B.Haydarov, E.Sariqov, A.Qo chqorov

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1 .Hydrov, E.Sriqov,.Qo chqorov GEMETRIY 9 zbekiston Respubliksi Xlq t limi vzirligi umumiy o rt t lim mktblrining 9-sinfi uchun drslik siftid tsdiqlgn «zbekiston milliy ensiklopediysi» vlt ilmiy nshriyoti Toshkent 0

2 U K 5.(075) K.5y7 X-8 Fizik-mtemtik fnlri doktori, professor. zmov thriri ostid. Tqrizchilr:. Nrmnov zbekiston Milliy universiteti geometriy v mliy mtemtik kfedrsi mudiri, fizik-mtemtik fnlri doktori. G. Yusupov zbekiston F Mtemtik institutining ktt ilmiy odimi, fizik-mtemtik fnlri nomzodi. M. krmov Toshkent viloyti Prkent tumnidgi 5-sonдшдлдлдлдli mktbning oliy toifli mtemtik o qituvchisi. M. Shniyzov Toshkent shhr Sirg li tumnidgi 00-sonli mktbning mtemtik o'qituvchisi 9-sinfd geometriyning plnimetriy qismini yssi geometrik shkllrning osslrini o rgnish dvom ettirildi. Und siz geometrik shkllrning o shshligi, uchburchkning tomonlri v burchklri orsidgi munosbtlr, yln uzunligi v doir yuzi, uchburchk v ylndgi metrik munosbtlr biln tnishsiz. Ushbu Geometriy drsligining mzmuni qt iy ksiomtik tizim sosig qurilgn. Und nzriy mterillr imkon borich sodd v rvon tild byon etilgn. rch mvzu v tushunchlrni turli-tumn hyotiy misollr orqli ochib berishg hrkt qilingn. Hr bir mvzudn so ng berilgn svollr, isbotlsh, hisoblsh v ysshg doir msl v misollr o quvchini ijodiy fikrlshg undydi, ung o zlshtirilgn bilimlrni chuqurlshtirishg v musthkmlb borishg yordm berdi. rslik o zining o zgch dizyni v drs mterilining ko rgzmli qilib tqdim etilishi biln hm jrlib turdi. Und keltirilgn rsm v chizmlr drs mterilini yshiroq o zlshtirishg izmt qildi. Respublik mqsdli kitob jmg rmsi mblg lri hisobidn ijr uchun chop etildi. ISN «zbekiston milliy ensiklopediysi» vlt ilmiy nshriyoti, 006, 00, 0 «Huquq v Jmiyt» MHJ shklidgi nshriyot, 0 M U N R I J 7 8-sinflrd o tilgnlrni tkrorlsh. Uchburchklr...8. Uchburchklr (dvomi) To rtburchklr.... To rtburchklr (dvomi)... I bob. shsh geometrik shkllr 5. Ko pburchklrning o shshligi shsh uchburchklr v ulrning osslri Uchburchklr o shshligining birinchi lomti Uchburchklr o shshligining ikkinchi lomti Uchburchklr o shshligining uchinchi lomti To g ri burchkli uchburchklrning o shshlik lomtlri shshlik lomtlrining isbotlshg doir msllrg ttbiqlri Msllr yechish.... ilimingizni sinb ko ring.... Geometrik shkllrning o shshligi shsh ko pburchklrning osslri Gomotetiy v o shshlik shsh ko pburchklrni yssh mliy mshg ulot Msllr yechish Msllr yechish I bobg doir qo shimch msllr v m lumotlr... 9 II bob. Uchburchk tomonlri v burchklri orsidgi munosbtlr. tkir burchkning sinusi, kosinusi, tngensi v kotngensi Msllr yechish zi burchklrning sinusi, kosinusi, tngensi v kotngensini hisoblsh Msllr yechish dn 80 gch bo lgn burchkning sinusi, kosinusi, tngensi v kotngensi sosiy trigonometrik yniytlr sosiy trigonometrik yniytlr (dvomi) ilimingizni sinb ko ring Uchburchk yuzini burchk sinusi yordmid hisoblsh Sinuslr teoremsi Kosinuslr teoremsi Sinuslr v kosinuslr teoremlrining b zi ttbiqlri Ikki vektor orsidgi burchkni hisoblsh... 78

3 5. Uchburchklrni yechish Msllr yechish Uchburchklrni yechish usullrining mliyotd qo llnishi II bobg doir qo shimch msllr v m lumotlr III bob. yln uzunligi v doir yuzi 0. ylng ichki chizilgn ko pburchk ylng tshqi chizilgn ko pburchk Muntzm ko pburchk Muntzm ko pburchkk ichki v tshqi chizilgn ylnlr Muntzm ko pburchkning tomoni biln tshqi v ichki chizilgn ylnlr rdiuslri orsidgi bog lnish ilimingizni sinb ko ring yln uzunligi yln yoyi uzunligi. urchkning rdin o lchovi oir yuzi oir bo lklri yuzi Msllr yechish III bobg doir qo shimch msllr v m lumotlr... IV bob. Uchburchk v ylndgi metrik munosbtlr 5. Kesmlr proyeksiysi v proporsionllik Proporsionl kesmlrniтпng osslri To g ri burchkli uchburchkdgi proporsionl kesmlr erilgn ikkit kesmg o rt proporsionl kesmni yssh ylndgi proporsionl kesmlr Msllr yechish IV bobg doir qo shimch msllr v m lumotlr... 8 V bob. Plnimetriy kursini tkrorlsh 59. Koordintlr usuli Koordintlr usuli v vektorlr yln v doir Tkrorlsh Tkrorlsh Tkrorlsh Tkrorlsh Tkrorlsh Ykuniy nzort ishi... Plnimetriyg oid sosiy tushunch v m lumotlr... 6 Jvoblr v ko rstmlr... 5 S ZSHI ziz o quvchilr! borot tenologiylri srid yshypmiz. Zmonviy trqqiyotd kechyotgn olmshumul o zgrishlr zmirid, lbtt, fn v tenik rivoji yotibdi. undy shroitd yoshlrning sosiy vzifsi buyuk jdodlrimizg munosib vlod bo lib, zmon biln hmnfs qdm tshlsh, ilm-fn cho qqilrini qunt biln egllshdn ibortdir. u bord mtemtikning tutgn o rni beqiyosdir. M lumki, mtemtik siz yoshlrning kmolg yetishingizd o quv fni siftid keng imkoniytlrg eg. U tfkkuringizni rivojlntirib, qlingizni peshlydi, mntiqiy fikrlsh, topqirlik isltlrini shkllntirdi v turli vziytlrd oqilon qror qbul qilish, thlil qilish hmd ulos chiqrish ko nikmlrini trbiylydi. Qo lingizdgi 9-sinf Geometriy drsligining sosiy vzifsi yssi geometrik shkllrning osslrini o rgtish biln bir qtord sizd izchil mntiqiy fikrlsh qobiliytini o stirib borish ntijsid qlingizni chrlshdn ibort. U o zlshtirilgn bilim, ko nikm v mlklrni kundlik turmushg ttbiq etishingizg ko mklshdi. rslikni yrtishd dunyod to plngn ilg or tjrib nmunlridn foydlndik. Shu biln bir qtord yurtimizg os shrqon v umrboqiy qdriytlrimizg, buyuk jdodlrimiz merosig hm muntzm murojt etishg hrkt qildik. Mzkur drslikdn t lim olr eknsiz, sizg bu ms uliytli, shu biln birg mroqli yo ld qunt v sbot tilb qolmiz. Geometriy soslri bo yich olgn sboqlringiz sizni brkmollik sri yetklb, Vtnimiz trqqiyoti yo lid izmt qilishg ko mkchi bo ldi, deb ishonch bildirmiz! 5

4 yngi kiritilyotgn geometrik tushunchning t rifi teoremning tvsifi nmun triqsid yechib ko rstilyotgn msl svol, msl v topshiriqlr o quvchilr folligini oshiruvchi mustqil yoki guruhlrd muhokm qilindign topshiriqlr 8. ykk trtibd yoki guruhlrd bjrildign mliy ish triiy m lumotlr v msllr qiziqrli msllr v boshqotirmlr Internetdn tvsiy etildign m lumotlr mnzili uyd yechishg tvsiy etildign msllr boshq rngd berilgn TKRRLSH Teorem yoki mslning semtik tlqini Teorem yoki msl shrtid berilgn m lumotlr Isbot qilinishi kerk bo lgn oss yoki topilishi tlb etildign elementlr 7 8-SINFLR TILGNLRNI TKRRLSH hizmlrd qbul qilingn lohid belgilr hizmlrd teng burchklr bir il sondgi yoychlr biln jrtildi. hizmlrd uzunligi teng kesmlr bir il sondgi chiziqchlr biln jrtildi. 7 8-sinflrd geometriydn o tilgn mvzulrni tkrorlb, olgn bilimlringizni esg olsiz v erishgn ko nikmlringizni musthkmlysiz. u sizg 9-sinfd geometriyni o rgnishni muvffqiytli dvom ettirishingizg zmin yrtdi. G E M I Y E TR Geometriyni ishg ol qilish uchun olg! 6 7

