a K O P H C T ~ ~ I TemuITe I Tpoyrna. 582. Ko~crpyamece n o ~ o h a nP MBC (MB= 7cm,LM = 370301,P = 75') sMc

n MB = { A } .

cpeuanrTe

/

,, ,

-6 , A

,

Ah-,

cn.14.

~ ( s , s Arue)

je S Kpyroan k , ( ~ , h , ) n k = { ~ k} a, ( ~ , h , ) n k = { M } ,a TasKa Cje npeceK npasux AM a BN. Bmu cnmy 14.

583. Konc~pyau~e ce Kpyr

B

cTpanaue

AB=6cm;

584. Ko~crpyame ce

IIOMO~HH 4

AA,C

(AA, = h, =3cm,LA1AC = 60°,

U A , C = 90' ) ~ ~ T UceMCA, nponyxn 3a nyrrm~yKpaKa b, ~j CA=CB. 585. H e ~ je a CD=5cm u AE=6cm. AKOje DF cpema nunxja npaBoyrnor rpoyrna ABE, ~ o m na e ce ~ o ~ c ~ p y a m e m a ~poyraoCDF. = 7cm,a = ~ O ' , L M= 45') 586. 1) Konc~pynmece ~ O M O ~ H EDI ,s, n M A = {c); 2 ) KoHc~py~lme ce ~ O M O ~ ~ H H D MAB (MA= 2cm,a =30°, LM = 1350), sMB n M A = { c } ; 3) Ko~crpyamece no~oiinunpasoyrnvr D BCD (BC=6cm je xnno~e~y3a, CD=hr=4cm je KaTe~a)a 3aTllM y TaYKll C : n l B D = {A}.

587. 1 ) K a ~ je o xmo~eny3ajenna~a~ J B O C ~ ~ Y K nonynpecIHEKy OM TO je OHa 5cm. Han XHnOTeHy30M ce KOHClpyvlIIIe nonyKpyr, Ha n p a ~ 0 A j B BHCHHa MN. Y Taqm N HopMana n l M N , n n k = {c,C, ). 2) KaKo je c = 2 . t , = 6cm h, = 3cm, K~H~rpyKq~jaje CnIlsHa npeTX0nHOM 3anaTKy; 588. Koncrpyame ce n o ~ o f i n uACC, ~ (AC = 3cm,LCAC, = 60' H

4 X

N

cn. 15.

Ta HopMana n nponasu K P O ~ nenocTaxan ~ p xjep , je TO ~peiia BXCElHa D MON. Bnnn CnHKY 15

591. Hem je A E OXH B E Oy , OHAa s,~,,~ ,n , , ,s = {s}~ o j anpunafia Tpame~ojceMeTpann L r o y ; 592. T a q ~ D e u E cxMeTparlHe .catrm A y OnHocy Ha npane a u b npenwajy npasoj BC. n p e ~ TOM% a TeMeHa B u C fio6njajy ce Kao npeceK Itpane DE ca npasaMa a u b.

Gpojennoj npasx; 602. 1) B

3

4

16 1 1 5 3. E < z ; 6 0 6 . - < - = Pm, ururca 17. l338. H e ~ a je 0 UeaTap ynuca~or Kpyra 1 1 1 1 -m+-b.r =-r .(a+b+c). 1339. Iz=4cn~, 2 2 2 2 ~=14,4cn1'; 1340. h=5cm, P=12,5cn12; 1341. H e ~ aje MN cpeuba n ~ ~ a j a

&,,

.

.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ 1

1

1

i

1

m~P,MN=P,M~,N=-m.-h=-mlaP,AMD=2.PMN=-m.h=>~,w,=-~A,o;' 2 2 4 2 2

1342. P,,,=-JcD(.1

1 4,8=-~,,,.,,=>~,,=9,bcd.

2

P,,

KaKo je

= P,,,

h a P,,

2

b

MCD

=

-21 1 ~ ~ 1 h. =>

~ ~ ~ ~ J ( H H TTpoyrao KII 3 PbArD- PPMcD=

Pbaa, - P ~ M I X~: P ~ A = MPH(~M D

l343. ( c n u ~ a18.) 1

P,,,~=-AB.(~-.T)+~ 2 1

PbACD=P2+-ICqx 2

a,.I R

K a ~ oje P,=P,

TO

'

2

je IOx=48-6x*x=3=

2

P, =30cm23 P, +P, = 60cm2.

d .d W44. P = -= 108cm2. 1345. H e ~ je a M CpeAaurTe KpaKa BC a MN 2

M nOBYI(llM0 pllAD TaKO ua p n A D = ( E } H p n c ~ (=F } . Tana je P,,, = P, a KaKO je AEFD napanenorpaM 3 PA,, = P = 12 . 5 = 60cm2. 1346. K a ~ o jc ~ ~ C T O ~KpaKOBa ~ B C II N E

A D , Kp03 TaYKy

,,, ,

d2

= -=

2 PA,,.,, =2.PAlRifj

2r)2 = 2r2 2

,,,,. 1347. H e ~ aje

a P,,

=

(2r)'

