www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ·¦(&/%&#&/;&3-÷,*** #FO[FS·ÀHFOMFSJO\"MBOMBS\"SBTOEBLJ÷MJöLJ ÖRNEK 2 TANIM A 2k B [ AE] a [ BD] = {C} 4S 2xn #FO[FSJLJпHFOJOBMBOMBSPSBOCFO[FSMJLPSB- <\"#><%&> OOOLBSFTJEJS C A & 3n A^ ABC h [DE] // [BC] & = 4 9S 9 DE D 3k A^ CDE h | |BD =DN E | |:VLBSEBLJWFSJMFSFHÌSF #$ æ=YLBÀDNEJS BC 4 AB BC BC = 2n CD = 3n 2 == 9 DE DE A^ ADE h CD = f p 5n =DN BC A^ ABC h n =DN x =DN ÖRNEK 1 ÖRNEK 3 A ABC üçgen A ABC üçgen m ( A%ED ) = m ( A%BC ) 3 [ ED] // [ BC] 9S E D3 | |DE =DN 9S | |AE =DN | |BC =DN | |EB å=DN 7S E D 3k 2 16S B 5k C B4 C & & A^ AED h A^ ADE h :VLBSEBLJWFSJMFSFHÌSF PSBOLBÀUS :VLBSEBLJWFSJMFSFHÌSF PSBOLBÀUS A^ EDCB h A^ BDEC h AE ED 3 \"%&~\"$# == DE 2 & AB BC 5 A^ ADE h 9 ED 2 A^ AED h f p= = 9 & 16 == BC A^ ABC h BC 2 A^ ABC h 25 A^ AED h 9S 9 & == A^ ADE h 9 A^ EDCB h 16S 16 = A^ BDEC h 7 9 99 9 1. 2. 6 3. 7 16
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ÖRNEK 4 ÖRNEK 5 A ABC bir üçgen A ABC üçgen 3k 3n [DE] // [FG] n D 2|FC| = 5|FB| D F E 2|AE| = |EB| |AD| = 3 |BD| | | | |AD = k DC 15S 15A |AC| = 4 |FC| 2n k n S A B E KG C F 2k B 3k C && & :VLBSEBLJWFSJMFSFHÌSF LLBÀUS A^ DBE h + A^ FGC h = k A^ ABC h PMEVôVOB HÌSF L LBÀUS 2k DC n · · =1 [\",] // [DE] // ['(] BD = 1 & A^ BDE h = 1 5k AD 2n AB 4 A^ ABK h 16 5 |\"%| = |DC| & A^ ABC h = 4+\" 1 k= 5 && & A ^ DBE h + A ^ FGC h = k.A ^ ABC h 4+\" =L 4+\" 1 k= 16 ÖRNEK 6 A ABC üçgen <\"&>WF<$%> 4 LFOBSPSUBZ N [ AC] m BC] D 3k |AK| = |KE| n | |AN =DN .FOFMBVT5FPSFNJ K Gk TANIM 2n 2k A B 2m E 2m C R Q | |:VLBSEBLJWFSJMFSFHÌSF GC LBÀDNEJS P C ( \"#$OJOWF B , \"%$OJOBôSMLNFSLF[J 2n DN 3k PB CR AQ =1 · · =1 .. 3n 4 k | |DN =DN PC RA QB veya AR CB PQ =1 .. AC BP QR 1 100 1 4. 5. 6. 2 16 5
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 7 ÖRNEK 9 A E ôFLJMEF CJS CJMBSEP NBTBT WFSJMNJõUJS , OPLUBTOEB CV- k MVOBOUPQCBOEB¿BSQULUBOTPOSB#OPLUBTOEBLJUPQB ¿BSQBSTBQVBOBMONBLUBES .F 2k A B C DE F Bm C mD ba Z b b | | | | | |ABC ve BDF birer üçgen, AF + AB = 2 FB X T | | | | | | | |BC + BD = 3 CD FD =DNPMEVôVOBHÌSF Y a | |&' LBÀDNEJS a R a K L M NP k m ED #VBUõZBQNBLJ¿JOTUSBUFKJHFMJõUJSFOCJSCJMBSEPDVCFO- 3k · m · EF = 1 [FSпHFOMFSJLVMMBOBSBLEPóSVTBMCJSZPM¿J[FOUPQMBCV BUõHFS¿FLMFõUJSFCJMFDFóJOJIFTBQMBNõUS 3|&'| = |ED| ÷ML PMBSBL [\"'] CBOEOEBLJ CJS OPLUBZ LFOEJOF TF |'%| = 4|&'| 9 ÀFOCJMBSEPDVOVOQVBOBMNBTJÀJOCVCBOUUBLJIBO |&'|= 2 HJOPLUBZBUPQVBUNBTHFSFLJS ÖRNEK 8 5PQVO¿BSQUóEPóSVMUVJMFHFSJTFLUJóJEPóSVMUVBSBTO- EBLJB¿EJS #JSTJUFZËOFUJNJ ZÐSÐZÐõZPMVZFõJMBMBOWFIBWV[PMBSBL CFMMJLTNMBSPMVõUVSNBLJTUJZPS#VBMBOMBSEBO'&$п- ,OPLUBTOEBCVMVOBOUPQ%OPLUBTOB[KD]EPôSVMUVTVO HFOTFMCËMHFTJJMF\"#$пHFOTFMCËMHFTJCJSCJSJOFCFO[FS JLJпHFOTFMCËMHFEJS4BCJUI[MBZÐSÐZFO4FSIBU IBWV- EBWVSVMEVôVOEBÑÀCBOEBÀBSQBSBL#OPLUBTOBHFMJS [VO¿FWSFTJOJCFZB[õFSJUMFSJJ[MFZFSFLFOB[ELEBZÐ- SÐSLFOTBEFDF,-.пHFOTFMCËMHFTJOEFLJZÐSÐZÐõZP- ÖRNEK 10 MVOEBLJCFZB[õFSJUMFSJLVMMBOBSBL\"#$CËMHFTJOJO¿FWSF- TJOJELEBZÐSÐZPS A ABC ve ACF birer üçgen 8 K 2k S F [DE] // [BC] D A [EF]JMFJ¿B¿PSUBZ E :ÐSÐZÐõZPMV 15 | |k 4 G AF =DN F :FõJM | |B C CF =DN BMBO Havuz \" #$&% =DN2PMEVôVOBHÌSF, & hLBÀDN2 A^ ADE B DE C LM EJS )BWV[JÀJOBZSMBOCÌMÑNN2JTFZFöJMBMBOJÀJOBZ SMBOCÌMÑNÑOBMBOOCVMVOV[ \"ÀPSUBZEBO AF AE 8 == FC EC 4 Ç^ FEC h 12 3 & == A^ ADE h 2 =d 2 Ç^ ABC h 20 5 n A^ FEC h 3 2 9 45 & 3 A^ ABC h =d n = = A^ ADC h 5 25 ? 4S = \" \"#$ =N2 9 S + 15 :FöJMBMBO=N2 4=DN2 9 101 9. D 10. 12 7. 8. 80 2
TEST - 45 ·ÀHFOEF#FO[FSMJL*** 1. A ABC ve DBF 4. A \"#$пHFOJOEF пHFOMFSJOEF D 15 <\"%>J¿B¿PSUBZ x |AD| = |DB| 10 E | |BC =DN 3 | |CF =DN E | |EF =DN x <\"%>m<%&> # C F 9 BD | |C AB =DN | | | |DE =DN AE =DNPMEVóVOBHËSF | |EC =YLBÀDNEJS | |PMEVôVOBHÌSF DE =YLBÀDNEJS \" # $ % & \" 3 # $ 5 % & 7 2 2 2 2. A ABC üçgen 5. A ABC üçgen E |AE| = |EB| D AD = 3 ve F |EF| = |FC| E AB 8 | |DC =DN & h DN2 A^ DEF B F C [ DE] // [BC] # Y% C & ADE PMEVôVOBHÌSF A^ hLBÀDN2EJS | |[AD] a[CE] = {F} PMEVôVOBHÌSF #% =YLBÀ \" # $ % & DNEJS \" # $ % & 3. A \"#$пHFOJOEF 6. A ABC üçgen \" - %EPóSVTBM K L K E [DE] // [BC] L |AL| = |LD| D P4 [-.] // [AC] |BD| = |DC| [KF] // [AB] BD | |KC =DN 9 M BF C C & = 4 br2 ve & = 9 br2 PMEVóVOB A^ KPE h A^ PFM h | |PMEVôVOBHÌSF KL LBÀDNEJS HËSF \" 1&$. LBÀCJSJNLBSFEJS \" # $ % & \" # $ % & 1. D 2. C 3. C 102 4. D 5. \" 6. #
·ÀHFOEF#FO[FSMJL*** TEST - 46 1. A \"#$ пHFOJOEF # / 4. 0LUBZ\"NDBhOOLBSFõFLMJOEFCJSUBSMBTCVMVONBL- WF%OPLUBMBSEPóSV- UBES5BSMBTOEBUFLUÐSMÐÐSÐOZFUJõUJSEJóJOEFZFUFSJ LBEBS LB[BO¿ TBóMBZBNBEóO IFTBQMBNõUS #V- TBMES OVOJ¿JOUBSMBTOпCËMHFZFBZSQIFSCJSJOEFGBSLM CJSÐSÐOZFUJõUJSFDFLUJS#VOVOJ¿JOCJSLºóEBUBSMB- 12 [\")] m [BC] TONPEFMMFZJQIBOHJCËMHFMFSEFIBOHJÐSÐOMFSJZF- D UJõUJSFDFóJOJIFTBQMBZQFOZÐLTFLLB[BODFMEFFU- NFLJTUJZPS 5N x |AD| = |DC| BH C 2*#)* = *)$* ,ºóEBUBSMBTO\"#$%LBSFTJõFLMJOEFNPEFMMJZPS 4POSBCVLBSFJ¿JOEFCJS&OPLUBTBMBSBL% $WF# *BN * = 5 br ve *AN * = 12 br * *PMEVôVOBHÌSF HC =YLBÀCSEJS LËõFMFSJJMF&OPLUBTOCJSMFõUJSJZPS[DE] m [EC]PM- \" # $ % & | |EVóVOVWF EC V[VOMVóVOV3 2 cNPMEVóVOVHË- 2. A SÐZPS &$# пHFO õFLMJOEFLJ CËMHFZF NTS FLUJóJOEF FO ZÐLTFLLB[BODFMEFFEFDFóJOJIFTBQMZPS 0LUBZ\"NDBNPEFMMFEJôJL»ôUUBLJNTSFLNFTJ HFSFLFOCÌMHFOJOBMBOOLBÀDN2CVMNVöUVS xD \" # $ % & K E N 4 BF C ABC üçgeninin [AC] LFOBS п [BC] LFOBS JLJ Fõ QBS¿BZBBZSMNõUS * * \" , /WF'OPLUBMBSEPôSVTBM /' =CSPM * *EVôVOBHÌSF \", =YLBÀCSEJS \" # 15 $ % 19 & 2 2 3. A [AB] m [CD] 5. A ABC üçgen <&%><#$> 55° *DF* = *FC* D D B *AF* = 2*FE* E <&'><\"#>EJS F % = 55° & =DN2 m ( BAE ) A^ EDG G h a FC & h =DN2 B EC A^ EGC :VLBSEBLJWFSJMFSFHÌSF % = a LBÀ EF m ( ABC ) SFDFEJS :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS ABC \" # $ % & \" 125 # 121 $ % 100 & 95 33 33 1. # 2. E 3. D 103 4. # 5. \"
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ·¦(&/%&.&3,&;-&3 ÷À5FôFU¦FNCFSJO.FSLF[J ÷À.FSLF[ %ö5FôFU¦FNCFSJOJO.FSLF[J TANIM TANIM #JSпHFOJOJ¿B¿PSUBZMBSOOLFTJNOPLUBTп- #JSпHFOJOJLJEõB¿PSUBZJMFCVB¿MBSBLPN- HFOJOJ¿UFóFU¿FNCFSJOJONFSLF[JZBEBJ¿NFS- õVPMNBZBOB¿OOJ¿B¿PSUBZCJSOPLUBEBLF- LF[JPMBSBLBEMBOESMS TJõJS #V OPLUB пHFOJO Eõ UFóFU ¿FNCFSJOJO A NFSLF[JEJS cb A O I C BC B a #JSпHFOJOпUBOFEõUFóFU¿FNCFSJWBSES #JSпHFOJOUÐNLFOBSMBSOBFõJUV[BLMLUBCVMV- OBOOPLUBJ¿UFóFU¿FNCFSJOJONFSLF[JEJS A rIr r BC ö¿UFóFU¿FNCFSJOJONFSLF[JJMFLËõFMFSJOPMVõ- UVSEVóV пHFOTFM CËMHFMFSJO BMBOMBS пHFOJO LFOBSMBSJMFPSBOUMõFLJMEFEBóUMS A c b \"ôSML.FSLF[J I TANIM B aC A ( B&IC ) A ( A&IC ) A ( &AIB ) #JS пHFOEF LFOBPSUBZMBS CJS OPLUBEB LFTJõJS == #VOPLUBпHFOJOBóSMLNFSLF[JEJS abc A ö¿UFóFU¿FNCFSJOZBS¿BQSPMNBLÐ[FSF G u= a+b+c & C 2 A^ ABC h = u.r 3k %JLпHFOEFLJ & B k A A^ ABC h = m.n G 2k O rr O K 2k rr k Ir Bm OC 104