5 UHURHKLR Mzkur bo limdgi msllr 7 8-sinflrd o rgnilgn geometrik shkllr v ulrning osslrini yodg olish uchun berilmoqd. Msllrni yechish uchun drslikning oirid keltirilgn sosiy geometrik shkllrg oid m lumotlr hmd ulrning osslrini ifodlovchi formullrdn foydlnishingiz mumkin. -msl. uchburchkning blndligi o tkzilgn (-rsm). gr = 0, 0 = 5 bo ls, uchburchkning v uchidgi burchklrini toping. Yechilishi. ) To g ri burchkli uchburchkd 5 = 0 v uchburchk ichki burchklrining yig indisi 80 g teng bo lgni uchun = 80 ( ) = 50. ) To g ri burchkli uchburchkd = 5 bo lgni uchun = 80 ( ) = 5. = + bo lgni uchun = = 85. Jvob: 50, 85. -msl. Ikki prllel to g ri chiziqni kesuvchi biln kesgnd hosil bo lgn ichki bir tomonli burchklrning bissektrislri orsidgi burchkni toping. y y Yechilishi. to g ri chiziq v prllel to g ri chiziqlrni -rsmd tsvirlngndek E kesib o tgn bo lsin. Ichki bir tomonli v burchklrning bissektrislri E nuqtd kesishgn bo lib, E =, E = y bo lsin. Und, burchk bissektrissining t rifig ko r = + =, = y + y = y. bo lgni uchun ichki bir tomonli burchklr osssig ko r, + y = 80, + y = 90. Endi, E uchburchk ichki burchklri yig indisi 80 g teng bo lgni uchun E = 80 ( + y) = = 90. Jvob: 90. -msl. uchburchkning tomoni 6 sm, v burchklri, mos rvishd, 0 v 60 bo ls, uchburchk yuzini toping. Yechilishi. Uchburchkning burchgini topmiz: = 80 ( + )=80 (60 +0 )=90. emk, to g ri burchkli uchburchkning gipotenuzsi 6 sm v burchgi 0 0 ekn. To g ri burchkli uchburchkd 0 li burchk qrshisidgi ktet gipotenuzning yrmig teng bo lgni uchun, = sm (-rsm). Pifgor teoremsidn foydlnib ktetni topmiz: = = 6 = 7, = sm. Endi uchburchk yuzini topmiz: S = = = 9 (sm ). Jvob: 9 Svol, msl v topshiriqlr. uchburchkd = 7, = 8 bo ls, uchburchkni uchinchi ichki burchgini v tshqi burchklrini toping.. Ktetlri 5 sm v 0 sm bo lgn to g ri burchkli uchburchk gipotenuzsig tushirilgn blndligini toping.. uchburchkning tomonig prllel to g ri chiziq v tomonlrni mos rvishd E v F nuqtlrd kesib o tdi. gr EF= 65 v EF= 5 bo ls, uchburchk burchklrini toping.. uchburchk bissektrislri I nuqtd kesishdi. gr = 80 v = 70 bo ls, I, I v I burchklrni toping. 5. Teng yonli uchburchkning bitt tshqi burchgi 70 g teng. Uchburchk burchklrini toping. 6. uchburchkning K bissektrissi o tkzilgn. gr K= 7 v K= 0 bo ls, uchburchk burchklrini toping. 7*. uchburchk blndliklri H nuqtd kesishdi. gr = 50, = 60 bo ls, H, H v H burchklrni toping. 8. Uchburchkning o rt chiziqlri uni to rtt teng uchburchklrg jrtishini isbotlng. 9*. uchburchkd medin dvomig bu meding teng E kesm qo yilgn. F medinning dvomig F meding teng FH kesm qo yilgn., H, E nuqtlr bitt to g ri chiziqd yotishini isbotlng. 0. teng yonli uchburchkd (=) N v K bissektrislr o tkzilgn. ) KN kesm tomong prllel eknini ko rsting. b) K=KN=N tenglik o rinli bo lishini isbotlng. 60 sm. 8 9

6 UHURHKLR (dvomi) А -msl. nuqtdn m to g ri chiziqq uzunliklri sm v sm bo lgn ikkit og m tushirilgn. gr birinchi og mning m to g ri chiziqdgi proyeksiysi 8 sm bo ls, ikkinchi m В 8 og mning proyeksiysini toping. Yechilishi. m to g ri chiziqdn tshqridgi nuqtdn shu to g ri chiziqq v А og mlr hmd perpendikulyr tushirilgn bo lib, = sm v = sm b o l s i n (-rsm). Und msl shrtig ko r =8 sm bo ldi v kesm uzunligini topish kerk. ) Pifgor teoremsidn foydlnib to g ri m burchkli uchburchkning ktetini В topmiz. = = 8 = 80, = 80 sm. ) To g ri burchkli uchburchkdn Pifgor teoremsidn foydlnib kesm uzunligini topmiz. = = ( 80) = 80 = 6, =9 sm. Jvob: 9 sm. -msl. Tomonlri, v 5 g teng bo lgn uchburchk yuzini v blndliklrini toping. Yechilishi. Geron formulsidn foydlnib, tomonlri =, b =, c = 5 bo lgn uchburchk yuzini topmiz: p = = ++5 = =, S = p(p )(p b)(p c) = ( ) ( ) ( 5) = = = = 7 = 8. Endi, uchburchk yuzini hisoblsh formulsi S= h dn foydlnib, uchburchkning h blndligini topmiz: h = S = 8 = 68 =. Xuddi shuningdek h b v h c blndliklrni topmiz. Jvob: 8; ; ; 5. -msl. uchburchkning v uchlridgi tshqi burchklrining bissektrislri nuqtd kesishdi. nuqtning burchk bissektrissid yotishini isbotlng. F Isbot. nuqtning, v to g ri chiziqlrdgi proyeksiylri mos rvishd, E v F nuqtlr bo lsin (-rsm). Und, birinchidn nuqt burchkning bissektrissid yotgni uchun =F bo ldi. Ikkinchidn, nuqt E burchkning bissektrissid yotgni uchun E F=E bo ldi. Shuning uchun, =F=E. emk, nuqt burchk tomonlridn teng uzoqlikd joylshgn ekn. Shuning uchun nuqt burchk bissektrissid yotdi. Svol, msl v topshiriqlr. Tomonlri 5, 6 v 7 bo lgn uchburchk yuzini toping.. erilgn nuqtdn to g ri chiziqq uzunliklrining yirmsi 6 g teng bo lgn ikkit og m tushirilgn. g mlrning to g ri chiziqdgi proyeksiylri 7 v 5 g teng. erilgn nuqtdn to g ri chiziqqch bo lgn msofni toping. *. uchburchkning v uchlridgi tshqi burchklrining bissektrislri nuqtd kesishdi. gr =75 bo ls, uchburchkning burchgini toping.. sosi bo lgn teng yonli uchburchkd bissektris o tkzilgn. burchk: ) 60 ; b) 75 g teng bo ls, uchburchk burchklrini toping. 5. ir kteti 7 sm g, gipotenuzsi es 5 sm g teng bo lgn to g ri burchkli uchburchkning gipotenuzsig tushirilgn blndligini toping. 6. uchburchkning blndligi o tkzilgn ( nuqt kesmg tegishli). gr =, =5 v =6 bo ls, uchburchk perimetrini v yuzini toping. 7. Teng yonli uchburchkning yon tomoni 0 sm, sosi es 0 sm. Uchburchkning sosig tushirilgn blndligini, yuzini v burchklrini toping. 8. tkir burchkli uchburchkk tshqi chizilgn yln mrkzi nuqtd bo lib = 0, = 0 bo ls, uchburchk burchklrini toping. 9. gr uchburchkning medinsi tomondn ikki mrt kichik bo ls, burchkni toping. 0. uchburchkning blndliklri nuqtd kesishdi. gr = 60, = 80 bo ls, burchkni toping.. uchburchkning v uchlridgi tshqi burchklrining bissektrislri nuqtd kesishdi. gr = 80 bo ls, burchkni toping. 0

7 T RTURHKLR -msl. gr prllelogrmmning bir ) 55 uchidn uning ikki tomonig tushirilgn blndliklri orsidgi burchk 55 g teng bo ls, prllelogrmning burchklrini toping. E Yechilishi. Prllelogrmmning F v E blndliklri orsidgi burchk 55 bo lsin F (-rsm). Rsmd tsvirlngn ikki hol: ) E b) blndlik tomong; b) E blndlik tomon dvomig tushgn bo lishi mumkin. ) hold EF to rtburchk burchklrining F K yig indisi 60 bo lgni uchun, E = 60. undn, = 5. b) hold E blndlik tomon biln kesishgn nuqt K bo lsin. Und, KE = KF = = 5. Uchburchk tshqi burchgining osssig ko r, = KE + KE = = 5. emk, hr ikkl hold hm =5. Und, = =80 =55, = =5. Jvob: 55, 5, 55, 5. -msl. To rtburchk tomonlrining E o rtlri prllelogrmm uchlri bo lishini F isbotlng. L Yechilishi. to rtburchkning,, v tomonlri o rtlri mos rvishd E, F, K v L nuqtlr bo lsin. digonlni o tkzmiz (-rsm). K EFKL prllelogrmm eknligini ko rstmiz. EF kesm uchburchkning, KL kesm es uchburchkning o rt chizig i bo ldi. Und, uchburchk o rt chizig ining osslrig ko r, EF, KL, EF =, KL =. undn EF KL v EF =LK. Shuning uchun, prllelogrmm lomtig ko r, EFKL prllelogrmm. -msl. to g ri to rtburchk v burchklrining bissektrislri tomond kesishdi. gr = sm bo ls, bu to g ri to rtburchk yuzini toping. Yechilishi. To g ri to rtburchk v burchklrining bissektrislri kesishgn nuqt E bo lsin (-rsm). Und, =90, E = 5 bo lgni uchun E = = 5. Y ni, E teng yonli uchburchk.und, = E = (sm). 5 E Xuddi shung o shsh E = = (sm) eknligini ko rstish mumkin. undn = E +E =8(sm) v 5 S = = 8 = (sm ). Jvob: sm. Svol, msl v topshiriqlr. To rtburchkning ucht burchgi 7, 8 v 0 K g tengligi m lum. Uning to rtinchi burchgini P toping.. Prllelogrmmning ikki burchgi yig indisi 56 g teng. Uning burchklrini toping.. To g ri to rtburchk digonllri orsidgi burchk 7. Uning bir digonli biln tomonlri 5 orsidgi burchklrni toping.. Teng yonli trpetsiyning ikkit burchgi P yirmsi 0 g teng. Uning burchklrini toping. 5. Romb burchklridn biri ikkinchisidn uch mrt ktt. Rombning burchklrini toping. K 6. to g ri to rtburchkning burchgi bissektrissi tomonini sm v 6 sm g teng kesmlrg jrtdi. To g ri to rtburchk perimetrini toping. 7. Tomonlri sm v 6 sm, ktt tomonlri orsidgi msof es sm bo lgn prllelogrmm ysng. 8. prllelogrmmning digonlid P v K nuqtlr tnlngn (-rsm). gr P ==K bo ls, KP to g ri to rtburchk bo lishini isbotlng. 9*. rombning ktt digonlid P v K nuqtlr tnlngn (5-rsm). gr =P = K bo ls, PK to rtburchk kvdrt eknligini isbotlng. 0*. prllelogrmmning digonlid P v K nuqtlr tnlngn. gr P =K bo ls, PK to rtburchk prllelogrmm eknligini isbotlng.