=

4r2

TO

je

a = 12cn 3 b = 6cm 3 a . h , = b . hh 3

h h = 2 h , ~ h c = 1 , 5 h , a c.h,=a.hO=,c=8cm.

iYCb

+ 1997p je ~ r a p 6po.i a ~ > 2 =. cnoxeH 6 p j 3a p = 3

l997p je 361'1pUBa aerraplra 61)oja jc nap31 6 p j > 2 =) cnoxeH 6poj; U5L K a ~ oje AB= AD= i A D B = LABD =36" I< K ~ K Oje L A C B = A D B = ~ ~*BCD ] ie#EdKoKp?lX; l352. K ~ p ~ C CTal lll3 1353. o, = 2

p

ac;

29

1354. -x = 87 3 n = 240knl; 1355. Z E {-2,-1,0,1,2,3,4); 80 -14 -12 1356. a = 40°, j = 50' ; 1357. y = 100' > a = 40' = a > b ; 1358. K a ~ oce nyna uo6~1jaKao II~OMSBOJJ je,q~e2 B j e n ~ e5, nonofino jt. npe6pojxrl.a cse rleTarlc, a bnx m a 100:5=20.K a ~ cy o ~ c b BnMa y n 25.50, 75 PI 100 ~ o j icy l qebaua ca 25 TO 1 ' 2 ' 3 ' 4....... ' 100 canpm 20t4=24 IleTaqa na ce 6poj saspurana ca 24 Hpne. 1359. K a ~ o je a=-1800. b=-2196 spejlHocT aspasa je -2. 1360. 3 8 8 8 . ~ = 2 ~ . 3 ~TO. xje, x=314 6poj je 108'. 3 8 ~ 8 . ~ = 2 ~ . 3 ' TO . y ,je I y = 2 - 2 - 3 11 6poj je 36'. 1361. Aqa 3a je,qa~flaw ypanH - nocna. 3a 6 naHa 10

6 6 4 ypanahe - nocna ca JLHKOM. 3 ~ a U wa J a n ~ oypa,qa 1- - = - nocna 10 i0 10 2 1 3a 2 Uana na 3a 1nan ypanvl - o,qnoc~o- nocna, na hfy je n o ~ p e 6 5~ naHa o 10 5 sa ~ a nocao. j 1362. &map q y a S je Ha mepm Lroy H D OAM z D OEM (CCY)a = U62 1;poj je neibns ca 9 a 4 a b E {0,4,8} a 360r

IOAI

/OBI;

~eJl,MBOCTI4Ca 9

3

a E {7,3,8); 1364.

D

ACD PI

D

BCE

< 5 a .r E {1,2,3,4,5}. 1366. H e ~ aje

CY ~ ~ ~ J H ~ K o M K ~ ~ K H

ke~a TemurHa ~ y cew x x m o ~ e s yy D. necnna y E, a cmelpana npanor y n a y M. Taus je D ACD j e n n a ~ o ~ p (aA~D. = BD = So)3 LDCA = 40°, D BCE je 3x-15 5 0 3 x

npasoyrnn LECM = 45'

3

K a ~ o je

LBCE = 40'.

- 40' = 5' = L DCF

.

LCAB =40',

LBCM = LACM = 45' =.

BD< A D = C D < D E ; ~ ~ I ~ . ~ A E D ~ B C F ( C Y C ) > L ~ E D = L B ~ = P j e w a ~ o ~ p m U76.128:25=5(3)= ; I F p m e o m np~y n m j a 6ap jtZaH pa3pu ca 6 p o j e ~ e on 25 b y q e m ; 1377.35 Ci3~6ona; Hem je LBAC=fABC=x, KaKO je LAMC Tyn y a o =LACM=x,aLBMC = 2 x 3 LBCM ~ o x e6 m x m 2x. AKO je LBCM = I = 4x = 180' 3 x =45' w ABC j e j q m o ~ p a ~npamyrm. o AKOje LBCM = 2x a 5x = 180' a x=36' aLACB=IO$. 1379. LMCN=49; 1380. K ~ K O m o 16 noma men3nja Im x lm a 15 p p , m . m o m &pmeomr npmqnna n m j jerpro ~ lIOJbe y KOMe HeMa HIlTU j e m pyllil. 138t A=7ElH x&, ki3 a AON -4 BOM (CYC)>(OMI = u L40N=LBOMu LMON=LAOB=

ON^

180' -2.30' = 120'; W83. He, .ua 60 KrmKepe nouenmn ~pe6am n a j m 0 + 1 + 2 + 3 + ...+ 14=7.15=105 > 1 O O , m w a n a ~ a h e 6 a p ~ j n q a ~ 0 6 ~ jeuna~6pojmepa;l384.Ka~oje180=1.4.5.9=1.6.6.5 ~ 3 ~ ~ e m c 2 1 a 2 1 b 2 1 96pojje 1 5 6 6 ; m K a ~ o j -