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ¦FWSFM¦FNCFSJO.FSLF[J ¶¿HFOHFOJõB¿MпHFOJTFEJLMJLNFSLF[Jп- TANIM HFOJOEõCËMHFTJOEFEJS Bir üçgenin LFOBSPSUBEJLNFMFSJCJSOPLUBEB K: ABC üçgeni- LFTJõJS #V OPLUB пHFOJO LËõFMFSJOEFO HF¿FO K ¿FNCFSJO ¿FWSFM¿FNCFS NFSLF[JEJS OJO EJLMJL NFSLF- [JEJS A A O BC BC A 0¥FWSFM¿FNCFSJONFSLF[J H3 K H2 r ¶¿HFOEBSB¿MпHFOJTF0OPLUBTпHFOJO J¿CËMHFTJOEFEJS B H1 C r ¶¿HFOHFOJõB¿MпHFOJTFOPLUBTпHFOJO EõCËMHFTJOEFEJS ,\"#$пHFOJOJOEJLMJLNFSLF[J r ¶¿HFO EJL пHFO JTF 0 OPLUBT IJQPUFOÐT , )1 )2 )3 пHFOJOJO J¿ UFóFU ¿FNCFSJOJO Ð[FSJOEFEJS A NFSLF[JEJS BC ÖRNEK 1 I OPLUBT \"#$ пH FOJOJO O J¿UFóFU¿FNC FSJOJONFSke- A [JEJS %JLMJL.FSLF[J a TANIM #JS пHFOEF ZÐLTFLMJLMFS CJS OPLUBEB LFTJõJS I #VOPLUBZBEJLMJLNFSLF[JEFOJS 3a A ,%JLMJLNFSLF[J BC K % = % PMEVôVOB HÌSF m ( % ) LBÀ BC 3m ( BAC ) m ( BIC ) BAC %BSB¿MпHFOEF,OPLUBTJ¿CËMHFEFEJS ¶¿HFOEJLпHFOJTFEJLMJLNFSLF[JEJLLËõFEF- EFSFDFEJS EJS |#*|WF|*$|BÀPSUBZ a 3a = 90° + 2 a = 36° 105 1. 36
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ÖRNEK 2 ÖRNEK 5 A 1 OPLUBT \"#& п #JSFõLFOBSпHFOJOEõUFóFU¿FNCFSMFSJOEFOCJSJO JOZB- 7P HFOJOJO J¿ UFóFU ¿FN S¿BQ 6 3 DNPMEVôVOBHÌSF CVFöLFOBSÑÀHFOJO 3 CFSJOJONFSLF[JEJS ÀFWSFTJLBÀDNEJS BC 4.D E A 30-60-ÑÀHFOJOEFO ¦ \"#$ =DN | |[AB] m [AE] [PC] // [AB] [PD] // [AE] AB = DN 12 12 | |AE =DNPMEVôVOBHÌSF ¦ 1$% LBÀDNEJS B 6 6 60° C 1JTBHPSEBO|#&| = JÀUFSTBÀMBSEBO 60° 60° |#$| = |PC|WF|PD| = |DE| 60° 63 30° ¦ 1$% = |#$| + |CD| + |DE| =DN D ÖRNEK 3 A 1 OPLUBTOO ABC üçgeninin LFOBSMB 33 SOB PMBO V[BLMLMBS FõJUUJS D E a P 11–a Ba F 11–a C | |AD =DN | |BC =DN PMEVôVOBHÌSF ¦ \"#$ LBÀDNEJS 1OPLUBTJÀUFôFUÀFNCFSJONFSLF[J ¦ \"#$ =DN ÖRNEK 4 ÖRNEK 6 A \"#$ пHFO 1 OPL- #JSEJLпHFOEF EJLLËõFOJOпHFOJOBóSMLNFSL F[JOF V[BLMóDNPMEVóVOBHËSF CVÑÀHFOJOÀFWSFMÀFN 6 UBT%#&пH FOJOJO CFSJOJOZBSÀBQLBÀDNEJS D EõUFóFU ¿FNCFSJOJO P NFSLF[JEJS ·ÀHFO EJL PMEVôVOEBO ÀFWSFM ÀFNCFSJO NFSLF[J 4 <%&><\"$> 6 IJQPUFOÑTÑ[FSJOEFEJS \"ôSMLNFSLF[JLÌöFZF B E C | EC | DN O CSJTFLFOBSBCSV[BL 2 MLUBES | | | |DA =DNPMEVôVOBHÌSF \"$ LBÀDNEJS 4G 6 |GO| = 2 ÷ÀUFSTBÀMBSEBO\"%1WF1$&JLJ[LFOBSÑÀHFOMFSEJS S=DN |\"%| = |\"1| = 6 |EC| = |PC| = 4 |\"$|=DN 2. 4 3. 28 4. 10 106 5. 36 6. 6
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 7 % OPLUBT \"#$ ÖRNEK 9 1 OPLUBT \"#$ п üçgeninin çevrel A ¿FNCFSJOJO NFSLF A 2 7 x=7 [JEJS geninin EJLMJL NFSLF [JEJS P g & ABC 3x – 6 m ( % ) = 3x - 6 B ABC B 2 H 5D 7 C E C .y D P | |[\")] m [ BC] [#)] =DN \") = 2 7 DNPMEVôVOB | |HÌSF \"% =YLBÀDNEJS :VLBSEBLJ WFSJMFSF HÌSF Y JO BMBCJMFDFôi FO CÑZÑL UBNTBZEFôFSJJMF en küçük UBNTBZEFôFSJOJOUPQ ^ 2 7 h2 = 2.^ 2 + y h jZ=DNÌLMJE MBNLBÀEFSFDFEJS |\"%| = |#%| = |DC| j |DC| =DNNVIUFöFNÑÀMÑ 90° < 3x - 6 < 180° 96° < 3x < 186° 32 < x < 62° 33° + 61° = 94° ÖRNEK 8 A 0 OPLUBT \"#$ ÖRNEK 10 üçgeninin çevrel 3a 2a x ¿FNCFSJOJONFSLF[J A 8 % O m ( OAB ) = 3a 3a 2a D8 E a a % = a P B m ( OBC ) C K 7 6 | |% BF C m ( OCA ) = 2a AB =DNPMEVôVOBHÌSF 1OPLUBT\"#$пH FOJOJOEJLMJLNFSLF[ JEJS | |\"$ = x LBÀDNEJS | |[ BE] a [ AF] a [ DC] = {P} DE =DN A 12a = 180°j a = 15° | | | | | |EF = DN DF = DN PMEVôVOB HÌSF DK LBÀ |\"$|= 4 6 DN DNEJS 8 43 46 1OPLUBT%&'ÑÀHFOJOJOJÀUFôFUÀFNCFSJOJONFSLF[J 60° 45° DK 8 j 3 |DK| = 4 |,'| B C = KF 6 |KE|BÀPSUBZOEBO |DK| =DN 7. 7 8. 4 6 107 9. 94 10. 4
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ÖRNEK 11 ÖRNEK 14 A , OPLUBT \"#$ п A 1 OPLUBT \"#$ 6a HFOJOJO EJLMJL NFSLF 45° üçHFOJOJO EJLMJL [JEJS 43 b 15° NFSLF[ JEJS K [ BD] a[ AE] = {K} D x =4 6 % b b | |AK =DN m ( PAB ) x = 15° 45° 12 % m ( PBA ) a P 45° = 30° B8 C E2 | |C BE =DN 30° 45° 45° | |BC = 12 2 DN | | | |EC =DNPMEVôVOBHÌSF KE = x LBÀDNEJS B F 12 2 #&,_\"&$ | |:VLBSEBLJWFSJMFSFHÌSF \"1 =YLBÀDNEJS x2 |#1|WF|\"'|V[BUMSTB = 8 6+x 45-45-ÑÀHFOMFSJWF- 60-ÑÀHFOJPMVöVS Y +Y = 16 j x =DN x = 4 6 cm ÖRNEK 12 A ,OPLUBT\"#$п- x 5k HFOJOJOEJLMJLNFSLF[J 10 a C [ AD] a [ KC] = {D} ÖRNEK 15 90°+a a \"õBóEBLJBõBNBMBSJ[MFOFSFLCJSHFPNFUSJL¿J[JNZBQ- 4k B MZPS | |AB =DN 6 D 3k | |BD =DN K % r #JS\"#$пHFOJ¿J[JMJZPS | |m(ABC) = 90 + % PMEVôVOB HÌSF \"$ = x r \"#$пHFOJOJOJ¿UFóFU¿FNCFSJOJONFSLF[JIPMNBL m ( ACB ) Ð[FSF #I$пHFOJ¿J[JMJZPS LBÀDNEJS |#$| BÀPSUBZES r IOPLUBTOEBO#I$пHFOJOJO¿FWSFM¿FNCFSJOF¿J- [JMFOUFóFUEPóSVTV\"#LFOBSO.OPLUBTOEB \"$ DC 6 LFOBSO/OPLUBTOEBLFTJZPS = , 4k = 16 j k = 4, x =DN AC 10 #VOBHÌSF MA PSBOLBÀUS ÖRNEK 13 NA A ABC üçgen A a x b <1%>m<#$> m ( I%CB ) = m(M%IB) %% E <1&>m<\"#> m^ IBC h = m^ NIC h P |AE| = |EB| M a+b a+b N |BD| = |DC| b a BZOZBZHÌSÑZPSMBS a b 25° 25° m ( P%BD ) = 25° MA =1 B DC a b Ba b NA % C BAC PMEVôVOBHÌSF m ( ) = x LBÀEFSFDFEJS |1\"|WF|PC|ÀJ[JMFSTFJLJ[LFOBSÑÀHFOMFSPMVöVS 2a + 2b + 50°= 180° a + b = 65° j a = 65° 11. 2 12. 20 13. 65 108 14. 4 6 15. 1
·ÀHFOEF.FSLF[MFS TEST - 47 1. A 4. A D rM 60° EK a BF C BD C ôFLJMEF \"#$ CJS пHFO & J¿ B¿PSUBZMBSO LFTJN .NFSLF[MJ¿FNCFS\"#$EJLпHFOJOJOJ¿UFóFU OPLUBT m % = 60°PMEVôVOBHÌSF m % (AED) (AFD) ¿FNCFSJEJS%EFóNFOPLUBT| BC | =CSWF¿FN- LBÀEFSFDFEJS CFSJOZBS¿BQCSPMEVóVOBHËSF \" # $ % & | |\"%LBÀCJSJNEJS \" 2 10 # $ % & 65 2. A 5. A D E K C B F E 7 5 D B4 C ôFLJMEFLJ\"#$пHFOJOJOEõUFóFU¿FNCFSJOJONFS- , \"#$пHFOJOJOBóSMLNFSLF[JEJS LF[J&EJS #VOBHÌSF , %&'ÑÀHFOJOJOOFTJEJS | AB | =CJSJN | AC | =CJSJNWF| BC | =CJSJN \" %JLMJLNFSLF[J # \"óSMLNFSLF[J | |PMEVôVOBHÌSF #% LBÀCJSJNEJS $ ö¿UFóFU¿FNCFSJOJONFSLF[J % ¥FWSFM¿FNCFSJOJONFSLF[J \" # $ % & & ö¿CËMHFTJOEFIFSIBOHJCJSOPLUB 3. A I 6. \"#$пHFOJOEFs_WBi > 90°EJS#VпHFOJOZÐLTFL- B HN C MJLMFSJOJOUBõZDMBSE1 E2 E3EPóSVMBSES E1 E2 E3 EPôSVMBS % OPLUBTOEB LFTJöUJôJOF \"#$пHFOJOEF*J¿B¿PSUBZMBSOLFTJNOPLUBTES HÌSF # OPLUBT \"%$ ÑÀHFOJOJO IBOHJ FMFNBO PMVS | A | = 2 | IN | | AB | + | AC | =DN \" \"óSMLNFSLF[J # ¥FWSFM¿FNCFSJONFSLF[J [\")] m [BC] |\")| =DNPMEVôVOBHÌSF $ ö¿UFóFU¿FNCFSJONFSLF[J % %JLMJLNFSLF[J A & LBÀDN2EJS & %õUFóFU¿FNCFSMFSJOEFOCJSJOJONFSLF[J (ABC) \" # $ % & 1. \" 2. D 3. D 109 4. # 5. # 6. D
TEST - 48 Üçgende Merkezler 1. A 4. A 70° 40° I E F B DC x 25° 25° B DC | | | | | |ABC üçgen, AB , AC ` Z+, BC = 8 br ABC üçgeninde, m ( % ) = 70°, m ( % ) = 40° BAD DAC * OPLUBT JÀ UFôFU ÀFNCFSJO NFSLF[J PMEVôVOB % = % = 25°, ma % k = x m ( ACB ) m ( ECA ) EDA AI :VLBSEBLJWFSJMFSFHÌSF YBÀTLBÀEFSFDFEJS göre, PSBOen azLBÀUS ID A) 20 B) 25 C) 30 D) 40 E) 45 A) 1 B) 9 C) 5 D) 11 E) 3 8 4 82 5. DaSB¿MCJS\"#$пHFOJOEF 2. A K, ABC üçgeninin r [CD], [AF] ve [BE] yükseklikleri çiziliyor. D, E, F diklik merkezi ZÐLTFLMJLBZBLMBSES | AC | = 6 br r % & 'OPLUBMBSCJSMFõUJSJMFSFL%&'пHFOJPMVõ- UVSVMVZPS 6 | KC | = 5 br K | BK | = 3 br r [DF] m [BE] = {M} 35 | || DE | = 6 br, |EF| =CSPMEVôVOBHÌSF DM , | |MF OJOLBÀLBUES BC #VOBHÌSF \"MBO #$\", LBÀCJSJNLBSFEJS A) 6 6 B) 6 3 C) 5 5 D) 12 E) 4 10 A) 2 B) 1 3 E) 3 C) D) 2 32 3. K 6. A KE 25 14 F3 a t b x B t A B D 6C L 25 M ABC bir üçgen, m ( B%EA ) = m ( B%ED ) KLM bir üçgen; \"BóSMLNFSLF[J #J¿UFóFU¿FNCF- || || % ED = 3 cm, DC = 6 cm, m^ KCA h = a rin merkezi, | KL | = | LM | = 25 br, | KM | = 14 br ma % k = b PMEVôVOB göre, a PSBO LBÀB KCB b | |PMEVôVOBHÌSF AB =YLBÀCSEJS FöJUUJS 2 3 5 6 7 A) 1 B) 1 C) 1 D) 2 E) 3 A) B) C) D) E) 3 2 3 4 2 5 2 1. B 2. C 3. B 110 4. E 5. A 6. C