8 T RTURHKLR (dvomi) E E F -msl. trpetsiydgi kichik sosning uchidn tomong prllel to g ri chiziq o tkzilgn. Ntijd, hosil bo lgn uchburchkning perimetri sm g teng. gr trpetsiyning perimetri 6 sm bo ls, tomon uzunligini toping. Yechilishi. Msl shrtig ko r, o tkzilgn to g ri chiziq kesmsi E bo lsin, E nuqt tomond yotdi (-rsm). kesm trpetsiyni E uchburchk v E prllelogrmmg jrtdi. Xususn, =E v =E. Msl shrtig ko r, P = = E + E = + E+ E + = = P E + = + = 6 (sm). undn, = yoki = 6 sm eknligini topmiz. Jvob: 6 sm. -msl. Rombning digonllridn biri sm, tomoni es 5 sm. Romb yuzini toping. romb, S = sm, = 5 sm. =? Yechilishi. Romb digonllri kesishish nuqtsi bo lsin (-rsm). Und, romb osssig ko r, (sm), = 90. Pifgor teoremsidn foydlnib kesmni topmiz: = =5 7 =576 yoki = sm. Und = = =8 (sm). Romb yuzini hisoblsh formulsig ko r, (sm ). Jvob: 6 sm. -msl. Teng yonli trpetsiyning yon tomoni 0 sm, soslri es sm v 6 sm. Trpetsiy yuzini toping. Yechilishi. trpetsiyd = = 0 sm, = sm, = 6 sm bo lsin. Trpetsiyning E v F blndliklrini o tkzmiz (-rsm). Und, EF = = (sm), (sm). To g ri burchkli E uchburchkk Pifgor teoremsini qo llb, E blndlikni topmiz: E = E = 0 = 56 yoki E = 6 sm. Trpetsiyning yuzini topmiz: Jvob: 8 sm. = (sm ). Svol, msl v topshiriqlr. trpetsiyning kichik sosi 7 sm g teng. Uning uchidn tomonig prllel to g ri chiziq o tkzilgn. Hosil bo lgn uchburchk perimetri 6 sm g teng. Trpetsiy perimetrini toping.. To g ri chiziqni kesib o tmydign kesmning uchlri bu to g ri chiziqdn 8 sm v 8 sm uzoqlikd joylshgn. Kesm o rtsidn to g ri chiziqqch bo lgn msofni toping.. Tomonlri sm v 5 sm, yuzi es 0 sm bo lgn prllelogrmm ysng.. Rombning digonllridn biri 80 sm, tomoni es 8 sm. Romb yuzini toping. 5. Prllelogrmmning 5 g teng bo lgn o tms burchgi uchidn tushirilgn blndligi sm g teng bo lib, u o zi tushgn tomonni teng ikkig bo ldi. ) Shu tomonni toping. b) Prllelogrmmning o tms burchklri uchlrini tutshtiruvchi digonli biln tomonlri orsidgi burchklrni toping. d) Prllelogrmm perimetri v yuzini toping. 6. Rombning o tms burchgi uchidn tushirilgn blndlik romb tomonini teng ikkig bo ldi. gr rombning tomoni 6 sm bo ls, romb yuzini toping. 7. To g ri burchkli uchburchkning gipotenuzsi sm, ktetlrining yig indisi es 7 sm. Uchburchk yuzini toping. 8*. To g ri burchkli trpetsiyning bir burchgi 5 g, o rt chizig i es 8 sm g teng. gr trpetsiy soslri nisbti :8 g teng bo ls, trpetsiyning yon tomonlrini toping. 9*. ( ) trpetsiy mrkzli ylng tshqi chizilgn. = 90 eknligini isbotlng. 5

9 MTEMTIK MSLLR XZINSI M lumki, keyingi pytlrd borot kommuniktsiy tenologiylri jud tez sur tlr biln rivojlnib bormoqd. Internet to ri borgn sri olis qishloqlrni hm qmrb olmoqd. Shu kung kelib, Internetning World-Wide-Web Jhon borot trmog id shunchlik ko p borot mnblri joylshtirilgnki, bu zindn foydlnish hr bir odm uchun hm qrz, hm frz hisoblndi. Jumldn, bir-biridn qiziq shundy web-shiflr borki, ulrdn itiyoriy fnni, jumldn, geometriyni o rgnish jryonid smrli foydlnish mumkin. Quyid shu borot mnblrining mnzillrini berishni lozim topdik. u web-shiflrdn siz o zbek, rus, ingliz v boshq tillrd mtemtik olmidgi eng oirgi yngiliklr, elektron resurs mrkzlrid sqlnyotgn ko plb elektron drsliklr, mtemtikdn msofdn turib t lim olish kurslri v ulrning mterillri, mzkur drslik shiflrig kirgn v kirmgn turli-tumn nzriy mterillr, mtemtikdn drs berib kelyotgn tjribli o qituvchilrning drs ishlnmlri v metodik tvsiylri, son-snoqsiz msllr, misollr v ulrning yechimlri, turli dvltlrd o tkzilyotgn mtemtik ko rik tnlovlr v olimpidlr to g risidgi m lumotlr v ulrd tqdim etilgn msllr hmd ulrning yechimlri, qiziqrli mtemtik msllr v ulrning yechimlri biln tnishishingiz mumkin. Xlq t limi vzirligi borot-t lim portli Xlq t limi vzirligi qoshidgi Multimedi mrkzi syti Xlq t limi vzirligi syti liy v o rt msus t lim vzirligining t lim portli Pedgogik t lim mussslri portli T lim mussslri portli Mtemtikdn qo shimch mterillr syti borot-t lim resurslri trmog i Msofdn turib o qitish syti (rus tilid) Msofdn turib o qitish syti portli (ingliz tilid) T limni rivojlntirish instituti syti (rus tilid) Umumt lim portli (rus tilid) Internetdn t lim portli (rus tilid) Rossiy dvlt kutubonsi portli (rus tilid) Test olish mrkzi serveri (rus tilid) Internet resurslri elektron kutubonsi (rus tilid) «Æèâàÿ ãåîìåòðèÿ» dsturini qo llb-quvvtlsh syti Mtemtikdn v informtikdn sirtqi tnlov (rus tilid) nline-drsliklr (rus tilid) Hmmsi e v π sonlri hqid (rus tilid) Elektron qo llnmlr kutubonsi. Sirtqi mtemtik olimpidlr Mtemtikdn 000 dn ortiq msllr (rus tilid) Mtemtik gimnstik. Mtemtik msllr v boshqotirmlr Mtemtik kleydoskop (rus tilid) Internetdgi mtemtik jurnl (rus tilid) Mtemtikdn msllr (rus tilid) I XSHSH GEMETRIK SHKLLR Ushbu bobni o rgnish ntijsid siz quyidgi bilim v mliy ko nikmlrg eg bo lsiz: ilimlr: o shsh shkllrning t rifini v belgilnishini bilish; uchburchklrning o shshlik lomtlrini bilish; gomotetiy tushunchsini bilish. mliy ko nikmlr: ikkit o shsh uchburchklrdn mos elementlrni top olish; uchburchklrning o shshlik lomtlrini isbotlshg v hisoblshg oid msllrni yechishd qo lly olish; gomotetiydn foydlnib,o shsh ko pburchklrni ysy olish. 6 7

10 5 K PURHKLRNING XSHSHLIGI ) b) Kundlik turmushd teng shkllrdn tshqri shkli (ko rinishi) bir il, lekin o lchmlri turlich bo lgn shkllrg ko p duch kelmiz. Tri v geogrfiy fnlrid turli msshtbd ishlngn ritlrdn foydlngnsiz. Sinf dosksig ilindign v drsliklrd tsvirlngn respublikmizning ritlri turli o lchmd, lekin ulr bir il shkld (ko rinishd). Shuningdek, bitt fototsmdn turli o lchmdgi fotosurtlr tyyorlndi. u surtlrning o lchmlri turlich bo ls-d, bir il ko rinishd, y ni ulr bir-birig o shydi (-rsm). Mshq. -rsmd to rtt romb tsvirlngn. Ulrdn fqt d) v e) romblr bir il ko rinishg eg. u romblr nimsi biln boshq romblrdn jrlib turibdi? Keling, buni birglikd niqlylik.. Rsmdn ko rinib turibdiki, =, =. Rombning tomonlri teng bo lgni uchun, d) e) tenglikni hosil qilmiz. u holtd romblrning mos tomonlri proporsionl deb yuritildi.. v romblrning mos burchklri o zro teng. Hqiqtn hm, = = 5, = = 5, = = 5, = = 5. Shundy qilib, bu romblrning bir-birig o shshligining sbbi mos tomonlrining proporsionlligi v mos burchklrining tengligi dey olmiz. Itiyoriy ko pburchklr o shshligi tushunchsi hm shu sosd kiritildi. urchklri soni bir il (demk, tomonlrining soni hm bir il) bo lgn ko pburchklr bir il nomli ko pburchklr deb yuritildi. Ikkit bir il nomli E v E ko pburchklrning burchklri mn bu trtibd teng bo lsin: =, =, =, =, E= E. undy burchklr mos burchklr deb yuritildi. U hold, v, v, v, E v E, E v E tomonlr mos tomonlr deyildi. T rif. ir il nomli ko pburchklrdn birining burchklri ikkinchisining burchklrig mos rvishd teng, mos tomonlri es proporsionl bo ls, bundy ko pburchklr o shsh ko pburchklr deb tldi (-rsm). Ko pburchklr o shshligi F F Mos burchklr teng =, =, =, =, E= E = = = E = E =k E E Mos tomonlr proporsionl belgisi biln ko rstildi. shsh ko pburchklrning mos tomonlri nisbtig teng bo lgn k son o shshlik koeffitsiyenti deyildi. Svol, msl v topshiriqlr. shsh ko pburchklr t rifini yting.. shshlik koeffitsiyenti nim v u qndy niqlndi?. gr v EF uchburchklrd =05, =5, E =05, F =0, =, sm, = 5, sm, = 7,6 sm, E =5,6 sm, F =,8 sm, EF =, sm bo ls, ulr o shsh bo ldimi?. -rsmd tsvirlngn ) v b) romblr nim sbbdn o shsh ems? b) v d) romblr-chi? 5. -rsmdgi v uchburchklr o shsh bo ls,, kesmlr uzunligini v o shshlik koeffitsiyentini toping rsmd. =, = 8, = = 0, = 5, = 5., v kesmlrni toping. 7*. uchburchk v tomonlrining o rtlri mos rvishd P v Q bo lsin. PQ eknligini isbotlng. F E 5 F 6 E 0 8 9