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ·¦(&/%&\"-\"/* TANIM ÖRNEK 2 A ABC üçgen #JSпHFOJOBMBO CJSLFOBSOOV[VOMVóVJMFP 10 < AD>B¿PSUBZ LFOBSBBJUZÐLTFLMJóJO¿BSQNOOZBSTES 6 A B D6 < AC> m < BC> c ha b | |AB =DN | |DC =DN C hb hc :VLBSEBLJWFSJMFSFHÌSF A^ & h LBÀDN2EJS ABD BDC |\"%|BÀPSUBZEBOBÀLPMMBSOBJOEJSJMFOEJLNFMFSFöJUUJS a & 10.6 A^ ABD h = & a.ha b.hb c.hc 2 = 30 2 A^ ABC h = cm = = 222 %JL·ÀHFOJO\"MBO ÖRNEK 3 C A ABC üçgen a & a.h = b.c b A^ ABC h = 22 <\"%>B¿PSUBZ h BI=CD 22–x x <\"$>m<#$> A cB 4 | AB | + | AC | =DN &öLFOBS·ÀHFOJO\"MBO | |DC =DN A B D4C :VLBSEBLJWFSJMFSFHÌSF A^ & h LBÀDN2EJS ABC 30° & a2 3 |\"%|BÀPSUBZEBOBÀLPMMBSOBJOEJSJMFOEJLNFMFSFöJUUJS A^ ABC h = a4 && & a A^ ABD h + A^ ADC h = A^ ABC h a3 2 4.^ 22 - x h + x.4 = 44 2 BaD 22 cm 2 60° aC 2 ÖRNEK 4 ÖRNEK 1 A A ABC üçgen < AB> m < AC> 10 |AB| = |AD| =DN 16 D <%&>m<#$> 8 10 | |BD =DN || ||8 10 AB =DN DC =DN B 6 12 6 D | |DC =DN | |B x E 6 C EC =DN 4C :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS | |:VLBSEBLJWFSJMFSFHÌSF #& =YLBÀDNEJS ADC \"EBOEJLJOEJSFMJN A^ BDC h = 10.16 = ^ 6 + x h.8 = 14 cm 22 & 4.8 = 16 2 A^ ADC h = 2 cm 1. 16 111 2. 30 3. 44 4. 14
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ÖRNEK 5 ÖRNEK 7 A ABC üçgen A ABC bir üçgen [ AD> m< BC>å <\"#>åm<#%> 4 10 <\"%>m<$&> 8E | |E AE =DN | |AB =DN | |BC =DN x | |BC =DN | |CD =DN Bx D 10–x C B2C 4 D :VLBSEBLJWFSJMFSFHÌSF UBSBMBMBO LBÀDN2 EJS | |:VLBSEBLJWFSJMFSFHÌSF CE =YLBÀDNEJS && A^ ACD h = 10.x = 4.8 16 A^ ABE h + A^ AEC h =5BSBMBMBO & x = cm 4.x + 4.^ 10 - x h = 20 2 22 5 22 cm %m/*m A E BD C BC . AE ÖRNEK 8 A^ ABEC h = 2 A ABC bir üçgen ÖRNEK 6 % = 30° m ( ABC ) A 10 8 | |AB =DN 17 5 | |CD =DN 8 \"#$EJLпHFO xH <\"%>åB¿PSUBZ 30° C | |6 D AD =DN | |AB å=DN B | |AC =DN B 8k D 17k C #VOBHÌSF $OPLUBTOO[\"%]OBPMBOFOLTBV[BL MôLBÀDNEJS 15 :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS \"EBO|#%|ZFEJLJOEJSJMJQ- 60 -PMVöUVSBMN ABD A^ ACD h = 6.5 = 8.x 15 & x = cm \"ÀPSUBZEBO|#%| = 8k 22 4 |DC| =L1JTBHPSUFPSFNJOEFO |#$| = 15 3 25k = 15 , k = 5 8· 8.3 & 5 = 96 2 A^ ABD h = 2 5 cm 5. 20 96 112 16 15 6. 7. 8. 5 54
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 9 ÖRNEK 11 A (FPNFUSJ EFSTJOEF CJS FULJOMJL ZBQBO ¥BóMB ±óSFUNFO BõBóEBLJ BENMBS J[MFZFSFL CJS HFPNFUSJL ¿J[JN ZBQU- 4 SZPS 4 r [AB] m [AC]PMBDBLõFLJMEFCJS\"#$пHFOJ¿J[JMJZPS bb a B Eb aC r E ` [AC]PMBDBLõFLJMEF[AC]Ð[FSJOEFCJS&OPLUB- 2 TTF¿JMJZPS D r D ` [BC]PMBDBLõFLJMEF[BC]Ð[FSJOEFCJS%OPLUB- [AB] m [AC] [AD] m [DC] m ( % ) = m ( % ) \"#$WF TTF¿JMJZPS ACB BCD | | | |r AB = 2 DE =DN r m ( A%EB ) = m ( B%ED ) :VLBSEBLJWFSJMFSFHÌSF A ( B%ED )LBÀDN2EJS | | | |\"%$CJSFSпHFO DC = AB =DN A :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS |#&| BÀPSUBZ PMEV AEC ôVOEBO 8E #EFOJOEJSJMFOEJLNF \"#&JLJ[LFOBSÑÀHFO|\"&| = 4 4 BD C MFSFöJU|\"#| = |#'| A^ AEC h = 4.2 = 4 2 |#'| = 8 2 cm 8 F A^ BED h = 8.4 = 16 2 2 cm ÖRNEK 10 A ABC ve BDE üçgen D < AB > m < BC > ÖRNEK 12 25 #%&FõLFOBSпHFO Bir ABC üçgeni çizJMJQ< AB >EPóSVTVÐ[FSJOEFCJS%OPL- 83 . | |BC =DN 20 E UBTTF¿JMJZPS%OPLUBTJMF$OPLUBTCJSMFõUJSJMJZPS h % m ( ADC ) 45° | | | |DC DN = AB DN PMNBL B 15 | |C AB =DN Ð[FSF \"#$ÑÀHFOJOJOBMBOLBÀDN2EJS PMEVôVOBHÌSF A^ & h LBÀDN2EJS A BDE A^ ABC h = 20.15 = 25.h j h =DN 14 E EDC 45-45-90 üçgeni 22 D 45° |EC| = 5 2 B |DE| = 8 3 DN 52 12 10 A^ ABC h = 14.5 2 = 35 3 DN2 43 A^ BDE h = 12.8 3 2 2 B C = 48 3 cm 60° D 43 E 9. 4 10. 48 3 113 11. 16 12. 35 3
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr A ÖRNEK 13 ÖRNEK 15 D A 30° A3 23 30° 3 10 10 9 E43 6 6 60° 63 6 30° 3C B C B 5D F 23 C 54 \"#$FõLFOBS #'%EJLпHFO [DF] m [BC] B5 D ôFLJM* | | | |DE = DN WF FC = 2 3 DN PMEVôVOB HÌSe, B' & A^ ABC h LBÀDN2EJS ôFLJM** \"EBO|#'|WF|%'|ZFEJLNFMFSJOEJSJQ \"#$пHFOJOEFUBSBM\"#%пHFOJ[AD]CPZVODBLBUMBO- 30-60-ÑÀHFOMFSJFMEFFEFMJN EóOEB#OPLUBTõFLJM**EFLJ#hOPLUBTOBHFMNFLUFEJS A^ ABC h = 9.6 3 = 27 2 2 3 cm | | | | | |C ` [\"#h] AB =DN AC =DN BD =DN :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS DCBl [\"#], [\"%]CPZVODBLBUMBOEôOEB[\"$]EPôSVTVZMBÀB LöUôOBHÌSF |#h$| = 10 - 6 = 4 ÖRNEK 14 A^ DCB h = 3.4 2 A 2 = 6 cm 105° x 8E F3 4 ÖRNEK 16 45° A \"#$EJLпHFOJOEF B 6 D 42 C 6 D [ BA ] m < CA > | | | | | |\"#$пHFO BD = DN DC = 4 2 DN AF = DN x+2 45° b 10 D ` <AC> || % % b–45° 6 E ` < BC > AE = x DN m ( ABC ) = 30° m ( BAC ) = 105° ve b–45° b | |\" \"'%& =DN2 PMEVôVOBHÌSF \"& =YLBÀDN B x E 2 10 8 C m ( B%DE ) = 45° EJS | | | | | |AD =DN EC =DN DC =DNPMEVôVOB %OPLUBTOEBO|\"'|WF|\"$|ZFEJLNFMFSJOEJSJQ HÌSF A ( B&DE )LBÀDN2EJS 30-60-WF-45-FMEFFEFMJN %&$ JLJ[LFOBS ÑÀHFOJOJO UBCBO BÀMBS b PMTVO EJôFS && BÀMBSbDJOTJOEFOZB[MSTB|#%|BÀPSUBZES A^ AFD h + A^ ADE h =\" \"'%& 8.3 4.x Y+ 2 + 162 = +Y 2 x =DN + = 18 A ( B&DE ) = x.6 = 10.6 =DN2 22 x =DN 22 13. 27 3 14. 3 114 15. 6 16. 30
·ÀHFOEF\"MBO* TEST - 49 1. A 4. A ABC üçgen [DF] m [AB] [DE] m [AC] D | |DF =DN BHC | |F E DE =DN | |\"#$пHFOJOEF [\")] m [BC] BC = 12 br ve BD | |AB =DN | |AD = 9 br PMEVôVOB HÌSF UBSBM BMBO LBÀ CS2 C EJS \" # $ % & | |AC =DN :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS ABC \" # $ % & 2. A ABC üçgen 5. A [AB] m [BC] |AB| = |AD| =CS 8 | |BE =DN | BD | = 16 br | |EC =DN 17 17 | DC | = 8 br | |DB =DN | |AD =DN B 16 D8 C D F C 4 B3E 6 :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀCS2EJS ADC \" # $ % & :VLBSEBLJ WFSJMFSF HÌSF UBSBM BMBOMBS UPQMBN LBÀDN2EJS \" # $ % & 3. A 10 6. A D 8 60° BC m ( A%BD ) % % BD C m ( DBC ), m ( ACB ) = = 60° | |\"#$пHFO [AD] m [AC] BD =DN | |CD = DN m(A%BC) = 2m(A%CB) | | | |AB =DN DC =DN #VOBHÌSF A^ & hLBÀDN2EJS & LBÀDN2EJS ABD ADC h \" 12 3 # 16 3 $ 18 3 :VLBSEBLJWFSJMFSFHÌSF A^ % 20 3 & \" # $ % & 1. C 2. E 3. D 115 4. D 5. C 6. #
TEST - 50 ·ÀHFOEF\"MBO* 1. A 4. NFUSF DN DN 15° 6 C ôFLJM* ôFLJM** B \"#$пHFOJOEF [AC] m [AB] m ( % ) = 15° ve #JSBMõWFSJõNFSLF[JOJOPUPNBUJLB¿MSLBQTÐ[F- ABC SJOEFSFLMBNJ¿JOPMVõUVSVMBOJLJ[LFOBSпHFOõFL- MJOEF CJS QMBUGPSN PMVõUVSVMNVõUVS ,BQMBS Fõ WF | |BC = DN PMEVôVOB HÌSF A^ & h LBÀ DN2 ôFLJM*EFLBQBMIBMJWFSJMNJõUJSôFLJM**EFJTFLBQ- ABC MBSOB¿MNõIºMJWFSJMNJõUJS EJS \" # 7 $ % 9 & 2 2 2. ,BQMBS LBQBM JLFO QMBUGPSNVO BMBOOO LBQMBS BÀMEôOEBLJ QMBUGPSNVO HÌSÑOFO CÌMHFMFSJOJO BMBOMBSOBPSBOLBÀUS \" 9 # 16 $ 25 % 3 & 9 25 25 92 4 A B D C ôFLJMEFLJ ¿BUOO \"#$ пHFOJ õFLMJOEFLJ CËMHF- 5. A TJ BõONBEBO EPMBZ CPZBOBDBLUS #PZBDMBS \"#$ пHFOTFMCËMHFOJOBMBOOIFTBQMBNBLJ¿JOËM¿ÐN- DE | | | |MFSTPOVDVOEB AB =N AC =N B C | | | |2 BD = 3 DC PMEVóVOVHËSÐZPS[AD] m [AC] ôFLJM* #VOBHÌSF \"#$ÑÀHFOTFMCÌMHFTJOJOBMBOLBÀ NFUSFLBSFEJS \" # $ % & D E 3. A F 17 B CK ôFLJM** B 27 D 6 C ôFLJM*EFWFSJMFO\"#$пHFOJ$LËõFTJOEFO * *\"#$пHFO m(A%BC) = 2m(A%CB) AB =DN | |[DE] // [AB]PMBDBLõFLJMEF DE EPóSVTVCPZVODB * * * *BD =DN DC =DNPMEVóVOBHËSF LBUMBOEóOEB\"WF,OPLUBMBS¿BLõNBLUBES & LBÀDN2EJS A^ ADC h | | | | | | | |3 DF = 2 FK DE =DN BD =DN \" # $ % & 0 PMEVôVOBHÌSF %&,ÑÀHFOJOBMBOLBÀDN2EJS \" # $ % & 1. D 2. E 3. # 116 4. C 5. C
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, TANIM ·¦(&/%&\"-\"/** ÖRNEK 2 öLJLFOBSJMFCVLFOBSMBSBSBTOEBLJB¿TCFMMJ A \"#$ WF %\"& FõLF- OBSпHFOMFSEJS PMBOпHFOJOBMBO 3300°° m ( A%DB ) = 90° A 30° 83 | AB | = 16 br cb 16 12 8 3 60° E 43 K 60° B DC B aC :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀCS2EJS AKE A^ ABC h = 1 AB . AC . sin A 2 30-60-ÑÀHFOMFSJOEFO A^ AKE h = 12.4 3 = 24 3 = 1 AC . BC . sin C 2 2 = 1 AB . BC . sin B 2 A D a BH C ÖRNEK 3 AD . BC . sin a A ABC üçgen A^ ABCD h = 2 x ba m ( B%AC ) = 135° ÖRNEK 1 x% 828 m ( AEC ) = 45° A D ABC üçgen 3k 8 45° b | AD | = | AC | <\"&>B¿PSUBZ | |B E 8 C | AB| = | EC| AE = 8 2 br | BD | = DN :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀCS2EJS DAE 8 | AC | = DN \"EBOEJLNFJOEJSJQ-45-ÑÀHFOJPMVöUVSBMN k % B 3 D 45° C m ( BCE ) = 45° & 1 2 · sin b A^ DAE h = x·8 2 8 3k sin b = E x & A^ DAE h = 1 8 = 32 2 ·x.8 2· & 2x 2 br DEC PMEVôVOBHÌSF A^ hLBÀDN2EJS A^ DEC h = 1 · 8 ·3k· sin 45° 2k \" %&$ = 6 2 DN2 1. 6 2 117 2. 24 3 3. 32 2
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ÖRNEK 4 ÖRNEK 6 A ABC üçgen A ABC üçgen a b [BA] m [AC] n |AD| = |DE| = |EB| kS F 12 [AD] m [BD] D n |AF| = |FK| = |KL| = |LC| 20 14 k K | |AD =DN D E 5S n \" %&-' =DN2 k L b 16 | |AC =DN 6S n C BC BD =DN | |B & ABC hLBÀDN2 :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2 EJS :VLBSEBLJWFSJMFSFHÌSF A^ EJS ADC 1JTBHPSEBO|\"#| = 20 sin b = 12 A^ ADF h = 1 ·k.n. sin A = S 2 20 A^ ADC h = 1 ·12.14. sin b = 1 ·12.14. 12 A^ AEL h = 1 ·2k.3n. sin A = 6S 2 2 2 20 A^ ABC h = 1 ·3k.4n. sin A = 12S = 252 2 2 5 cm 4= 35 j4= 7 12 . 7 =DN2 ÖRNEK 7 ÖRNEK 5 A A ABC üçgen 4 6 ACE ve ABF üçgen 2 B | |AD =DN 5 E | |AB =DN AD | |DC =DN Dx | |BC =DN E | |BC =DN | |AE =DN | |EF =YDN 3 && AS A^ ABC h = A^ DFC h && 4.A ^ ACE h = 3.A ^ ABF h Fx B4 C CF | |:VLBSEBLJWFSJMFSFHÌSF '# =YLBÀDNEJS :VLBSEBLJWFSJMFSFHÌSF YLBÀDN EJS && 4Aa A&CE k = 3.A^ A&BF h A^ ABC h = A^ DCF h 4· 1 ·9·6· sin XA = 3· 1 ·4·^ 6 + x h. sin XA 1 ·5.4. sin C = 1 ·3.^ 4 + x h. sin C 22 22 20 = 12 + 3x 4.9.6 = +Y 8 x =DN x = cm 3 252 8 118 6. 84 7. 12 4. 5. 5 3
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 8 \"ZO UBCBOB BJU пHFOMFSJO BMBOMBS PSBO ZÐLTFLMJL PSBOMBSOBFõJUUJS A \"#$пHFOJOEF AD <\"$> FõJU m <#$> å FõJU QBS¿BZB D m BZSMNõUS 12S m 9S m B H1 C H 3S B n n En n n Fn C & & AH1 A(DEF) A^ ABC h #VOBHÌSF & PSBOLBÀUS &= A(ABC) A^ BCD h DH m (XC) TBCJUUVUVMVQÑÀHFOMFSJOLFOBSMBSOOÀBSQNOO Benzer iki üçgenin benzerlik PSBOOOLBSFTJBMBOMB- PSBOMBSBMBOMBSPSBOOWFSFDFLUJS SPSBOOBFõJUUJS & 1 ·n.3m. sin C = 3S Am B [AB] // [DE] A^ DFC h = C 2 & A^ ABC h m2 & 1 ·4n.3m. sin C = 12S = k2 A^ DCE h = & 2 A^ DCE h & 1 A^ ABC h = ·4m.6n sin C = 24S 2 & A^ DEF h 9S 3 = = & 24S 8 A^ ABC h D kE \"MBOJMF÷MHJMJ²[FMMJLMFS \"ZOZÐLTFLMJóFTBIJQпHFOMFSJOBMBOMBS UBCBOMBS ÖRNEK 9 JMFPSBOUMES A ABC üçgen A 7k [AB] m [BC] 9 7S E 5 |AE| = | EC| S 3S 2S 5k | |AB =DN 5S | |B Bk 3k 2k C D8 C DC =DN C DE d1 E1E2 :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS EDC \"%$ÑÀHFOJOJOBMBO+ 55 12S = 9.8 j4= 3 2 d2 A^ & h =4=DN2 EDC AB &&& A^ ABC h = A^ ABD h = A^ ABE h 3 119 9. 15 8. 8
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ÖRNEK 10 ÖRNEK 13 A \"#$пHFOJOEF A ABC üçgen (BóSMLNFSLF[J 35 | |AD =DN | |DB =DN [BE] a[CD] = {G} D 3S | |AE =DN | |EC =DN D2 G 3E % E m ( BGC ) 4 30° 3 = 150° 12 12S 12S 150° | |DG =DN C 6 | |B C GE =DN B PMEVôVOBHÌSF A^ & h LBÀDN2EJS :VLBSEBLJWFSJMFSFHÌSF A (BDEC) PSBOLBÀUS ABC & (BôSMLNFSLF[J|GC| = 4 A(ABC) & 1 ·3.4. sin 30° = 2 |#&| LFOBSO ÀJ[FMJN WF LFOBS PSBOMBSOB HÌSF BMBO A^ GEC h = 2 3 cm PSBOMBSOZB[BMN A^ BDEC h 24 8 A^ & h =DN2 ABC == A^ ABC h 27 9 ÖRNEK 11 A ABC üçgen 6 | AD| = | DE| D 12 DF = 1 FC 3 3 F | BE| = 2 | EF| 6 9 E 18 & h = DN2 A^ DEF BC :VLBSEBLJWFSJMFSFHÌSF & LBÀDN2 EJS ÖRNEK 14 A(ABC) ,FOBSPSBOMBSOBHÌSFBMBOMBSöFLJMÑ[FSJOEFHÌTUFSFMJN A ABC üçgen A^ & h =DN2 (BóSMLNFSLF[J ABC [DE] // [BC] ÖRNEK 12 4S 2n G ` < DE > A ABC üçgen D 2k E 5k G [BD]B¿PSUBZ n 5S D 5S 5 |BE| = |ED| | |AB =DN B 3k C | |BC =DN 5S 7k A (BDEC) E :VLBSEBLJWFSJMFSFHÌSF & PSBOLBÀUS 14S A(ABC) B7 C (BôSMLNFSLF[J|\"#| = 2n |EC| =OPMTVO & & A(ABE) A^ ADE h PMEVôVOBHÌSF & PSBOLBÀUS =d 2 2 4 A(ABC) & 3 n= 9 A^ ABC h ,FOBS PSBOMBSOB HÌSF BMBOMBS öFLJM Ñ[FSJOEF HÌTUFSFMJN A^ BDEC h 5 = & & 9 A^ ABE h 5 A^ ABC h = & 24 A^ ABC h 10. 18 5 120 8 5 11. 54 12. 13. 14. 24 9 9
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 15 ÖRNEK 17 A Bh A \"#$%EËSUHFO 2x 12 [BC]å[AD] | | | |3x 2 AD = 3 BC & =DN2 A^ BCD h h D C BC :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2 EJS ABD \"#$ пHFOTFM CËMHFTJ õFLMJOEF CJS UBSMBT CVMVOBO 3F- #EFO|\"%|ZFWF%EFO|#$|ZFEJLJOEJSFMJN õBU\"NDBhOOBSB[JTJOJEFWMFUJTUJNMBLFEFSFL< BC >LFOB- 2x.h 3x.h & = 12 , = A^ ABD h 22 SOBQBSBMFMZFOJCJSZPMEBIBZBQBDBLUS:FOJZPMVOÐTU A^ & h =DN2 ABD UBSBGOEBLBMBOCËMHFOJOBMBOEËOÐN BMUUBSBGOEBLB- MBOCËMHFOJOBMBOJTFEËOÐNEÐS:BQMBOZPMUBSMBOO Ð[FSJOEF%` < AB > &` < AC >PMBDBLõFLJMEF< DE >EPó- ÖRNEK 18 SVTVJMFHËTUFSJMJZPS A \"#$пHFO #VOBHÌSF AD PSBOLBÀUS DB 2k 2n (BóSMLNFSLF[J D 2 A 12 E [DE] // [BC] G n AD p= 4 f 4 DE & =DN2 A^ DEF h AB 25 21 k6 BC AD 2 AD 2 BF C =, = & AB 5 BD 3 ABC :VLBSEBLJWFSJMFSFHÌSF, A^ hLBÀDN2 EJS A^ & h = 12 2 PMVS ADE cm ÖRNEK 16 2 :ÑLTFLMJLMFSJPSBO PMEVôVJÀJO A 3 ABC üçgen & 2k AD = 2 A^ ADE h 2 E 2n D DC 3 =d 2 4 = 12 2 n= 2x & 3 9A & A = 27 cm 5x F 2S 3k A^ ABC h 5S [ED]å/ [BC]å ÖRNEK 19 B 5n C & h =DN2 A A^ ABC 5 DF PMEVôVOBHÌSF UBSBMBMBOLBÀDN2EJS \"#$EJLпHFO B4 E AD ED AE [AB] m [BC] == [DE] // [AC] AC BC AB | |AD =DN ED FD | |BE =DN = C BC BF & A^ ABC h 5k 175 :VLBSEBLJWFSJMFSFHÌSF A^ DEF hLBÀDN2 EJS == A^ BDC h 3k 7S j4=DN2 4= 105 j4= 15 , A^ & h =DN2 'OPLUBT\"OPLUBTOBUBöOSA^ DEF h = 5.4 2 DFC = 10 cm 2 2 16. 30 121 15. 17. 18 18. 27 19. 10 3
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ÖRNEK 20 ÖRNEK 22 A ABC üçgen A ABC üçgen 4k 2n [CD]B¿PSUBZ & =DN2 16 A^ BDF h D4 D 2k E & 20 [AE] m [CD] 3k A^ ADE =DN2 3n E 9 h 12 4 20 2k 3| AC| = 2| BC| BF B kK C & =DN2 C ADE A^ h :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2 EJS PMEVôVOBHÌSF %&$'QBSBMFMLFOBSOOBMBOLBÀDN2 ABC EJS [\"&] EPôSVTV V[BUMODB JLJ[ LFOBS ÑÀHFO FMEF FEJMJS A&DE ` D&BF [DK]CJSMFöUJSJMFSFLBMBOEBôUMS A^ & h =DN2 AD 16 4 ABC = = DB 93 A^ & h =d 4 2 16 ADE n= & 7 49 ÖRNEK 21 Aa ABC k A A \" %&$' =DN2 F BD C DE C ôFLJM* ôFLJM** | AB | = | AC | |= AF | = | FC | | DE | = | EC | ÖRNEK 23 ôFLJM*EFLJEJLпHFOTFMCËMHFTJCJ¿JNJOEFLJLºóU A ABC üçgen | | | |BD = DC PMBDBLCJ¿JNEF\"%CPZVODBLBUMBOZPS BS- k [AD] a [BE] = {K^ [AD]åLFOBSPSUBZ EOEBOôFLJM**EFLJ&'CPZVODBLFTJMJQTBóUBSBGUBLJп- 3S S E HFOCJ¿JNJOEFLJQBS¿BBUMZPS n | AK|å= | KD|å #VOB HÌSF TPO PMBSBL FMEF FEJMFO L»ôU BÀMEôOEB 3n K 2S 2k PMVöBOöFLMJOBMBOLBÀCJSJNLBSFEJS 3S 3S B DC ,FOBSPSBOMBSOBHSFBMBOMBSOPSBO & :VLBSEBLJWFSJMFSFHÌSF A(AKB) PSBOLBÀUS A A kk Alan (KDCE) F' S SF 20 kk %OPLUBTOEBO|#&|ZFQBSBMFMÀFLFMJN 3k 3k 20 3k A^ AKB h 3 3S 4S 4S 3S 3S 10S = 3k k A^ KDCE h 5 3S B E' D E C BC n k 16S = 20.20 & S= 25 3n 2n k 22 25 2 mm \" \"'&&h'h =4= 10· = 125 cm 2 20. 60 21. 125 122 22. 24 23. 3 5