11 6 XSHSH UHURHKLR V ULRNING XSSLRI Eng sodd ko pburchk bo lmish uchburchklr o shshligini o rgnmiz. Teorem. Ikkit o shsh uchburchk perimetrlrining nisbti o shshlik koeffitsiyentig teng. u teoremni mustqil isbotlng. Teorem. Ikkit o shsh uchburchk yuzlri nisbti o shshlik koeffitsiyentining kvdrtig teng. (- rsm), k o shshlik koeffitsiyenti Isbot. Teorem shrtig ko r, S :S =k. emk, ko pburchklr o shshligi t rifig ko r, =, =, = v. ) ( E ) = eknligidn foydlnib, ulrni -b rsmdgidek ustm-ust qo ymiz v tegishli yssh hmd belgilshlrni mlg oshirmiz. Quyidgi uchburchklr yuzlrini topmiz v ulrning nisbtlrini qrymiz: S =. E ; S =. E ; S =. F ; => S =. F ; () tenglikni hdm-hd () tenglikk bo lsk, teng burchkk eg bo lgn uchburchklr yuzlrining nisbti uchun () tenglikni hosil qilmiz. b) F S = (), S S => = S (). () u yerd shrtg ko r, eknligini hisobg olsk, tenglik kelib chiqdi. Teorem isbotlndi. -msl. shsh uchburchklrning mos tomonlri nisbti shu tomonlrg tushirilgn blndliklr nisbtig tengligini isbotlng (-rsm).,, blndliklr = Yechilishi. erilgn uchburchklrning o shshlik koeffitsiyenti k bo lsin. Und, : = k; S : S =k () bo ldi. Ikkinchi tomondn,. () () v () tengliklrdn yoki. Shundy qilib, nisbt hm, nisbt hm k g teng, y ni. Svol, msl v topshiriqlr. shsh uchburchklr yuzlri nisbti hqidgi teoremni yting v isbotlng.. Ikkit o shsh v uchburchklr berilgn. gr S = 5 sm v S = 8 sm bo ls, o shshlik koeffitsiyentini toping.. Ikkit o shsh uchburchk yuzlri 65 m v 60 m. irinchi uchburchkning bir tomoni 6 m bo ls, ikkinchi uchburchkning ung mos tomonini toping.. erilgn uchburchk tomonlri 5 sm, 5 sm v 0 sm. gr perimetri 5 sm bo lgn uchburchk berilgn uchburchkk o shsh bo ls, uning tomonlrini toping. 5. v bu uchburchklrning mos tomonlri nisbti 7:5 g teng. gr uchburchk yuzi uchburchk yuzidn 6 m g ortiq 0 F bo ls, bu uchburchklr yuzlrini toping rsmd berilgnlrdn foydlnib, uchburchklrning o shsh yoki o shsh emsligini E niqlng. 0

12 7 UHURHKLR XSHSHLIGINING IRINHI LMTI Follshtiruvchi mshq -rsmd tsvirlngn uchburchklr ichidn o shshlrini niqlng. Ulrning o shshligini qndy niqldingiz? T rifg ko r, ikkit uchburchkning o shshligini niqlsh uchun ulr burchklrining tengligini v mos tomonlrining proporsionl eknligini tekshirish lozim bo ldi. Uchburchklr uchun bu ish nch osonlshr ekn. Quyid keltirildign teoremlr shu ususd bo lib, ulr uchburchklr o shshligining lomtlri deb nomlndi. Teorem. (Uchburchklr o shshligining lomti). gr bir uchburchkning ikkit burchgi ikkinchi uchburchkning ikkit burchgig mos rvishd teng bo ls, bundy uchburchklr o shsh bo ldi (-rsm).,, =, = Isbot.. Uchburchk ichki burchklri yig indisi hqidgi teoremg ko r, =80 ( + ), = =80 ( + ) ИЦиИ emk, v uchburchklrning burchklri mos rvishd teng.. Shrtg ko r, =, =. Teng burchkk eg bo lgn uchburchklr yuzlrining nisbti hqidgi teoremg ko r v. u tengliklrning o ng qismlrini tenglb, bir il hdlr qisqrtirils, tenglik hosil bo ldi. Xuddi shu singri, = v = tengliklrdn foydlnib, tenglikni olmiz. Shundy qilib, v uchburchklrning burchklri teng v mos tomonlri proporsionl, y ni bu uchburchklr o shsh. Teorem isbotlndi. Msl. uchburchkning ikki tomonini kesib o tuvchi v uchinchi tomonig prllel bo lgn E to g ri chiziq uchburchkdn ung o shsh uchburchk jrtishini isbotlng (-rsm). Isbot. v E uchburchklrd umumiy, = E ( v E prllel to g ri chiziqlrni kesuvchi biln kesgnd hosil bo lgn mos burchklr teng bo lgni uchun) (-rsm). emk, uchburchklr o shshligining lomtig E ko r, E. Svol, msl v topshiriqlr. Uchburchklr o shshligining t rifi v lomtini o zro solishtiring.. Uchburchklr o shshligining lomtini isbotlng.. Rsmdgi m lumotlr sosid ni toping. ) b) d) 6 0. Rsmdgi m lumotlr sosid ni toping. ) b) d) prllelogrmmning tomonid E nuqt olingn. E v nurlr F nuqtd kesishdi. ) gr E = 8 sm, E = sm, = 7 sm, E = 0 sm bo ls, EF v F ni; b) gr = 8 sm, = 5 sm, F = sm bo ls, E v E ni toping. 6. Rsmd trpetsiy. Rsmdgi m lumotlr sosid ni toping. ) b) d), *. ittdn o tkir burchklri teng bo lgn ikkit to g ri burchkli uchburchklr o shsh eknligini isbotlng. 8*. uchburchkning tomonid nuqt olingn. gr = bo ls, v uchburchklr o shsh eknligini isbotlng. Shuningdek, = v = 9 sm bo ls, kesmni toping. 0

13 8 UHURHKLR XSHSHLIGINING IKKINHI LMTI Teorem. (Uchburchklr o shshligining TT lomti). gr bir uchburchkning ikki tomoni ikkinchi uchburchkning ikki tomonig proporsionl v bu tomonlr hosil qilgn burchklr teng bo ls, bundy uchburchklr o shsh bo ldi (-rsm).,, =, = Isbot. =, = b o l d i - gn qilib uchburchk ysymiz (-rsm). U lomt bo yich uchburchkk o shsh bo ldi. ( ) (shrtg ko r). u ikki tenglikdn, = eknligini niqlymiz. Und, uchburchklr tengligining TT lomtig ko r, =. Xususn, =. Lekin ysshg ko r, = edi.emk, =.U hold, = v = bo lgni uchun, uchburchklr o shshligining lomtig ko r,. Teorem isbotlndi. Msl. v kesmlr nuqtd kesishdi, = sm, = sm, =0 sm, =0 sm bo ls, v uchburchklr yuzlri nisbtini toping. Yechilishi: Shrtg ko r, emk, uchburchkning ikki tomoni uchburchkning ikki tomonig proporsionl v bu tomonlr orsidgi mos burchklr vertikl burchklr bo lgni uchun: =. Shuning uchun, uchburchklr o shshligining TT lomtig ko r, v o shshlik koeffitsiyenti. k= =. Endi o shsh uchburchklr yuzlrining nisbti hqidgi teoremni qo llymiz: = k =9. Jvob: 9. Svol, msl v topshiriqlr ). Uchburchklr o shshligining t rifi v TT lomtini o zro solishtiring.. Uchburchklr o shshligining TT lomtini isbotlng.. Uchidgi burchklri teng bo lgn teng yonli uchburchklrning o shshligini ) ; b) TT,5 lomtdn foydlnib isbotlng. b). -rsmd tsvirlngn v uchburchklr o shsh bo ldimi? gr o shsh bo ls, bu uchburchklr perimetrining nisbtini toping. 5. v nurlr nuqtd kesishdi. gr : = : =, = 7 sm bo ls, 5,5,5 kesmni hmd v uchburchklr yuzlri nisbtini toping. d) 6. v uchburchklrd =, : = : = :. ) gr kesm dn 5 sm ortiq bo ls, 6 v tomonlrni toping. b) gr kesm dn 6 sm km bo ls, 5 v tomonlrini toping.,6 d) gr berilgn uchburchklrning yuzlri yig indisi 00 sm bo ls, hr qysi uchburchkning yuzini toping. 7. gr bir to g ri burchkli uchburchkning ktetlri ikkinchi to g ri burchkli uchburchkning mos ktetlrig proporsionl bo ls, bu uchburchklr o shsh bo lishini isbotlng. 8. uchburchkd = 5 m, = 0 m, = m. Uchburchkning tomonig =9 m kesm, tomonig es E= m kesm qo yildi. E kesmni toping. 9. Ktetlri dm v dm bo lgn to g ri burchkli uchburchk biln bir kteti 8 dm v gipotenuzsi 0 dm bo lgn to g ri burchkli uchburchk o shsh bo lishini isbotlng. 0*. kesm v l to g ri chiziq nuqtd kesishdi. l to g ri chiziqq v perpendikulrlr tushirilgn. gr = sm, = sm v = sm bo ls,, v kesmlrni toping. 5