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 24 ÷À5FôFU¦FNCFSWF\"MBO÷MJöLJTi %m/*m A \"#$CJSEJLпHFO A u= a+b+c 2 12–x N ` < AB > .` < BC > r =ö¿UFóFU¿FNCF- SJOJOZBS¿BQ x P /WF.OJO#ZFPMBO c b& N x V[BLMLMBSFõJUUJS r A^ ABC h =VS x 4=VS BxM | |C AB =DN 20 | |BC =DN BC a /#. пHFOJOJO <./> ZF HËSF TJNFUSJóJ BMOBSBL /1. & пHFOJPMVõUVSVMN VõUVS A A^ ABC h =NO #VOBHÌSF A^ & hLBÀDN2EJS NMP O 5FNFMPSBOU5FPSFNJ Bm D nC 12 - x x A 414243 =B = b:c 12 20 cb 15 x= 2 A^ NMP h = 225 8 S3 I S2 S1 ÖRNEK 25 Ba C \"#$пHFOJOEF m (A%BC) = m, m (A%CB) = n, | | | |AB =DN BC =DNWFO+N=PMEVôV ¦FWSFM¦FNCFSWF\"MBO÷MJöLJTJ OBHÌSF A^ & hLBÀDN2EJS %m/*m ABC A 3 ¥FWSFM ¿FNCFSJO \"LÌöFTJOEFOOLBEBSMLBÀBMBMN ZBS¿BQ A & 2n +N= 90 A^ ABC h = a.b.c 4n +N= 180 4 n cO b 4.R 6 x R A Ba m 8–x 2n xn C B 5 3C 8 A -Y 2 = x2 + 42 1JTBHPSUFPSFNJOEFO n mb cD x =DN F p 56 = z 3A 18 B xE yC A= a 5 & 48 & A^ ABC h = 5 A^ DEF h x.z.n + y.m.p &= A^ ABC h a.b.c 225 48 123 24. 25. 8 5
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ÖRNEK 28 A U= a+b+c A \"#$пHFOJOJO<#$>ZF cb 2 HËSF TJNFUSJóJ BMO BSBL 13 13k #1$ пHFOJ PMVõUVSVM- B 13S NVõUVS B aC K <\")>m<1$> & 5S 18S 5 | |), =DNWF A^ ABC h = u.^ u - a h^ u - b h^ u - c h P 8k | |H 5k C KA =DN & LBÀDN2 EJS PMEVôVOBHÌSF A^ KAB h ÖRNEK 26 A 3 12k = 18 j k = j 8k = 12 9 2 B6 A ABC üçgen 13k 36S = 18.12 j4= 3 18 | |AB =DN 2 5 A^ & h =DN2 ABK | |AC =DN | |BC =DN H 5k C C :VLBSEBLJWFSJMFSFHÌSF A^ & LBÀDN2EJS ABC h 9+6+5 u = = 10 2 A^ ABC h = 10.^ 10 - 6 h^ 10 - 9 h^ 10 - 5 h = 10 2 ÖRNEK 27 ÖRNEK 29 A | |AB å=DN | |AC å 12 cN A ABC üçgen 9 12 7 | |AB =DN 3S I 4S 5 | |BD =DN 135° | |DC =DNWF 5S 66 36 B 15 C \"#$пHFOJOJO#WF$B¿MBSOOJ¿B¿PSUBZMBSIOPLUB- B6 D3 C | |AD =DN TOEBLFTJõJZ PS & % ) = 135° PMEVôVOB HÌSF UBSBM BMBO LBÀ DN2 :VLBSEBLJWFSJMFSFHÌSF A^ ADC hLBÀDN2 EJS m ( BIC EJS % 12S = 9.12 9 \"#%ÑÀHFOJOJOÀFWSFTJDN V=DN m ( BAC ) = 90° , & S= A^ ABD h = 9.^ 9 - 7 h^ 9 - 6 h^ 9 - 5 h = 6 6 22 A^ ADC h = 3 6 63 2 7S = cm 2 26. 10 2 63 124 28. 39 29. 3 6 27. 2
www.aydinyayinlari.com.tr ÜÇGENLER 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 30 ÖRNEK 32 A ABC üçgen A ABC üçgen | AB | = | AC | =DN 3k 2|AE| = 3 |BE| | |BC =DN |BD| = |DC| 3t 15 15 9S r r E 2k 6S F 5S t 5S B9 9C B m D mC & :VLBSEBLJ WFSJMFSF HÌSF JÀ UFôFU ÀFNCFSJO ZBSÀBQ LBÀDNEJS A^ AFE h :VLBSEBLJWFSJMFSFHÌSF & PSBOLBÀUS A^ DFC h 1JTBHPSUFPSFNJOEFO\"EBOJOEJSJMFOEJLNFDNEJS & 18.12 = 108 A^ ABC h = 2 CD BE AF · · = 1.FOFMBWTUFPSFNJ & A^ ABC h =VS= 108 CB EA FD 15 + 15 + 18 1 2 AF =1 u = = 24 ·· 2 3 ED 2 9 |\"'| = 3 |ED| S= 108 j r = cm 2 ÖRNEK 31 & A^ AFE h 9 = & 5 A \"MJ TVOUBCJSUBIUBÐ[FSJOF A^ DFC h F \"WF#OPLUBMBSBZOEPó- K H SV Ð[FSJOEF PMBDBL CJ¿JN- ÖRNEK 33 L E EFBSBMBSOBFõJUBSBMLMBSMB G D EËSU¿JWJ¿BLZPS\"ZOJõMF- NJ \" WF $ OPLUBMBS BZO B C EPóSV Ð[FSJOEF PMBDBL CJ- ¿JNEFBSBMBSOBп¿JWJ¿B- LMZPS A \" # WF $ OPLUBMBS пHFOTFM CJS CËMHF CFMJSUNFLUFEJS \"MJGBSLMJQMFSLVMMBOBSBL' &WF\"OPLUBMBSOEBLJ¿JWJMF- D 2S 45° E 2S SFHFSHJOCJSJQTBSBSBL\"'&пHFOJOJ\" (WF%OPLUB- MBSOEBLJ¿JWJMFSFHFSHJOCJSõFLJMEFTBSBSBL\"(%пHFOJ- OJ \" #WF$¿JWJMFSJOFHFSHJOCJSJQTBSBSBL\"#$пHF- 45° OJOJOFMEFFEJZPS A^ GDEF h SS 45° 5BIUBEBPMVöBOTPOöFLJMJÀJO A^ BCGD h PSBOLBÀUS B 12 F 12 C 12 S. A \" BÀTO LVMMBOBSBL PMVöBO | | | | | |\"#$пHFOJOEF AD = DE = EB k ÑÀHFOMFSJO LFOBSMBSOO ÀBSQ =| | | | | |ACCF= BF = 12DN % = 45° PMEuôV F 2S 2m N JMF EPôSV PSBOUM öFLJMEF & m ( ACD ) BEF nBHÌSF A^ hLBÀDN2 EJS 3k 10S E BMBOMBSZB[MS [&'] // [DC] // [\"4] PMBDBL öFLJMEF |\"4| ÀJ[JMJSTF JÀUFST 8S m G A^ AFE h = 1 ·k.2m. sin A = 2S BÀMBSEBO\"$4ÑÀHFOJ-45-PMVS k D 2 B m 12.24 A^ AGD h = 1 ·4k.3m. sin A = 12S 6S = C 2 A^ GDEF h 10 5 2 j4=DN2 == A^ ABC h = 1 ·5k.4m. sin A = 20S 2 A^ BCDG h 8 4 9 5 125 9 30. 31. 32. 33. 24 2 4 5
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÜÇGENLER www.aydinyayinlari.com.tr ÖRNEK 34 ÖRNEK 35 A ABC üçgen kD C 45° 30° | AD| = 2 | BD| 2k 2m 3|AE| = 2|EC| k k2 2k 75° 6 4S E 2|BF| = 3|FC| 75° D E 45° 30° 15° 45° 30° A 2k B k 3m \"#$%CJSEËSUHFO [AD] m [BD] m ( % ) = 15° 3A 6B CAD B 3n F 2n C | |% % m ( CAB ) = m ( ACD ) = 30° BC =DN & PMEVôVOBHÌSF A^ EBC hLBÀDN2EJS & A^ DEF h :VLBSEBLJWFSJMFSFHÌSF & LBÀDN2EJS \"EBO|DC|OJOV[BOUTOBEJLÀFLJMJSTF A^ ABC h 45-45-WF-60-ÑÀHFOJPMVöVS A^ EBC h = 1 ·6.6. sin 30° = 9 2 2 cm ZPM ²[FMMJLLVMMBOMSTB & A^ DEF h 2k.3n.3m + k.2n.2m 22 == & 3k.5n.5m 75 A^ ADC h ÖRNEK 36 **ZPM A )FSCJSLÌöFZFHÌSFTJOÑTMÑBMBOGPSNÑMÑOÑÑÀHFOLF ABC üçgen OBSMBSOOÀBSQNZMBJÀMFSJOFPSBOBUBSBLCVMBMN [AB] m < AC > & 1 15 20 A^ ADE h = ·2k.2m. sin A = 4S 12 [AK] m < BC > 2 | |AB =DN & 1 ·5m.3k. sin A = 15S | |C AC =DN A^ ABC h = 2 9 16 & 1 ·k.3n. sin B = 3A A^ BDF h = BK 2 ôFLJM* & 1 ·5n.3k. sin B = 15A A^ ADC h = 2 &LJO ôFLJM * EFLJ LBSUPO пHFOJ [AK] EPóSVTV CPZVODB & 1 ·3m.2n. sin C = 6B LFTJZPS\"#,WF\",$пHFOMFSJOJ,OPLUBTFUSBGOEBT- A^ CEF h = SBTZMBTBóBWFTPMBEËOEÐSEÐóÐOEFôFLJM**ZJFMEFFEJ- 2 ZPS & 1 ·5m.5n. sin C = 25B A^ ABC h = 2 && A #VOBHÌSF &LJO A^ ABC h = 15A =4=# j A^ ABC h =U A' A^ AKA' h 4=UjU= 20 12 12 PSBOOLBÀ A^ KBC h \"=UjU=U 15 a CVMNVöUVS Kb #=UjU=U & 9 B 16 A^ DEF h =U A^ & h = 72 DEF Aa & k 75 ôFLJM** C ABC a + b = 180° sina =TJO# A^ AKA' h 1 = ·12.12. sin a A^ KBC h 2 =1 1 ·9.16. sin b 2 22 126 35. 9 36. 1 34. 75
·ÀHFOEF\"MBO** TEST - 51 1. A B [AE] a[BD] = { C } 4. A C [AB] // [DE] 4D 2|AC| = 3|CE| & A^ ABC h =DN2 B6 C DE \"#$пHFOJOEF %J¿UFóFU¿FNCFSJONFSLF[JEJS & | | | |AB & h =DN2 :VLBSEBLJWFSJMFSFHÌSF A^ CDE h LBÀDN2EJS =DN BC =DN A^ BDC \" # $ % & :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2æEJS ABD \" # $ % & 2. A [BD] a[AE] = { C } 5. A ABC üçgen [ AB ] m [ BD ] [DE] m [AB] 12 [DF] m <AC> | |8 AB =DN | | | |E F AB = AC =DN 10 D | DE | + | DF | = 6 cN BC | |AC =DN BD C | |9 CE =DN | |E CD = 1DN :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS ABC :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2æEJS \" # $ % & CDE \" # $ % & 6. A 2 3. A ABC ve DBC D E 3 пHFOMFSJOEF B 4 C xF D E [AB] // <EF> // <DC> |AB| = 3|DC| \"#$пHFOJOEF [DF] aå[AC] = {E} & | | | |& & A^ EFC h A^ ADE h = A^ ECF h AD =DN DB =DN = DN2 PMEVôVOB HÌSF \"#$%& CFö | | | |BC = DN PMEVôVOB HÌSF $' = Y LBÀ DN HFOJOJOBMBOLBÀDN2EJS EJS \" # $ % & \" # 7 $ 8 % & 10 33 3 1. \" 2. \" 3. E 127 4. D 5. C 6. C
TEST - 52 ·ÀHFOEF\"MBO** 1. A [AB] m [BC] 4. A 8 E6 <%&><#$> N M | |AB =DN L D K | DE | =DN B D EF C BC \"#$CJSпHFO [AC]FõQBS¿BZB [BC]FõQBS- PMEVôVOBHÌSF A^ & hLBÀDN2EJS ¿BZBCËMÐONÐõUÐS ADC \" # $ % & A^ & h =DN2PMEVôVOBHÌSF UBSBMCÌMHF ABC OJOBMBOLBÀDN2EJS \" # $ % & 2. A \"#$пHFOJOEF A DE 5. (BóSMLNFSLF[J [DF] // [BC] D F G [DB] // [AC] K F & A^ DBE h = 6 DN2 B E GL C BC PMEVôVOBHÌSF A^ & hLBÀDN2EJS \"#$пHFOJOEF [KL] // [FG] // [DE] // [AC] ABC |AD| = |DB| |CL| = |LG| = |GE| & \" # $ % & A^ ABC h = 144 DN2 PMEVôVOB HÌSF \" ,-(' LBÀDN2EJS \" # $ % & 3. A 6. A \"#$%EËSUHFOJOEF K 3 [AB] m [BC] E G D [AD] // [EC] L | |AE =DN BD EC | |BC DN BC B 4C \"#$CJSпHFO (BóSMLNFSLF[J DE = 2 KL = AB A^ & h =DN2 PMEVôVOBHÌSF ABC & 3 :VLBSEBLJWFSJMFSFHÌSF A^ DEC hLBÀDN2EJS UBSBMBMBOMBSUPQMBNLBÀDN2EJS \" # $ % & \" 2 5 # 3 3 $ 4 5 % & 1. # 2. D 3. D 128 4. C 5. # 6. D
·ÀHFOEF\"MBO** TEST - 53 1. A \"#$пHFOJOEF 4. A G ( BóSML NFS- D kezi 45° 60° <%&><#$> B D C E <('><\"$> B FC \"#$CJSпHFO [AD]B¿PSUBZ m ( A%BC ) = 45° PMEVôVOBHÌSF & PSBOLBÀUS % = 60°EJS A(ADE) m ( ACB ) A (GECF) A(A&BD) :VLBSEBLJWFSJMFSFHÌSF A(A&DC) PSBOLBÀUS \" # 3 $ % 5 & 2 2 \" 2 6 # 6 $ 3 % 2 & 6 2 2. A D A#$пHFOJOEF K F |AF| = |FE| = |EC| [DE] // [BC] E A^ & h =DN2 5. \"#$EJLпHFOEF< AB > m < AC > < BC >OJOZÐLTFL- DBK MJóJ< AD >EPóSVTV ( \"#$пHFOJOBóSMLNFSLF[J BC | | | |BD =DN DC =DNPMEVôVOBHÌSF #(% & ÑÀHFOJOJOBMBOLBÀDN2EJS FBC PMEVôVOBHÌSF A^ hLBÀDN2EJS \" 10 # 8 % 4 & 2 33 33 \" # $ % & $ 3. A F 6. \"#$EJLпHFOJOEF[AB] m [AC] [AB]Ð[FSJOEFTF- D ¿JMJ&OPLUBTWF[CE] $B¿TOOB¿PSUBZ BE C | | | |BE = BD PMBDBLõFLJMEF[BC]Ð[FSJOEF%OPLUB- | | | |\"#$WF%&' CJSFSпHFO AF = FC TBMOZPS | | | | | | | |AD = DB EC = EB EJS | | | |ED =DN EC =DNPMEVôVOBHÌSF & & = DN2 & A^ EDC hLBÀDN2EJS A^ ABC h PMEVôVOBHÌSF A^ DEF hLBÀ DN2 EJS \" 6 2 # 6 3 $ 12 2 \" # $ % & % 12 3 & 24 2 1. C 2. \" 3. \" 129 4. E 5. D 6. C
TEST - 54 ·ÀHFOEF\"MBO** 1. A ABC üçgen 4. A 30° <%&><#$> D DE m ( B%AE ) = 30° BE FC B * AE * = 6 br * AB * = 10 br C :VLBSEBLJWFSJMFSFHÌSF \" \"%$& LBÀCS2EJS * * * *\"#$CJSпHFO 2 AE = 3 DF [AE] m [BC] [DF] m [BC]PMEVóVOBHËSF AD PSBOLBÀUS \" # $ % & DC \" 1 # 2 $ % 3 & 2 3 2 2. A D 5. ôFLJM*EF\"#$пHFOTFMCËMHFTJCJ¿JNJOEFLJLºóU E [AD]CPZVODBLFTJMJZPS 15 F A 10 A B 20 C 4 K B' C ABC ve BCD CJSFSпHFO [AB] m [BC] B 3D 8C D m (K%CB) = m (A%CB), m (A%CE) = m (E%CD) * *# ' & %OPLUBMBSEPóSVTBM AB =DN ôFLJM* A' ôFLJM** * * * *EC =DN BC =DNPMEVóVOBHËSF \"#% пHFOJOJO % LËõFTJ FUSBGOEB OFHBUJG ZËOEF A^ & h + A^ & hLBÀDN2 EJS EËONFTJZMFZFOJPMVõBOOPLUBMBSJMF#h` [DC] ABF EFC WF\"hOPLUBTõFLJM**EFHËTUFSJMNJõUJS \" # $ % & õFLJM**EFPMVöBO\" \"h$#h LBÀDN2EJS \" # $ % & 3. D C \"#$%LBSF 6. C \"#$%CJSEËSUHFO E8 6 *DK * = 6 br D [DE] // [BC] K [DK] m [KA] 6 * EB * = 8 br A * DA * = 6 br AB [AD] m [AB] B :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀCS2EJS :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀCS2EJS DKC DEC \" # $ % & \" # $ % & 1. D 2. C 3. D 130 4. \" 5. # 6. D
·ÀHFOEF\"MBO** TEST - 55 1. A A 3. )BTBO±óSFUNFO BõBóEBLJBENMBSJ[MFZFSFLCJS B CB C HFPNFUSJL¿J[JNZBQUSZPS r #JS\"#$пHFOJ¿J[JOJ[%` [BC]PMBDBLõFLJMEF [AD] m [AC]¿J[JOJ[ r [DC]OJOPSUBOPLUBT& r [\"&ZJEPóSVTBMCJSõFLJMEFV[BUQ$OPLUBTOEBO [\"&ZFBUMBOEJL'OPLUBTES ôFLJM* ôFLJM** | | | | 'n [\"&], #% =CS '$ =CSPMEVôVOBHÌ ôFLJM * EFLJ HËSTFMJ WFSJMFO ¿BESO ËOZÐ[Ð ôFLJM ** SF A ( & ) LBÀCJSJNLBSFEJS BAD EFLJ\"#$пHFOJõFLMJOEFEJS¥BESOGFSNVBS[AD] EPóSVTVEVS%` [BC]WFGFSNVBSV[VOMVóVNFU- \" # $ % & SFEJS¥BESO[AC]LFOBS 6 3 NFUSFEJS'FSNVB- SO[AC]JMFZBQUóB¿WFm ( % ) = 60°EJS 4. A ABC пHFO õFL- ABC MJOEFLJ LºóU # D LËõFTJOEFO \"& | | #VOB HÌSF ÀBESO \"# V[VOMVôV LBÀ NFUSF EPóSVTV CPZVO- BEF DB LBUMBOEóOEB EJS # JMF % OPLUBMBS ¿BLõNBLUBES 5 # 18 $ 20 % 22 & 24 \" C 7 7 777 2. %JMFL±óSFUNFOLFOBSPSUBZV[VOMVLMBSWFSJMFOп- | | | |[DF] // [AE] AD = DC PMEVôVOBHÌSF HFOJOBMBOOCVMNBOOCJSZPMVOVBõBóEBLJHJCJPM- & EVóVOVËóSFODJMFSJOFTËZMÐZPS A ( DFC ) ±SOFóJO LFOBSPSUBZ V[VOMVLMBS L N O PMBO & PSBOLBÀUS пHFOJOBMBO A ( ABE ) A \" # 9 D 3 % 4 & 5 4 2 94 A 5. L D ¥PDVL ZFõJM QBSL F n GN E BMBO N n T ZFõJM F n BMBO L B $BGF E 30° BD C C %OPLUBTOEBO[BE]ZFQBSBMFM¿J[JOJ[ ôFLJMEFCJSQBSLONPEFMMFONJõIºMJWFSJMNJõUJS1BS- LO \"%' LTN ¿PDVL QBSL &'$ LTN DBGF EJ- | | | |DT ve GT CVMVOEVóVOEBO(%5пHFOJOJOBMB- óFSCËMHFMFSJZFõJMBMBOPMBSBLZBQMNõUS OOIFSPOGPSNÐMÐJMFIFTBQMBZQJMF¿BSQBSTBO[ | | | |[AE] m [DC] ve AF =N BE =N пHFOJOBMBOOCVMVSTVOV[ % | |DF = 3 N m ( DCB ) = 30° PMEVôVOBHÌSF DB #VOB HÌSF %JMFL ²ôSFUNFO LFOBSPSUBZ V[VO GFWFÀPDVLQBSLOOBMBOMBSUPQMBNLBÀNFUSF MVLMBS PMBOÑÀHFOJOBMBOOLBÀCJSJN LBSFEJS LBSFCVMNVöUVS 29 3 33 3 \" 36 6 # 48 6 $ 60 6 \" # 15 3 $ 2 2 & 20 3 % 72 6 & 84 6 % 35 3 2 1. E 2. D 131 3. # 4. # 5. \"
KARMA TEST - 1 4. Üçgenler A 1. A ABC üçgen zy | |AB =DN | |AC =DN | |18 12 BC =DN Bx C B 15 C \"#$ÀFöJULFOBSCJSÑÀHFOIx < hZ, n# < nCPM 5BSBM CÌMHFOJO ÀFWSFTJ DN PMEVôVOB HÌSF EVôVOBHÌSF XA XB XC BÀMBSOOTSBMBOöBöBô UBSBMPMNBZBOCÌMHFOJOÀFWSFTJLBÀDNEJS EBLJMFSEFOIBOHJTJEJS \" # $ % & \" WB > WA > XC # WA > WB > XC $ WA > XC > WB % XC > WA > WB & XC > WB > WA 2. Peynir 5. A ABC bir üçgen 45° 30° [AC] m [BC] Beyaz Siyah Fare Fare 18 |BF| = |FD| õFLJMEFLJ TJZBI GBSFOJO QFZOJSF VMBöNBL JÀJO F2E | |D AB =DN BMBDBô ZPMVO CFZB[ GBSFOJO QFZOJSF VMBöNBL JÀJOBMBDBôZPMBPSBOLBÀUS 4 x | |BF =DN B C \" # 3 $ 2 % 3 & | |FE =DN 2 % % m ( BAC ) = m ( CAD ) | |:VLBSEBWFSJMFOMFSFHÌSF '$ =YLBÀDNEJS \" # $ % & 3. A ABC üçgen m (C%AB) =120° E |BD| = |AC| |AE| = |EB| 6. #JSEJLпHFOEFEJLLFOBSMBSOV[VOMVLMBSDNWF B DC DNEJS [ED] m [AB] :VLBSEBLJ WFSJMFSF HÌSF m % LBÀ EFSFDF %JLMJLNFSLF[JWFÀFWSFMÀFNCFSJONFSLF[JBSB (EDB) TOEBLJV[BLMLLBÀDNEJS EJS \" # $ % & 17 17 17 & \" 5 # $ % 32 1. D 2. C 3. D 132 4. # 5. C 6. C
Üçgenler KARMA TEST - 2 1. A \"#$%EËSUHFO 4. A ôFLJMEF 60 x | |AB =DN x [BC] m [AC] 20 D | |BC =DN 15° | |AD =YDN E D *BE* = *ED* 60 12 C 10 B [ AD ] m [ DC ] % m ( DBC ) = 30° % % 30° ABC BAD B m ( A%ED ) = 15° C m ( ) = 60° ve m ( ) = 60° PMEVôVOBHÌSF *BC * =DN YLBÀDNEJS * *:VLBSEBLJWFSJMFSFHÌSF \"& =YLBÀDNEJS \" # C % & \" 4 3 # 5 2 $ 3 6 % 4 5 & 2. A D 84 B6 C 5. A ABC üçgen | |ôFLJMEF [AB] m [BC] [BD] m [DC] AB =DN F 60° 4 3 <#&>B¿PSUBZ || || & B E m (B%AC) = 60° BC =DN CD =DNPMEVóVOBHËSF A^ ABD h 2*AF* = 3*CD* LBÀDN2EJS * *AE = 4 3 cm # 40 $ 41 % & 43 CxD \" 33 3 & h+ & h =DN2 PMEVóVOBHËSF A^ AFE A^ ECD * *CD =YLBÀDNEJS \" 2 3 # $ % 3 5 & 4 3 3. A 69 DE 24 B xC 6. #JS,-.пHFOJOEF m (WK) - m (VL) = 90° ve üçge- ABCпHFOJOJOLFOBSMBSCJSFSUBNTBZPMNBLÐ[F- OJOZÐLTFLMJLMFSJOJOLFTJNOPLUBT)EJS | | | | | |SF AD =DN BD =DN AE =DN & = DN2 PMEVóVOB HËSF & | | | |EC =DNPMEVóVOBHËSF #$ =YJOFOCÑZÑL A^ HKL h A^ KLM h LBÀ UBNTBZEFôFSJLBÀUS DN2EJS \" # $ % & \" # $ % & 1. C 2. # 3. E 133 4. # 5. C 6. D