14 9 UHURHKLR XSHSHLIGINING UHINHI LMTI Teorem. (Uchburchklr o shshligining TTT lomti). gr bir uchburchkning ucht tomoni ikkinchi uchburchkning ucht tomonig mos rvishd proporsionl bo ls, bundy uchburchklr o shsh bo ldi.,, = = (-rsm) Isbot. uchburchkning tomonid = bo ldign qilib nuqtni E belgilymiz. nuqtdn tomong prllel qilib o tkzilgn to g ri chiziq tomonni E nuqtd kessin. Und uchburchklr o shshligining lomtig ko r, E v o shsh bo ldi. U hold teorem shrtig v t rifig ko r: v. mmo ysshg ko r, =. Und yuqoridgi tengliklrdn =E tenglik hosil bo ldi. Shundy qilib, uchburchklr tengligining TT lomtig ko r, E v teng v E. emk,. Teorem isbotlndi. Msl. gr ikkit teng yonli uchburchkdn birining sosi v yon tomoni ikkinchisining sosi v yon tomonig proporsionl bo ls, bu uchburchklrning o shsh eknligini isbotlng., =,, =, =. trpetsiyning v yon tomonlri dvom ettirils, E nuqtd kesishdi. gr = 5 sm, =0 sm, = 6 sm, =5 sm bo ls, E uchburchk yuzini toping. 5. Trpetsiyning soslri 6 sm v 9 sm, blndligi 0 sm. Trpetsiyning digonllri kesishgn nuqtdn soslrigch bo lgn msoflrni toping. 6. Istlgn ikkit teng tomonli uchburchk o shsh bo lishini isbotlng. 7. sosi sm, blndligi 8 sm bo lgn teng yonli uchburchk ichig kvdrt shundy ichki chizilgnki, kvdrtning ikkit uchi uchburchk sosid, qolgn ikki uchi es yon tomonlrd yotdi. Kvdrt tomonini toping. 8*. tkir burchkli uchburchkning v blndliklri o tkzilgn. eknligini isbotlng. 9. шйцуваошушушушушушуощшоikkit o shsh uchburchk yuzlri 6 v g teng. Ulrdn birining perimetri ikkinchisinikidn 6 g ortiq. Ktt uchburchkning perimetrini toping. Triiy lvhlr. u voqe miloddn vvlgi VI srd bo lgn. u vqtd yunonlr geometriy biln deyrli shug ullnishms edi. Yunon fylsufi Fles misr fni biln tnishish uchun tshrif buyurgn. Misrliklr ung qiyin msl berdi: ulkn pirmidlrdn birining blndligini qndy hisoblsh mumkin? Fles bu mslning sodd v jozibli yechimini topdi. U tyoqchni yerg qoqdi v shundy dedi: Qchonki shu tyoqch soysining uzunligi tyoqchning uzunligi biln teng bo ls, pirmid soysining uzunligi pirmid blndligi biln teng bo ldi. Fles fikrini soslshg hrkt qiling! ) b) e) f) d),5,5, Isbot. erilgn =, = tengliklr v = nisbtdn = = tengliklrni hosil qilmiz. emk, uchburchklr o shshligining TTT lomtig ko r,. Svol, msl v topshiriqlr. Uchburchklr o shshligining TTT lomtini yting v isbotini byon qiling.. = sm, = sm, = sm, = 8 sm, = sm, = 6 sm eknligi m lum. v uchburchklr o shsh bo ldimi?. -rsmdgi o shsh uchburchklr juftliklrini ko rsting. Quyosh nuri E 6 7

15 T G RI URHKLI UHURHKLRNING XSHSHLIK 0 LMTLRI M lumki, to g ri burchkli uchburchklrning bittdn burchklri to g ri burchkdn ibort bo ldi. Shuning uchun bundy uchburchklrning o shshlik lomtlri nch soddlshdi. -teorem. To g ri burchkli uchburchklrning bittdn o tkir burchgi mos rvishd teng bo ls, ulr o shsh bo ldi. -teorem. To g ri burchkli uchburchklrning ktetlri mos rvishd proporsionl bo ls, ulr o shsh bo ldi. -teorem. To g ri burchkli uchburchklrdn birining gipotenuzsi v kteti ikkinchisining gipotenuzsi v ktetig mos rvishd proporsionl bo ls, ulr o shsh bo ldi. u lomtlrdn dstlbki ikkitsining to g riligi o z-o zidn rvshn. Keling, uchinchi lomtni isbotlylik.,, =90, = 90, = Isbot. uchburchkning tomonig E = bo ldign qilib E kesmni qo ymiz v E ni o tkzmiz (-rsm). Und uchburchklr o shshligining lomtig ko r, E v o shsh bo ldi. shsh uchburchklr mos tomonlrining proporsionlligidn: E. Ysshg ko r, E =. emk, tenglik o rinli. oshq tomondn, teorem shrtig ko r, = () () v () tengliklrdn E = eknligini niqlymiz. v E uchburchklrni qrymiz:. E = (ysshg ko r);. E = (isbotlngn tenglik). To g ri burchkli uchburchklrning bittdn kteti hmd gipotenuzsi bo yich tenglik lomtig ko r, = E. Ikkinchi tomondn es E. U hold, bo ldi. Teorem isbotlndi. () Msl. gr ikkit teng yonli uchburchkdn birining yon tomoni v blndligi ikkinchisining yon tomoni v blndligig proporsionl bo ls, bu uchburchklrning o shsh eknligini isbotlng (-rsm). Isbot. To g ri burchkli v uchburchklrni qrymiz. Shrtg ko r, ulrning bittdn kteti v gipotenuzsi o zro proporsionl. emk, -teoremg sosn. Und. Teng yonli uchburchk sosig tushirilgn blndlikning bissektris hm bo lishini hisobg olsk, = = = bo ldi. Ntijd, v uchburchklrd = v tengliklrg eg bo lmiz. b) ) 5 emk, uchburchklr o shshligining TT 8 lomtig ko r,. So rlgn tsdiq isbotlndi. Svol, msl v topshiriqlr. To g ri burchkli uchburchklrning o shshlik lomtlrini yting v isbotlng. d) f). -rsmdn o shsh uchburchklrni toping. 0. Ktetlri m v m bo lgn to g ri burchkli uchburchkk o shsh uchburchkning bir kteti m bo ls, ikkinchi kteti nech m bo ldi?. Yuzlri m v 8 m bo lgn ikkit to g ri e) g) burchkli uchburchklr o shsh. gr birinchi uchburchkning bir kteti 6 m bo ls, ikkinchi 8 5 uchburchk ktetlrini toping. 5. ir ylng ikkit o shsh to g ri burchkli uchburchk ichki chizilgn. u uchburchklrning 0 tengligini isbotlng. 6*. Ktetlri 0 sm v sm bo lgn to g ri burchkli uchburchkk bitt burchgi umumiy bo lgn kvdrt ichki chizilgn. gr kvdrtning bitt uchi gipotenuzd eknligi m lum bo ls, kvdrtning tomonini toping. 7*. uchburchk berilgn. Ung EF romb shundy ichki chizilgnki,, E v F nuqtlr mos rvishd uchburchkning, v tomonlrid yotdi. gr = c, = b bo ls, romb toопmonini toping. 8 9

16 XSHSHLIK LMTLRINING ISTLSHG IR MSLLRG TTIQLRI E F -msl. Uchburchk bissektrissi o zi tushgn tomonni qolgn ikki tomong proporsionl kesmlrg jrtishini isbotlng., bissektris (-rsm) = Yechilishi. to g ri chiziqq E v F perpendikulrlr tushirmiz. Und F = E bo lgni uchun, to g ri burchkli F v E uchburchklr o shsh bo ldi. shsh uchburchklrning mos tomonlri proporsionlligidn Shung o shsh F E. F E. () v () tengliklrni solishtirsk, yoki bo ldi. u v kesmlr v kesmlrg proporsionl eknligini ngltdi. -msl. uchburchkning b i s s e k t r i - ssi uchburchkk tshqi chizilgn ylnni v P nuqtlrd kesdi. P eknligini isbotlng (-rsm). Yechilishi. P v d:. = P shrtg ko r;. = P chunki ulr bitt yoyg P tirlgn. emk, uchburchklr o shshligining lomtig ko r, P. Svol, msl v topshiriqlr. Uchburchk bissektrissi o zi tushgn tomond jrtgn kesmlri v uchburchkning qolgn tomonlri orsidgi proporsionllikni yozib ko rsting.. To g ri burchkli uchburchkning to g ri burchgidn blndlik o tkzilgn. = bo lishini isbotlng. Hosil bo lgn shkld necht () () o zro o shsh uchburchklrni ko rst olsiz?. -rsmdgi m lumotlr sosid ni toping.. uchburchklrning bissektrissi o tkzilgn. gr =,5 m; =,5 m v uchburchk perimetri m bo ls, uning v tomonlrini toping. 5. uchburchk medinlri N nuqtd kesishdi. gr uchburchk yuzi 87 dm bo ls, N uchburchk yuzi nimg teng? 6. uchburchk medinlri kesishgn N nuqtdn v tomonlrgch bo lgn msoflr mos rvishd dm v dm. gr = 8 dm bo ls, tomonni hisoblng. 7*. Trpetsiyning sosig prllel to g ri chiziq yon tomonlridn birini m:n nisbtd bo lishi m lum. u to g ri chiziq uning ikkinchi yon tomonini qndy nisbtd bo ldi? Qiziqrli msllr Geometriy v ingliz tili. Quyid ingliz tilid berilgn geometrik mslni yechib ko ring-chi! u biln hm ingliz tilidn, hm geometriydn nimg qodirligingizni sinb olsiz. ) issection Puzzle: Let M be the midpoint of the side of given tringle. The tringle hs been dissected into prts X, Y, Z long the lines MN nd MK pssing through M such tht MN is prllel while MK is perpendikulr to the bse (picture ). Show how the three pieces cn be fitted together to mke rectngle, respectively two different prllelogrms. ) Look t the picture 5 nd proof + + = 90. ) 9 7,5,5 b) d) Y M N X Z K 5 0

17 MSLLR YEHISH -msl. Uchburchklrning o shshligidn foydlnib, uchburchk o rt chizig i uchburchkning bir tomonig prllel v shu tomonning yrmig teng eknligini isbotlng.,mn o rt chiziq (-rsm): M=M,N = N M MN trpetsiy,, (-rsm) MN, MN = =. Yechilishi. -qdm. v uchburchklrning burchklrini tqqoslymiz. =, chunki bu burchklr ichki lmshinuvchi burchklr., chunki trpetsiy(ks hold, + = + =80, y ni bo lib, trpetsiy bo lmy qolr edi). U hold, = v =. -qdm. Endi v o shsh uchburchklrning mos tomonlri nisbtini yozmiz: N, bundn =.. Yechilishi. v MN uchun: umumiy,. Shuning uchun, uchburchklr o shshligining TT lomtig ko r, bu ikki uchburchk o shsh. Endi mushohdni mn bundy dvom ettirmiz:, -msl. gr soslri v bo lgn trpetsiyning digonli uni ikkit o shsh uchburchkk jrts, =. bo lishini isbotlng., Svol, msl v topshiriqlr. ) o yi 70 sm bo lgn odm soysining ) uzunligi m bo ls, blndligi 5, m bo lgn simyog och soysining uzunligini toping. b) Ikkit teng yonli uchburchkning uchidgi burchklri teng. irinchi uchburchkning yon tomoni 7 sm, sosi 0 sm g, ikkinchi uchburchkning sosi 8 sm g teng. Ikkinchi uchburchkning yon tomonini toping. b). -rsmdgi hr bir chizmdn o shsh uchburchklrni ko rsting.. uchburchkning P medinsi tomong prllel v uchlri v tomonlrd yotgn istlgn kesmni teng ikkig bo lishini isbotlng. d). Uchburchkning uchlri uning o rt chizig ini o z ichig olgn to g ri chiziqdn teng msofd yotishini isbotlng. 5. ylng ichki chizilgn to rtburchk digonllri nuqtd kesishdi. eknligini isbotlng. 6. uchburchk ichki sohsid nuqt v,, nurlrd mos rvishd E, F, K nuqtlr olingn (-rsm). gr EF v FK bo ls, v EFK uchburchklr o shsh eknligini isbotlng. 7*. Trpetsiyning digonllri kesishish nuqtsidn o tuvchi to g ri chiziq trpetsiy soslridn birini m:n nisbtd bo ldi. u to g ri chiziq ikkinchi sosni qndy nisbtd bo ldi? 8. gr uchburchkning yuzi S g teng bo ls, E 5-rsmd biln belgilngn soh yuzini toping. 5 ) b) d) E À E K K F K F F