KARMA TEST - 3 Üçgenler 1. A ABC bir üçgen 4. A ABC üçgen F [CF]B¿PSUBZ 2E [BE] ve [DE] D 8 B¿PSUBZ xF E |BE| = 2 |EF| | |C AD =DN 3 | |BD =DN 2 | |ED =DN 4 | |BC =DN BD C B | |AC =DN [ED] m [BC] PMEVôVOB HÌSF A^ & h LBÀ DN2 ACF EJS \" # $ % & | |:VLBSEBLJWFSJMFSFHÌSF %' =YLBÀDNEJS \" 14 # 15 $ 20 % 22 & 24 25 26 21 45 35 5. A ABC üçgen B 74°14° |AB| = |AC| 2. E 4 D ôFLJMEF _ |AD| = |BD| D [DE] m [EA] [EA] m [AB] % = 14° m ( BAD ) 17 C [AB] m [BC] C % = 74° m ( DAC ) 5 % | |ED =DN m ( ADC ) A9 B | |EA =DN :VLBSEBLJWFSJMFSFHÌSF = a LBÀ EF | | | |AB =DN BC =DNPMEVôVOBHÌSF $JMF% SFDFEJS OPLUBMBSBSBTV[BLMLLBÀDNEJS \" # $ % & \" # $ % & 6. A 8 4x C 3. A ABC üçgen BH 120° | |AB =DN D 10 | |AC =DN \"#$пHFO [\")] m [BC] 6 m ( % ) = 120° |AD| = |BD| = |CD| =DN, | AC | =DN BAC Bx C | | * *AB = DN PMEVôVOB HÌSF \") = Y LBÀ DN | | :VLBSEBLJWFSJMFSFHÌSF #$ =YLBÀDNEJS EJS \" 2 10 # 3 10 $ 4 6 % & \" # 10 $ % 8 & 4 3 33 1. \" 2. D 3. E 134 4. E 5. D 6. D
Üçgenler KARMA TEST - 4 1. A \"#$пHFOJOEF[DB] ve 4. A \"#$EJLпHFO [ EC ] \"&%пHFOJOJOEõ 9 y x [AB] m [BC] 3D B¿PSUBZMBSES 4 E *AD * =DN x 4 *AE * =DN B D8 *DC * =DN *ED * =DN C *DC *=DN C *AD * =ZDN * * :VLBSEBLJWFSJMFSFHÌSF \"$ =YJOBMBCJMFDFôJ B *EB * =YDN UBNTBZEFôFSMFSJUPQMBNLBÀUS :VLBSEBLJWFSJMFSFHÌSF Y+ZUPQMBNOOEFôF \" # $ % & SJLBÀUS \" # $ 67 % & 73 55 2. A ABC üçgen 5. A (BóSMLNFSLF[J 6 *AG *=DN 35 5D G *CG * =DN B 15 D x C * *8 GD =DN * * * * * *\"#$JLJ[LFOBSпHFO AB = AC AD = 3 5 CS BC * *BD = 15 br ve [BA] m [AD] PMEVôVOBHÌSF * *DC = xLBÀCSEJS :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS BGC \" # $ % & \" 5 5 # $ 6 5 % & 8 5 3. A ABC üçgen D 6 <\"#>m<#$> 6. \"B¿THFOJõB¿PMBOCJS\"#$пHFOJOEF%`[BC] 22 *AB* = *BD* = *BE* % = 90° m ( % ) = a m ( % ) = 3a ve *DE * = 2 2 cm m ( DAC ) BCA BAC * *B E C DC =DN * * * * * *BC - AB =DNPMEVôVOBHÌSF DC uzun :VLBSEBLJWFSJMFSFHÌSF A^ & hLBÀDN2EJS MVôVLBÀDNEJS DEC \" # 3 2 $ % 4 3 & \" # $ % & 1. C 2. \" 3. C 135 4. D 5. # 6. D
KARMA TEST - 5 Üçgenler 1. D 4. B \"#$%EËSUHFO A 25° 2IAEI = IECI E A 6 [ BC ] m [ DC ] E [AB] // [DE] B CF I ID 8 C BC = 6 br :VLBSEB WFSJMFO õFLJMEF [BD] J¿ B¿PSUBZ WF [CD] I IDC =CSWF A^ & h = 9 br2 PMEVôVOBHÌSF ABC EõB¿PSUBZES# $WF'OPLUBMBSEPóSVTBM % = 25° PMEVôVOBHÌSF m ( % ) LBÀEF- & m ( BDC ) BAD A^ DEC hLBÀCJSJNLBSFEJS SFDFEJS A) 6 B) 8 C) 10 D) 12 E) 15 A) 110 B) 115 C) 120 D) 125 E) 130 2. A \"#$FõLFOBSпHFO [ KD ] m [ BC ] 5. A K F ['&] m [ BC ] x I'$I = IKAI 4 BD 4 E I IC DE = 4 br B3D 11 C :VLBSEBLJ WFSJMFSF HÌSF UBSBM BMBO LBÀ CJSJN- I I ôFLJMEFLJEJLпHFOEF [ AB ] m [ AC ] AD =DN LBSFEJS I I I IBD =DN DC =DNPMEVóVOBHËSF A) 8 3 B) 7 3 C) 6 3 D) 5 3 E) 4 3 I IAB =YLBÀDNEJS 7 7 A) B) C) 4 D) 7 E) 8 4 2 3. A [AB] m [DC] 45° 4 2 m( % = 45° E BAC ) D I BD I = 6 br F I AE I = 4 2 br 6. \"#$EJLпHFOJOEF [ AB] m [ BC] [ AK] \"B¿TOO 6 J¿B¿PSUBZ [ DE] // [ BC]PMBDBLõFLJMEF[ AC]Ð[F- I'$I = 3I'%I SJOEF& [ AK]Ð[FSJOEF%OPLUBTTF¿JMJZPS BC I I I I I I I IAB = AE EC =DN BC = 16 cm PMEVôV- & I IOBHÌSF DE uzuOMVôVLBÀDNEJS :VLBSEBLJ WFSJMFOMFSF HÌSF A^ BEC h LBÀ CJ- SJNLBSFEJS A) 8 B) 7 C) 6 D) 5 E) 4 A) 32 B) 36 C) 40 D) 48 E) 60 1. B 2. A 3. B 136 4. D 5. B 6. C
Üçgenler KARMA TEST - 6 1. A \"#$EJLпHFO 4. A ABC üçgen D [ BA] m [ CA] 10 E *AD* = *AE* B F [ ED] m [ AB] B3 D 4 *AB *=DN [ EF] m [ AC] 6 *BD* =DN E C |AF| = |FC| * *C |BE| = |EC| DC =DN :VLBSEBLJ WFSJMFSF HÌSF BöBôEBLJMFSEFO IBO * *DE =DN m ( % ) = % PMEVôVOBHÌ HJTJEPôSVEVS DAE 2m ( BAD ) SF A^ & h kBÀDN2EJS ABC \" &OPLUBTZÐLTFLMJLMFSJOLFTJNOPLUBTES \" # $ % & # &OPLUBTLFOBSPSUBZMBSOLFTJNOPLUBTES 5. A $ [ BF]B¿PSUBZES % & OPLUBT LFOBS PSUB EJLNFMFSJO LFTJN OPLUBT ES & | FC | > | |BF EJS 2. & OPLUBT \"$% пHFOJOJO Eõ UFóFU ¿FNCFSJOJO 15° C NFSLF[J A % BH 2 m ( CAD ) = 150° 30° 150° *AE * = 2 br E *BE * = 4 br D 4 \"#$пHFO [BA] m [CA] [\")] m [BC] BC D * * * *% % ADH & m ( ACB ) :VLBSEBLJWFSJMFSFHÌSF A^ ACD hLBÀCJSJNLB = 15° m ( ) = 30° \") = )% SFEJS * * * *BC =DNPMEVôVOBHÌSF \"% LBÀDNEJS \" # $ % & \" 3 3 # 4 2 $ % 3 5 & 4 3 3. A [AB] m [BC] E <#&>m<&%> 6. \"#$пHFOJOEF m (WA) = 30°, m (XC) = 105° ve 6 |AE| = |EC| | |BC =DNPMEVôVOBHÌSF CVÑÀgeniOÀFWSFM Bx D4 C | AB | = 6 br ÀFNCFSJOJOZBSÀBQJMF\"$LFOBSOOV[VOMVôV | DC | = 4 br OVOUPQMBNLBÀDNEJS * *:VLBSEBLJWFSJMFSFHÌSF #% =YLBÀCSEJS \" 8 - 4 2 # 4 + 4 2 $ 4 + 4 3 \" # 2 5 $ % 4 3 & 2 13 % 8 + 4 2 & 8 + 4 3 1. D 2. # 3. E 137 4. \" 5. \" 6. #
KARMA TEST - 7 Üçgenler 1. A G ABC üçgen 4. A ABC bir üçgen % m ( E%AD ) = 45° 8 45° 72 m ( ABC ) > 90° B | |AB =DN B C 46 D x E | |BD = 46 DN D | |AE = 7 2 DN <\"&>m<$%> | |BC =DN <\"'>m<&$>EJS F E 10 C #VOBHÌSF BöBôEBLJMFSEFOIBOHJTJEPôSVEVS | | :VLBSEBLJWFSJMFSFHÌSF DE =YLBÀDNEJS \" | AG | = | |GC EJS \" 4 2 # 5 2 $ 5 3 % 6 3 & 7 2 # & OPLUBT \"#$ пHFOJOJO ¿FWSFM ¿FNCFSJOJO NFSLF[JEJS $ 'OPLUBT\"#$пHFOJOJOEJLMJLNFSLF[JEJS % | EG |2 + | GC |2 = | |EC 2EJS & # OPLUBT \"&$ пHFOJOJO J¿UFóFU ¿FNCFSJOJO NFSLF[JEJS 5. A 2. A \"#$CJSEJLпHFO 36 K 10 m ( A%DE ) = 45° 2 E B Cx D 45° 3 8 <\"#>m<#$> \"#%пHFO [ BA ] m [CA] AB = 3 6 cm <\"%>B¿PSUBZ BD C * * * *AC = =DN m % ) = % 10 cm, KC ( BCA m ( ACK ) | |DE =DN * *PMEVôunBHÌSF CD =YLBÀDNEJS | |EC =DNPMEVôVOBHÌSF ¦FWSF %&$ LBÀ \" 3 # $ % & 9 DNEJS 2 2 \" # $ % & 3. A ABC üçgen 7x [BA] m [CA] 6. \"#$CJSпHFO m ( % = 90° \"B¿TOOJ¿B¿PS- ABC ) % 24 m ( ADC ) = 2a UBZOB $ OPLUBTOEBO JOEJSJMFO EJLNFOJO BZBó % % ) = a \"%JMF#$OJOLFTJNOPLUBT&PMEVóVOBHËSF m ( ACB 2_ * *_ * * * *AE =DN ED =DNPMEVóVOBHËSF BD C AC =DN * * * *AB = DN PMEVôVOB HÌSF \"% = Y LBÀ DN * *\"$ V[VOMVôVLBÀDNEJS EJS \" 5 # 2 5 $ 3 5 \" # $ 25 % & % 4 5 & 5 5 2 1. D 2. C 3. C 138 4. # 5. D 6. D
Üçgenler KARMA TEST - 8 1. A \"#$FõLFOBSпHFO 4. A [AB] m [AD] D [ CD] a [ AE] = {F} 5 <\"&><%$> F 12 |AD| = |BE| | |x D AB = 12 br B | AD | = 5 br E 5 | DC | = 5 br BE C C |BE| = |EC| PMEVóVOB HËSF BöBôEBLJMFSEFO LBÀ UBOFTJ EPô | | :VLBSEBLJWFSJMFSFHÌSF \"& =YLBÀCJSJNEJS SVEVS \" 15 # $ 17 % & 19 * AD AF 2 2 2 = AE AB ** | DC | = | AE | * * * *5. \"#$JLJ[LFOBSпHFO AB = AC =DN [BC] *** | AD || AB | = | AC || BE | * * * *Ð[FSJOEF BE = FC =DNPMBDBLõFLJMEF&WF *7 m ( A%FC ) = 120° 'OPLUBMBSBMOZPS | | | | | | 7 AF ve DF V[VOMVLMBS WFSJMEJóJOEF BE %WF,TSBTZMB[\"#]WF[\"$]OJOPSUBOPLUBMBS V[VOMVóVIFTBQMBOBCJMJS PMEVôVOB HÌSF %,'& EJLEÌSUHFOJOJO BMBO LBÀ DN2EJS \" # $ % & \" # $ % & 2. A D FK 3 E 6. B C \"#$пHFOJOEF, ,BóSMLNFSLF[J [AB] m [AC] A B ôFLJM* *DE * *=CS EC * = * KE * = * FK *PMEVóVOBHËSF * *$# LBÀCJSJNEJS \" # $ % & 3. D \"#$%EËSUHFO A C D E FK LMN B C <\"$>B¿PSUBZ ôFLJM** A *CB * = 12 br ôFLJM * EF EJLEËSUHFO CJS SBGB ZBO ZBOB EJ[JMNJõ Fõ B *DC * = 5 br EJLEËSUHFOEPNJOPUBõMBSWFSJMNJõUJS#JSEPNJOPUB- õOOHFOJõMJóJ DNZÐLTFLMJóJDNEJS%PNJOP m ( % ) = m ( % ) + 45° PMEVóVOBHËSF UBõMBSEFWSJMFSFLõFLJM**EFLJLPOVNBHFMNJõMFSEJS ADC ABC && A^ ABC h - A^ ACD hGBSLBöBôEBLJMFSEFOIBOHJ TJEJS | | | |\"$ = DNPMEVôVOBHÌSF /# LBÀDNEJS \" 12 2 # 13 2 $ 14 2 \" 50 # 60 $ % & 120 13 13 13 % 15 2 & 16 2 1. E 2. \" 3. D 139 4. D 5. C 6. E