18 ILIMINGIZNI SIN K RING I. Testlr. Quyidgi t riflrdn qysi biri to g ri? ) Ikkit uchburchkning burchklri mos rvishd teng bo ls, ulr o shsh deyildi; ) Ikkit uchburchkning tomonlri mos rvishd teng bo ls, ulr o shsh deyildi; ) Ikkit uchburchkning mos tomonlri proporsionl v mos burchklri teng bo ls, ulr o shsh deyildi; E) Ikkit uchburchkning mos tomonlri v mos burchklri teng bo ls, ulr o shsh deyildi.. Ikkit o shsh uchburchk yuzlrining nisbti nimg teng? ) shshlik koeffitsiyentig; ) Ulrning mos tomonlri nisbtig; ) Ulrning perimetrlri nisbtig; E) shshlik koeffitsiyentining kvdrtig.. Quyidgi tsdiqlrdn qysi biri to g ri? ) Uchburchklrdn birining ikkit burchgi ikkinchisining ikkit burchgig teng bo ls, ulr o shsh bo ldi; ) Uchburchklrdn birining ikkit tomoni ikkinchisining ikki tomonig teng bo ls, ulr o shsh bo ldi; ) Ikkit uchburchkning bittdn burchklri teng v ikkitdn tomonlri proporsionl bo ls, ulr o shsh bo ldi; E) Ikkit uchburchkning bittdn burchklri teng v bittdn tomonlri proporsionl bo ls, ulr o shsh bo ldi.. To g risini toping. gr ikkit uchburchk o shsh bo ls, ulrning ) lndliklri teng bo ldi; ) Tomonlri proporsionl bo ldi; ) Tomonlri teng bo ldi; E) Yuzlri teng bo ldi. 5. shsh uchburchklrning perimetrlri nisbti nimg teng? ) Mos tomonlr nisbtining kvdrtig; ) shshlik koeffitsiyentig; ) shshlik koeffitsiyentining kvdrtig; E) Yuzlri nisbtig. II. Msllr. uchburchkning v tomonlri o rtlri mos rvishd E v F nuqtlr bo lsin. gr EF uchburchk yuzi sm bo ls, uchburchk yuzini toping.. uchburchkning tomonig prllel to g ri chiziq v tomonlrni mos rvishd N v P nuqtlrd kesdi. gr N =, N =, P =,6 bo ls, tomonni toping.. tkir burchkli uchburchkning tomonid K nuqt olingn. gr K=, K= v uchburchkning blndligi g teng bo ls, K nuqtdn kesmgch bo lgn msofni toping.. prllelogrmning tomoni o rtsidgi K nuqtdn o tkzilgn K nur biln nur F nuqtd kesishdi. gr =, K = 5 v = 5 bo ls, F uchburchk perimetrini hisoblng. 5. uchburchk ichki sohsid olingn nuqtdn v tomonlrg prllel to g ri chiziqlr o tkzilgn. u to g ri chiziqlr tomonni mos rvishd P v Q nuqtlrd kesdi. gr PQ =, = 7 v uchburchk yuzi 98 g teng bo ls, P Q PQ uchburchk yuzini niqlng (-rsm). K 6. trpetsiyning v soslrid mos rvishd K v L nuqtlr olingn. KL kesm trpetsiyning digonllri kesishgn nuqtdn o tdi. gr L=, L= 5 v K = bo ls, K kesmni toping (-rsm). L 5 III. zingizni sinb ko ring (nmunviy nzort ishi). trpetsiyning digonli uni o shsh v uchburchklrg jrtdi. und = m, = 9 m bo ls, digonl uzunligini hisoblng.. Ikkit o shsh uchburchkning yuzi 50 dm v dm, ulrning perimetrlri yig indisi 7 dm bo ls, hr bir uchburchkning perimetrini toping.. Rsmd tsvirlngn uchburchklrning o shshligini isbotlng. v F to g ri chiziqlrning o zro joylshishi to g risid nim dey olsiz?. (Qo shimch). tkir burchkli uchburchkning v E blndliklri o tkzilgn.. = E. bo lishini isbotlng. 5,5 5 6 F 0 0 E

19 GEMETRIK SHKLLRNING XSHSHLIGI ldingi drslrd ko pburchklrning o shshligi tushunchsi biln tnishdik. u tushunchni fqt ko pburchklr uchun ems, blki istlgn geometrik shkllr uchun hm kiritish mumkin. gr F v F* shkllr berilgn bo lib, F shklning F hr bir nuqtsig F* shklning biror nuqtsi mos F * qo yilgn bo ls v bund F* shklning hr bir nuqtsig F shklning fqt bitt nuqtsi mos kels, (-rsm) F shkl F* shklg lmshtirilgn deyildi. T rif. gr F shklni F * shklg lmshtirishd nuqtlr orsidgi msoflr bir il son mrt o zgrs, bundy lmshtirishg o shshlik lmshtirishi deyildi (-rsm). X Y F F F * u t rifni quyidgich tlqin qilish mumkin: ytylik, biror lmshtirish ntijsid F shklning itiyoriy X * X, Y nuqtlrig F* shklning X*, Y * nuqtlri mos qo yilgn bo lsin. gr X*Y *=k. XY, k>0 bo ls, bundy lmshtirishg o shshlik lmshtirishi Y * deyildi. und k brch X v Y nuqtlr uchun bir F * il son bo lib, u o shshlik koeffitsiyenti deb yuritildi. gr F v F * shkllr berilgn bo lib, bu shkllrdn birining ikkinchisig o tkzdign o shshlik lmshtirishi mvjud bo ls, F v F * shkllr o zro X *Y* = kxy k o shshlik koeffitsiyenti o shsh deyildi. Shkllrning o shshligi F F * kbi yozildi. gr o shshlik koeffitsiyenti k ni hm ko rstish lozim bo ls, F F * trzd hm belgilndi. gr o shshlik lmshtirishid X nuqtg X* nuqt mos qo yilgn bo ls, X nuqt X* nuqtg lmshdi yoki o tdi deyildi. Teorem. shshlik lmshtirishi ) to g ri chiziqni to g ri chiziqq; b) nurni nurg; d) burchkni (uning kttligini sqlgn hold) burchkk; e) kesmni (uzunligi bu kesmdn k mrt uzun bo lgn) kesmg o tkzdi. Isbot. ) shshlik koeffitsiyenti k bo lgn Z* Z lmshtirishd bir to g ri chiziqd yotgn turli X, Y Y Y* v Z nuqtlr mos rvishd X*, Y * v Z * nuqtlrg lmshsin (-rsm). X X, Y, Z nuqtlrdn biri, ytylik, Y qolgn X* ikkitsining orsid yotsin. U hold XZ =XY +YZ. shshlik lmshtirishi t rifig ko r: X*Z *=k XZ=k (XY+YZ )= k XY+ k YZ= X*Y*+Y*Z *. u tenglikdn X*, Y * v Z* nuqtlrning bir to g ri chiziqd yotishi kelib chiqdi. Teoremning isbotini fqt ) tsdiq uchun keltirdik. Qolgn tsdiqlrd isbotlshni sizg mshq triqsid qoldirmiz. Svol, msl v topshiriqlr. shshlik lmshtirishi nim?. Qndy shkllr o shsh deyildi?. Eni sm, bo yi sm bo lgn to g ri to rtburchkk o shsh, o shshlik koeffitsiyenti g teng bo lgn to rtburchk ysng.. -rsmd mktb hovlisining tri :000 msshtbd tsvirlngn. lchsh ishlrini bjrib, ) hovlining; b) mktb binosining; d) gulzorlrning; e) sport mydonining; f) bog ning hqiqiy o lchmlrini toping. 5. gr rit :50000 msshtbd tsvirlngn bo ls (5-rsm), bod v zod qishloqlri mrkzlri orsidgi msofni toping. 6. shshlik lmshtirishid nurlr orsidgi burchk sqlnishini isbotlng. 7*. shshlik lmshtirishid ) prllelogrmm prllelogrmmg; b) kvdrt kvdrtg; d) to g ri to rtburchk to g ri to rtburchkk; e) trpetsiy trpetsiyg lmshishini isbotlng. 8*. uchburchk o shshlik lmshtirishid *** uchburchkk lmshdi. gr o shshlik koeffitsiyenti 0,6 g v uchburchk perimetri sm g teng bo ls, *** uchburchk perimetrini toping rsmdn o shsh to g ri to rtburchklr juftliklrini toping v o shshlik koeffitsiyentlrini niqlng. sm 5 sm 6 ) 6 og Gulzor b) 6 bod qishlog i Mktb binosi d) e),5 Sport mydoni f) Gulzor stnsiy 6 zod qishlog i 6 7