<(1m1(6m/6258/$5 Üçgenler 1. 3. ôFLJMEF \"#$ пHFOTFM CËMHFTJ CJ¿JNJOEFLJ LºóU x WFSJMNJõUJS 70° AA xx 100° 150° EF #BTLFUCPMNB¿OEBWFSJMFONPMBEBLP¿PZVODVMBS- B CK D N OBUPQVOBTMPZVOBTPLVQCBTLFUBUBDBLMBSO UP- ôFLJM* ôFLJM** QVOPZVODVMBSBSBTOEBEPóSVTBMCJSõFLJMEFHJEJõJ- OJHËTUFSFSFLBOMBUNõUS r ôFLJMEFLJ LºóU # WF $ V¿MBSOEBO LBUMBOBSBL #VOBHÌSF YLBÀEFSFDFEJS \" # $ % & [BC] EPóSVTV Ð[FSJOEFLJ CJS % OPLUBTOEB ¿B- 2. LõNBLUBESMBSWFôFLJM**PMVõNBLUBES r # LËõFTJOEFLJ LBUMBNB JõMFNJ ZBQMSLFO [AB] EPóSVTVOEBPMVõBOLBUMBNBJ[JOJOCJUJõOPLUBT& r $ LËõFTJOEFLJ LBUMBNB JõMFNJ ZBQMSLFO [AC] EPóSVTVOEB PMVõBO LBUMBNB J[OJOJ CJUJõ OPLUBT 'PMNBLÐ[FSF m ( E%DF ) = 70°EJS #VOBHÌSF m ( % ) = x LBÀEFSFDFEJS BAC \" # $ % & ôFLJM** 4. \"#$FõLFOBSпHFOTFMCËMHFTJõFLMJOEFLJCJSLPõV ôFLJM* QJTUJOEF\"INFUJMF4FSIBUBZOTBCJUI[MBLPõBDBL- MBSES\"ZOOPLUBEBOGBSLMEPóSVMUVEBBZO[BNBO- EBCBõMBZQBZO[BNBOEBEVSBDBLMBSWFBSBMBSO- EBLJV[BLMóIFTBQMBZBDBLMBSES A ôFLJM*** CSEFOPMVõBOõFLJM*EFLJLBSFMJLºóUõFLJMMFSEF BC HËTUFSJMEJóJ HJCJ LBUMBOELUBO TPOSB ôFLJM *** UFLJ HJ- CJNBLBTZBSENZMBLFTJMJZPS\"MUUBSBGUBPMVõBOп- 1JTUJO PSUBTOEBLJ CFZB[ ¿J[HJMFS Ð[FSJOEFO LPõ- HFOTFMCËMHFBUMZPS,FTJMEJLUFOTPOSBLBMBOQBS¿B NBLUBESMBS#V¿J[HJOJOUPQMBNV[VOMVóVNFU- LBUMBOEóZFSMFSEFOUBNBNFOB¿MZPS SFWF\"INFUJMF4FSIBUhOI[NELES 4POEVSVNEBPMVöBOöFLMJOBMBOOOJMLEVSVN ,PöVCBöMBELUBOELTPOSBLPöNBZCSBLQ EBLJöFLMJOBMBOOBPSBOOFEJS EVSEVLMBSOBHÌSFBSBMBSOEBLJEPôSVTBMV[BLML LBÀNFUSFEJS \" 143 # 141 $ 143 % 1 & 151 \" # $ 180 192 192 2 192 % 60 2 & 60 3 1. D 2. C 140 3. \" 4. C
Üçgenler <(1m1(6m/6258/$5 1. \"LOBõBóEBLJBENMBSJ[MFZFSFLHFPNFUSJLCJS¿J- 3. A [JNZBQZPS F E G | | | |r AB = DN AC = DN PMBDBL õFLJMEF CJS B DC \"#$пHFOJ¿J[JZPS ôFLJM*EF(OPLUBTBóSMLNFSLF[J BMBOCJSJN- r [AB]LFOBSOBBJUEõUFóFU¿FNCFSJOJ¿J[JZPSWF LBSFPMBO\"#$пHFOJWFSJMNJõUJS ¶¿HFOMFSLFOBSPSUBZ¿J[HJMFSJJMFQBS¿BMBSBBZSMBSBL NFSLF[JOJ/JMFJTJNMFOEJSJZPS CJSLTNJMFôFLJM**PMVõUVSVMVZPS r [AC]LFOBSOBBJUEõUFóFU¿FNCFSJOJ¿J[JZPSWF K NFSLF[JOJ-JMFJTJNMFOEJSJZPS P | | | |r AN =YDN AL =ZDNPMEVóVOVËM¿FSFLCV- MVZPS \"LO ZBQUô ÀJ[JN TPOSBT CVMEVôV Y WF Z EF ôFSMFSJOJ ÀBSQBSBL IFTBQMBEôOEB TPOVDV LBÀ CVMNVöUVS \" # $ % & LM N ôFLJM**EFLJCFõHFOJGBEFTJ[LP]EPóSVTVCPZVODB LFTJMJZPS 4PO EVSVNEB PMVöBO ,-1 ÑÀHFOJOJO BMBO LBÀ CJSJNLBSFEJS \" # $ % & 2. &SFO ±óSFUNFO ËóSFODJMFSJOF HFPNFUSJ EFSTJOEF 4. (FPNFUSJËóSFUNFOJ\"S[VhZBBõBóEBLJBENMBSJ[- BõBNBM CJS FULJOMJL ZBQUSQ TPOVOEB ËóSFODJMFSJOF MFZFSFLHFPNFUSJLCJS¿J[JNZBQNBTOTËZMÐZPS CJSTPSVTPSVZPS r #JS\"#$пHFOJ¿J[NFTJOJ % r m ( ABC ) = 90°PMBDBLõFLJMEFCJSEJLпHFO¿J- r m % = 32° ( ACB ) [FMJN r \"LËõFTJOEFOCJSB¿PSUBZEPóSVTV¿J[FSFL[BC] r $ LËõFTJOEFO % ` [AB] PMBDBL õFLJMEF [CD] JMFLFTJõUJóJOPLUBZ%OPLUBTPMBSBLCFMJSMFNF- EPóSVTVOV¿J[FMJN TJOJ | | | |r 3 AD = AC PMBDBLõFLJMEFJõBSFUMFZFMJN | | | | | |r AB + BD = AC PMNBT HFSFLUJóJOJ CFMJSUJ- r 2m ( D%CB ) = m ( B%AC ) = 90° ZPS | | | |r 2 CE = ED PMBDBLõFLJMEF&` [CD]OPLUBT- #VWFSJMFSJLVMMBOBSBL¿J[JNZBQBO\"S[VhEBOËóSFU- NFOJ\"B¿TOCVMNBTOJTUFNJõUJS OJõBSFUMFZFMJN #VOBHÌSF \"S[VhOVOCVMEVôVEPôSVTPOVÀLBÀ * *r &OPLUBTOEBO CB ZJJLJFõQBS¿BZBBZSBDBL EFSFDFEJS | |EF =DNPMBDBLõFLJMEF'` [CB]OPLUBTO JõBSFUMFZFMJN | | #VOBHÌSF %# V[VOMVôVLBÀDNEJS \" # $ % & \" # $ % & 1. E 2. C 141 3. D 4. D
<(1m1(6m/6258/$5 2. Üçgenler C 1. D A 4 N 10 S 4 C AB 10 10 :PMLFOBSOEBBSB[JTJPMBOCJS¿JGU¿JBSB[JTJOJпHFO- TFMCËMHFMFSFBZSNõFõпFõLFOBSпHFOJOPMEV- B óVLTNBFLJNZBQQEJóFSCËMHFMFSJOBEBTBCSBL- ED NõUS ôFLJM* 5SBLUËSMF[AC]LËõFHFOJCPZVODBUBSMBZTÐSÐQZPM 0LVMEB CVMVOBO EVWBSMBSEBLJ LBSF QBOPMBS Ð[FSJO- UBSBGOB OPIVU EJóFS UBSBGOB JTF BZ¿J¿FóJ FLNFZJ EFPLVMBBJUFõLFOBSпHFOõFLMJOEFLJGMBNBMBSZFS QMBOMBNBLUBES BMNBLUBES #V QBOPMBSEBO CJSJOEF LËõFMFSJOEFO JóOF JMF UVUUV- | |\"# =DNPMEVôVOBHÌSF ZBQUôQMBOEPô SVMNVõGMBNBôFLJM*EFHËTUFSJMNJõUJS /LËõFTJOEFLJUPQMVJóOFEÐõFSFLGMBNBôFLJM**EF- SVMUVTVOEB ÀJGUÀJ OPIVU FLNFTJ HFSFLFO BMBO LJLPOVNBHFMNJõUJS LBÀ N2 CVMNVöUVS ,ËõFHFO CPZVODB ZBQMBO A SC UBSMBTÐSÐNÐOÐOBMBOBFULJTJPMNBZBDBLUS \" 100 3 # 125 3 $ 180 3 % 170 3 & 200 3 N B E D ôFLJM** ôFLJM**EFLJEFOHFEVSVNVOEB4OPLUBTOEBLJEÐ- 3. (FPNFUSJEFSTJOEFFULJOMJLZBQBO.VSBU±óSFUNFO õFZEPóSVMUVBóSMLNFSLF[JOEFOHF¿NFLUFWFGMB- BõBóEBLJBENMBSJ[MFZFSFLËóSFODJMFSJOFCJSHFP- NBMBSO[NB]LFOBSQBOPOVOBMULFOBSOBQBSBMFM- NFUSJL¿J[JNZBQUSZPS EJS r [AB] m [AC]PMBOCJSпHFO¿J[JOJ[ 1BOPOVO EöOEB LBMBO LTNOO BMBOO JÀJOEF r D ` [AC]BMQ[BD]¿J[JOJ[ LBMBO LTNOO BMBOOB PSBO BöBôEBLJMFSEFO IBOHJTJEJS r m ( A%BD ) = a ve % = b PMTVO m ( ACB ) \" 1 # 1 $ 3 % 3 & 4 50 49 49 50 49 | | | | | |r AB =CJSJN AD = CJSJN DC =CJSJN- EJS #VOBHÌSF a LBÀUS b \" 1 # $ 3 % & 7 2 2 2 1. # 142 2. \" 3. #
Üçgenler <(1m1(6m/6258/$5 1. ôFLJMEFHFOJõMJóJNFUSFPMBO/FIJShJOCJSLFOBSO- 3. 'JMPHSBGJ BIõBQZBEBTVOUBQBOPOVOÐ[FSJOF¿JWJ- EBBóB¿MBSEJóFSLFOBSOEBJTF&SDBOCVMVONBLUB- MFS¿BLMQBSBMBSOOJ[BNJCJSõFLJMEFUFMWFZBJQMFS- ES/FIJSJOLFOBSMBSOOCJSCJSJOFQBSBMFMPMEVóVCJS MFEPMEVSNBZBQMBO¿JWJTBOBUES CËMÐNHËTUFSJMNJõUJS Tahta A &SDBO OFIJSEF LBSõEBO LBSõZB HF¿FSLFO [PSMBO- BC NBNBLJ¿JOLZMBSBSBTOBJLJJQHFSNFZJEÐõÐOFSFL CVMVOEVóVLFOBSEBLJCJSOPLUBZBV[VOMVLMBSN \"INFUGJMPHSBGJTBOBUOLVMMBOBSBLËODFMJLMF WFNPMBOJLJJQTBCJUMJZPS [AB] m [BC]PMBDBLõFLJMEF\" #WF$OPLUBMBSOB /FISJOLBSõLFOBSÐ[FSJOEFCVMVOBOBóB¿MBSBJQ- MFSHFSHJOPMBDBLõFLJMEFCBóMBZBDBLõFLJMEFOFISJO ¿JWJMFS¿BLZPSWFJQMFHFSHJOCJSõFLJMPMVõUVSVZPS LBSõLZTOB¿LZPS | | | |AB =DN AC =DNEJS &SDBO JQMFSJ OFIJSJO LBSö LFOBSOEBLJ BôBÀMB SB CBôMBELUBO TPOSB JLJ BôBÀ BSBTOEB PMVöB %BIBTPOSBпHFOJOJ¿JOEFJLJOPLUBEBIBCFMJSMJZPS DBLNFTBGFMFSJOUPQMBNOOBMBCJMFDFôJEFôFSMF #JSJODJOPLUBпHFOJOJ¿UFóFU¿FNCFSJOJONFSLF[J SJ IFTBQMBNBL JTUFEJôJOEF CVMEVôV TPOVÀ LBÀ CVOPLUBZIIBSGJZMFHËTUFSJQ¿JWJTJOJ¿BLZPS NFUSFEJS öLJODJOPLUBJTFпHFOJOBóSMLNFSLF[J CVOPLUBZ \" # $ % & (IBSGJZMFHËTUFSJQCJS¿JWJ¿BLZPS \"INFU UBIUBOO Ð[FSJOEFLJ # I WF ( OPLUBMBSOB HFSHJOJQJTBSZPS % \"INFUhJOPMVöUVSEVôV#I(ÑÀHFOJOEFLJm ( BIG ) BÀTLBÀEFSFDFEJS \" # $ % & 2. B C E A' A D \"#$WF#%$пHFOMFSJOEF[AC] a [DB] = {E} ve | | | | | | | |AE = EC = BE = ED EJS\"&#пHFOJ[EB] CPZVODBLBUMBOEóOEB\"h&#пHFOJPMVõVZPS m ( A%'EC ) = 24° PMEVôVOB HÌSF m ( B%AC ) LBÀ EFSFDFEJS \" # $ % & 1. E 2. D 143 3. C
<(1m1(6m/6258/$5 Üçgenler AD 1. 5BOHSBNUBõ QMBTUJL UBIUB LºóUHJCJOFTOFMFSLVM- 2. C MBOMBSBLHFPNFUSJLQBS¿BпHFO LBSF QB- BE SBMFMLFOBSCJSBSBZBHFUJSFSFL¿FõJUMJHFPNFUSJLõF- LJMMFSPSUBZB¿LBSBOCJS[FLBPZVOVEVS DN DN DN FG DN ôFLJM* &OFSKJ ÐSFUNFL J¿JO UBTBSMBOBO п Fõ FõLFOBS п- \"LJG ±óSFUNFO TOGOEBLJ ËóSFODJMFSJ JMF UBOHSBN HFOEFO PMVõBO ZFMEFóJSNFOJOJ TBóMBNMBõUSNBL FULJOMJóJ ZBQNBL J¿JO ËóSFODJMFSJOF LºóU DFUWFM J¿JO\"# %&WF'(LFOBSMBSOBCJSCJSMFSJZMFLFTJõF- NBLBTWFB¿ËM¿FSEBóUQBõBóEBLJUBOHSBNõBC- DFLõFLJMEFEPóSVTBM¿UBMBS¿BLBSBLZFMEFóJSNFOJ- MPOVOEBOHFPNFUSJLõFLMJLFTNFTJOJJTUFNJõUJS OJTBóMBNMBõUSZPSWFôFLJM**PMVõUVSVMVZPS K AD C E B %BIBTPOSB\"LJG±óSFUNFOËóSFODJMFSJOEFOJLJLÐ- LF GM ¿ÐLWFPSUBCPZпHFOJBMBSBLCVпHFOMFËODFEJL CJSпHFOEBIBTPOSBJTFZJOFппHFOMFCJSLBSF- EFOGBSLMEJLEËSUHFOZBQNBMBSOJTUJZPS :BQMBO FULJOMJL TPOSBTOEB ÌôSFODJMFSJOJO ôFLJM** PMVöUVSEVôVEJLÑÀHFOWFEJLEÌSUHFOJOÀFWSF MFSJOJOGBSLMBSOONVUMBLEFôFSJLBÀDNEJS \" 4 - 2 2 # 8 - 4 2 $ 4 2 õFLJM**EFPMVöBO,-.ÑÀHFOJOJOBMBOOOõFLJM * EFLJ ÑÀHFOMFSJO BMBOMBS UPQMBNOB PSBO LBÀ % 6 - 4 2 & 2 2 - 8 US \" # $ % & 1. # 144 2. \"
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