20 5 XSHSH K PURHKLRNING XSSLRI -teorem. shsh ko pburchklr perimetrlrining nisbti o shshlik koeffitsiyentig teng. Isbot. Hqiqtn hm,... n v... n ko pburchklr o shsh v o shshlik koeffitsiyenti k bo ls, =k, =k,..., n =k n bo ldi. undn P= n = k +k +...+k n =k ( n )=k P tenglikni hosil qilmiz. Teorem isbotlndi. -teorem. shsh ko pburchklrni bir il sondgi o shsh uchburchklrg jrtish mumkin. Isbot. ytylik, E v E ko pburchklr o shsh bo lib, o shshlik koeffitsiyenti k bo lsin. zro mos v uchlrdn, E v, E digonllrni o tkzmiz. Ntijd, E E ko pburchklr bir il sondgi uchburchklrg jrldi. Hosil bo lgn uch juft mos uchburchklrning o shshligini ko rstmiz... hu n k i, b u uchburchklrd, shrtg ko r, =,. Uchburchklr o shshligining TT lomtig ko r,.. E E. u o shshlik -bnddgi kbi isbotlndi.. E E. Hqiqtn, E v E burchklrni qrymiz: E = E, E = E. u yerd, E = E (berilgn o shsh beshburchklrning mos burchklri). = (o shsh v uchburchklrning mos burchklri). emk, E = E. v E hmd v E tomonlrni qrymiz: = k, chunki ulr o zro o shsh v uchburchklrning mos tomonlri, E = k E, chunki ulr hm berilgn o shsh beshburchklrning mos tomonlri. emk, uchburchklr o shshligining TT lomtig ko r, E E. Itiyoriy o shsh ko pburchklr uchun hm shu kbi mushohdlr yroqli bo lishi rvshn. Teorem isbotlndi. -teorem. shsh ko pburchklr yuzlrining nisbti o shshlik koeffitsiyentining kvdrtig teng. Isbot. ytylik,... n v... n ko pburchklr o shsh v k o shshlik koeffitsiyenti bo lsin. U hold,,..., n- n uchburchklr S k S mos rvishd,,,..., n- n uchburchklrg o shsh bo lib, o shsh n- uchburchklr yuzlrining nisbti k g n- n teng bo ldi (-rsm): n S = k S, S =k S,..., S n- n = k S n- n. u tengliklrning mos qismlrini qo shsk, S... n = k S... n bo ldi. Teorem isbotlndi. Msl. Perimetrlri 8 sm v sm bo lgn ikkit o shsh ko pburchk yuzlrining nisbtini toping. Yechilishi. ) shsh ko pburchklr perimetrlrining nisbti o shshlik koeffitsiyentig teng eknligidn foydlnib, k = : 8 = : eknligini topmiz. ) shsh ko pburchklr yuzlrining nisbti o shshlik koeffitsiyentining kvdrtig teng bo lgni uchun izlngn nisbt k = g teng. Jvob:. Svol, msl v topshiriqlr. shsh ko pburchklr perimetrlrining nisbti nimg teng?. shsh ko pburchklr yuzlrining nisbti hqidgi teoremni shrhlng.. Uchburchk biln to rtburchk o shsh bo lishi mumkinmi?. Yuzlri 6 m v m bo lgn ikkit to rtburchk o shsh. shshlik koeffitsiyentini toping. 5. Ikkit ko pburchkning perimetrlri 8 sm v 6 sm g, yuzlrining yig indisi es 0 sm g teng. Ko pburchklr yuzlrini toping. 6. Perimetri 8 sm bo lgn uchburchkning bir tomonig prllel qilib o tkzilgn to g ri chiziq undn perimetri sm g v yuzi 6 sm g teng uchburchk jrtdi. erilgn uchburchk yuzini toping. 7. nuqtg nisbtn simmetrik shkllr o shsh bo ldimi? qq nisbtn simmetrik shkllr-chi? Ulrning o shshlik koeffitsiyenti nimg teng? 8. To rtburchk shklidgi pt mydoni ritd yuzi sm bo lgn to rtburchk biln tsvirlndi. gr rit msshtbi :000 bo ls, mydonning hqiqiy yuzini hisoblng. 9*. Yuzlri 8 sm v sm bo lgn ikkit o shsh uchburchk perimetrlrining yig indisi 8 sm g teng. Uchburchklrning perimetrlrini toping. 8 9

21 6 GMTETIY V XSHSHLIK F X X X N Y Y Z F M F * N * X * X * X * F* Y * Y * Z * M * Eng sodd o shsh lmshtirishlrdn biri gomotetiydir. ytylik, F shkl, nuqt v k musbt son berilgn bo lsin. F shklning istlgn X nuqtsi orqli X nur o tkzmiz v bu nurd uzunligi k.x bo lgn X* kesmni qo ymiz (- rsm). Shu usul biln F shklning hr bir X nuqtsig X* nuqtni mos qo ydign lmshtirish gomotetiy deyildi. und, nuqt gomotetiy mrkzi, k soni gomotetiy koeffitsiyenti, F v gomotetiy ntijsid F shkl lmshdign F * shkllr es gomotetik shkllr deyildi. Teorem. Gomotetiy o shshlik lmshtirishi bo ldi. Isbot. Itiyoriy mrkzli, k koeffitsiyentli gomotetiyd F shklning X v Y nuqtlri X* v Y* nuqtlrg o tsin (-rsm). U hold, gomotetiy t rifig ko r, XY v X*Y* uchburchklrd umumiy v =k bo ldi. emk, XY v X*Y* uchburchklr ikki tomoni v ulr orsidgi burchgi bo yich o shsh. Shuning uchun * * *, ususn, X*Y*=k. XY. Teorem isbotlndi. Msl. burchk tomonlrig urinuvchi itiyoriy ikki yln gomotetik bo lishini v nuqt bu gomotetiy uchun mrkz eknligini isbotlng. Yechilishi. Mrkzlri v bo lgn ylnlr burchk tomonlrig urinsin (-rsm). u ylnlrning gomotetik eknligini isbotlymiz. ylnlr nurg mos rvishd X v X* nuqtlrd uringn bo lsin (-rsm). U hold, X X * (chunki X = X * v X = X * = 9 0 ). * undn, ng tomondgi nisbtni k biln ) 0 <k < belgilymiz v koeffitsiyenti, mrkzi bo lgn gomotetiyni qrymiz. ytylik, bu gomotetiyd mrkzli ylnning istlgn M nuqtsi M* F F * n u q t g l m s h g n b o l s i n. U h o l d M *= k. * M yoki b) k undn, X= M bo lgni uchun M*= X* tenglikni hosil qilmiz. u M* nuqt mrkzi nuqtd, rdiusi F F * X* g teng bo lgn ylnd yotishini bildirdi. emk, qrlyotgn ylnlr o zro gomotetik ekn. Follshtiruvchi mshq -rsmd gomotetiy koeffitsiyenti ) 0<k <; b) k bo lgn gomotetik shkllr tsvirlngn. Gomotetiy koeffitsiyentining qiymtig qrb gomotetik shkllrning siqilishi yoki cho zilishi hqid qndy ulos chiqrish mumkin? Svol, msl v topshiriqlr. Gomotetiy nim? Gomotetiy mrkzi, koeffitsiyenti-chi?. Gomotetiy o shshlik lmshtirishi eknligini izohlng.. Uchburchk chizing. Uchburchk ) ichki sohsid; b) tshqi sohsid nuqt belgilng v koeffitsiyenti g teng bo lgn mrkzli gomotetiyni qrb, berilgn uchburchkk gomotetik uchburchk ysng.. Perimetrlri 8 sm v 7 sm bo lgn ikkit romb o zro gomotetik. u romblr tomonlri v yuzlrining nisbtlrini toping. 5. Gomotetiyd X nuqt X* nuqtg, Y nuqt Y * nuqtg o tdi. gr X, X *, Y, Y * nuqtlr bir to g ri chiziqd yotms, shu gomotetiy mrkzini toping. 6. Koeffitsiyenti g teng bo lgn gomotetiyd X nuqt X * nuqtg o tishi m lum. Shu gomotetiy mrkzini ysng. 7. ylng gomotetik shkl yln bo lishini isbotlng. 8. yln chizing. Mrkzi yln mrkzid v koeffitsiyenti ) ; b) ; d) ; e) g teng bo lgn gomotetiyd chizilgn ylng gomotetik bo lgn shkllrni quring. 9. urchk v uning ichki sohsid nuqt berilgn. urchk tomonlrig urinib, nuqtdn o tuvchi yln ysng. 0

22 7 ) b) XSHSH K PURHKLRNI YSSH Shu pytgch teoremlrni isbotlshd v msllrni yechishd turli o shsh uchburchklrni ysb keldik. shsh ko pburchklr qndy ysldi? Quyid shu biln tnishsiz. Msl. erilgn to rtburchkk o shsh, o shshlik koeffitsiyenti g teng bo lgn to rtburchk ysng (-rsm). Yssh. Tekislikd itiyoriy nuqtni olmiz. Undn v to rtburchkning uchlridn o tuvchi,, v nurlrni o tkzmiz. u nurlrd nuqtdn =, =, = v = kesmlrni qo ymiz. Hosil bo lgn to rtburchk izlngn to rtburchkdir. soslsh. eknligini isbotlymiz.. Mos tomonlrning proporsionlligi. ) ; () b). () () v () tenglikdn eknligini hosil qilmiz. To rtburchklrning boshq mos tomonlri proporsionlligini uddi shung o shsh isbotlsh mumkin.. Mos burchklrning tengligi. shsh uchburchklrning mos burchklri teng bo lgni uchun, =, =. U hold, = + = = + =, y ni to rtburchklrning mos v burchklri teng. Xuddi shung o shsh to rtburchklrning boshq mos burchklri tengligi isbotlndi. emk, v to rtburchklr d) o shsh. Tomonlri itiyoriy sond bo lgn ko pburchkk o shsh ko pburchk hm uddi shu kbi ysldi. Gomotetiy mrkzini bu msld to rtburchk tshqi sohsidn tnldik. Umumn olgnd, gomotetiy mrkzini to rtburchkning ichki sohsid (- rsm), biror uchid (-b rsm) yoki biror tomonid (-d rsm) yotdign qilib tnlshimiz hm mumkin edi. Gomotetiy mrkzini qyerd olmylik, berilgn to rtburchkk o shsh v o shshlik koeffitsiyenti g teng bo lgn to rtburchklr o zro teng bo ldi. Svol, msl v topshiriqlr. erilgn ko pburchkk o shsh ko pburchkni yssh ketm-ketligini yting.. ftringizg biror E beshburchk chizing. Gomotetiy yordmid bu beshburchkk o shsh, o shshlik koeffitsiyenti 0,5 g teng bo lgn beshburchk ysng. Gomotetiy mrkzi ) nuqtd; b) beshburchk ichid; d) tomond bo lgn hollrni lohid ko ring.. Ktklrni inobtg olgn hold, -rsmd berilgn shkllrni dftringizg chizing: ) yproqq o shshlik koeffitsiyenti g teng bo lgn yproq; b) bliqchg o shshlik koeffitsiyenti 0,8 g teng bo lgn bliqchni gomotetiy yordmid chizing.. F ko pburchk F ko pburchkk o shsh, k o shshlik koeffitsiyenti. P, P, S, S hrflr biln mos rvishd bu ko pburchklrning perimetrlri v yuzlri belgilngn. Quyidgi jdvlni dftringizg ko chiring v uni to ldiring. P P S S k ) b) 8 8 d) e) 0

23 8 MLIY MSHG ULT bo lsin.qog ozg =, =b bo lgn uchburchk ysymiz. Und v uchburchklr ikki burchgi bo yich o shsh bo ldi (- v -rsmlr). undn, yoki 750 m. lndlikni niqlsh. Yerd turib, Toshkent teleminorsining blndligini topylik. Minorning uchi nuqtning soysi nuqt bo lsin. EF tyoqni vertikl trzd shundy qoqmizki (-rsm), tyoqning E uchi soysi hm nuqtd bo lsin. Minorning sosini biln belgilymiz. Hosil bo lgn, to g ri burchkli v EF uchburchklr o shsh bo ldi. Shuning uchun, yoki E m, F msoflrni v EF tyoq uzunligini o lchb, hosil bo lgn formuldn teleminor blndligi kesm uzunligini topmiz. Msln, gr EF = m, =750 m, F = m ekni m lum bo ls, u hold =75 m bo ldi.. orib bo lmydign joygch bo lgn msofni o lchsh. ytylik, nuqtdn borish mumkin bo lmgn nuqtgch bo lgn msofni niqlsh lozim bo lsin (-rsm). nuqtdn borib bo ldign shundy nuqtni belgilymizki, undn qrgnd v nuqtlr ko rinib tursin hmd msofni o lchb bo lsin. sboblr yordmid v burchklrni o lchymiz. ytylik, = v = b F m msof v, kesmlrni o lchb, ntijd hosil bo lgn formul yordmid kesm hisoblndi. Hisoblsh ishlrini osonlshtirish mqsdid : nisbtni 00:, 000: kbi nisbtd olish mumkin. Msln, =0 m, =7, =58 bo ls, qog ozd uchburchkni =7, =58, =0 mm qilib chizmiz. kesmni o lchb, uning 5 mm eknligini topmiz. Und, izlngn msof 5 m bo ldi.. rol dengizi hqid mliy ish. -rsmd suv hvzsining kosmik kemdn olingn oldingi surti tsvirlngn. U sosid tegishli o lchsh v hisoblsh ishlrini bjrib, suv hvzsi yuzining tqribiy qiymtini toping. Svol, msl v topshiriqlr. gr bo yi,7 m bo lgn odm soysining uzunligi,5 m bo ls, soysining uzunligi 0, m bo lgn drt blndligi qnch bo ldi?. 5-rsmd tsvirlngn minor blndligini niqlng. 6 7 Msshtb: : m. 6-rsmdgi ikkit o shsh v uchburchklr yordmid dryoning kengligini (enini) niqlsh zrur. gr =00 m, = m v = m bo ls, dryoning eni ( ) ni toping.. nhor qirg og idgi E drtning suvdgi ksi nuqtdgi odmg ko rinypti. gr =65 sm, =0 sm, =,8 m bo ls, drt blndligini toping (7-rsm). 5. Hovlid biror drtni tnlng v uning blndligini niqlng. u ishni qndy bjrgningiz hqid hisobot tyyorlng.,8 m E 5,8 m 5

24 9 MSLLR YEHISH -msl. Uzunliklri mos rvishd v b bo lgn v ustunlr tik qilib o rntilgn. Ulrning musthkmligini oshirish mqsdid v, v uchlri nuqtd kesishuvchi po lt simlr biln mhkmlngn (-rsm). Rsmd berilgn m lumotlr sosid ) v tengliklrni isbotlng; b) tenglikning to g ri eknligini ko rsting v uni shrhlng. Yechilishi. ) Msl shrtig ko r:. E. Shuning uchun, y ni.. E. Shuning uchun, b y ni. () b) () v () tengliklrni hdm-hd qo shsk, E yoki m n d tenglikni hosil qilmiz. emk, ustunlr qndy o rntilmsin, po lt simlr kesishgn nuqt yerdn bir il blndlikd bo lr ekn. -msl. trpetsiyning v yon tomonlrid M v N nuqtlr olingn. und MN kesm trpetsiy soslrig prllel v trpetsiy digonllri kesishgn nuqtdn o tdi. gr =, = b bo ls, ) M; b) N; d) MN kesmlrni toping (-rsm). Yechilishi. ) v uchburchklr lomtg ko r o shsh, chunki =, =. undn, yoki ) v M uchburchklr hm lomtg ko r o shsh, chunki M =, = M. undn, yoki. () ) () v () tengliklrning o ng qismlrini tenglshtirib, () () tenglikni v undn eknligini topmiz. ) Yuqoridgidek yo l tutib () () tenglikni, keyin es () v () tengliklrning mos qismlrini qo shib tenglikni hosil qilmiz. Jvob: ) ; b) ; d). Esltm. u msl yechimidn M = = N eknligi kelib chiqdi. Svol, msl v topshiriqlr. uchburchkning v yon tomonlrid v E nuqtlr olingn. gr E, = 6, = v E = bo ls, tomonni toping.. Ikkit o shsh ko pburchkning yuzlri 8 dm v 7 dm g teng, ulrdn birining perimetri ikkinchisinikidn 6 dm g km. Ktt ko pburchkning perimetrini toping.. Perimetri m bo lgn uchburchk uchburchkning tomonlri o rtlrini, uchburchk uchburchk tomonlri o rtlrini, uchburchk es uchburchk tomonlri o rtlrini tutshtirishdn hosil qilingn bo ls, uchburchkning perimetri qnch bo ldi?. Ikkit o shsh uchburchkning perimetrlri 8 dm v 6 dm g, yuzlrining yig indisi 0 dm g teng. Ktt uchburchkning yuzini toping. 5. Romb tomonlrining o rtlri to g ri to rtburchk uchlri bo lishini isbotlng. 6. uchburchk ysng. u uchburchkk o shsh v yuzi uchburchk yuzidn 9 mrt kichik bo lgn uchburchkni ysng. 7*. E v F nuqtlr mos rvishd prllelogrmning v tomonlri o rtlri. F v E to g ri chiziqlr digonlni teng uch qismg bo lishini isbotlng (-rsm). M b F N E 6 7

25 0 MSLLR YEHISH. Teng yonli uchburchkning sosidgi burchk bissektrissi bu uchburchkdn o zig o shsh uchburchk jrtdi. Uchburchk burchklrini niqlng (-rsm, =, ).. yln ysng v und nuqt belgilng. Mrkzi nuqtd v koeffitsiyenti g teng bo lgn gomotetiyd berilgn ylng gomotetik bo lgn yln ysng.. Ikkit o shsh ko pburchk perimetrlrining nisbti : kbi. Ktt ko pburchkning yuzi 7 bo ls, kichik ko pburchkning yuzini toping.. -rsmd Quyoshning to l tutilgn holti tsvirlngn. gr Quyosh rdiusi km, y rdiusi 760 km v Yerdn ygch bo lgn msof 800 km bo ls, Yerdn Quyoshgch bo lgn msofni toping. 5. ) itt ylng ikkit o shsh ko pburchk ichki chizilgn. u ko pburchklr teng bo ldimi? b) itt ylng ikkit o shsh ko pburchk tshqi chizilgn. u ko pburchklr teng bo ldimi? 6*. ir kvdrtning tomonlri ikkinchi kvdrt tomonlrig prllel. gr kvdrtlr bir-birig teng bo lms, ulr gomotetik bo lishini isbotlng (-rsm). 7. uchburchkning v tomonlri to rtt teng kesmlrg bo lindi v bo linish nuqtlri tomong prllel kesmlr biln tutshtirildi. gr = sm bo ls, hosil bo lgn kesmlr uzunliklrini toping. 8. gr rsmlr yni bir pytd surtg olingn bo ls, berilgn m lumotlr sosid ikkinchi binoning blndligini toping (-rsm). 70 m 0 m -? 9 m I G IR Q SHIMH MSLLR V M LUMTLR I. Testlr. Ikkit o shsh uchburchk uchun noto g ri tsdiqni toping:. Yuzlri nisbti o shshlik koeffitsiyentig teng;. Mos medinlri nisbti o shshlik koeffitsiyentig teng;. Mos bissektrislri nisbti o shshlik koeffitsiyentig teng; E. Mos blndliklri nisbti o shshlik koeffitsiyentig teng.. Ikkit gomotetik ko pburchk uchun to g ri tsdiqni toping:. Ulr teng;. Ulr o shsh;. Ulr tengdosh; E. To g ri jvob yo q.. Uchburchk medinlri uchun noto g ri tsdiqni ko rsting:. ir nuqtd kesishdi;. Kesishish nuqtsid : nisbtd bo lindi;. ir-birig teng; E. Hr biri uchburchkni ikkit tengdosh qismg jrtdi.. Uchburchk bissektrislri uchun noto g ri tsdiqni ko rsting:. ir nuqtd kesishdi;. Kesishish nuqtsid : nisbtd bo lindi;. zi tushgn tomonni qolgn ikki tomong proporsionl kesmlrg jrtdi; E. zi chiqqn uchdgi burchkni teng ikkig bo ldi. 5. Ikkit o shsh ko pburchk uchun noto g ri tsdiqni toping:. Ulrning tomonlri soni teng;. Ulrning burchklri soni teng;. Mos tomonlri proporsionl; E. Yuzlrining nisbti o shshlik koeffitsiyentig teng. II. Msllr. soslri 6 m v m bo lgn trpetsiy digonllri kesishgn nuqtdn soslrg prllel to g ri chiziq o tkzilgn. To g ri chiziqning trpetsiy ichidgi qismi uzunligini toping.. uchburchkd ==0, =8. gr v uchburchk bissektrislri bo ls, kesmni toping.. nuqtdn borib bo lmydign nuqtgch bo lgn msofni niqlsh uchun tekis joyd nuqt tnlndi. Keyin msof, v burchklr o lchndi v uchburchkk o shsh uchburchk ysldi. gr = m, = 6, sm, = 7, sm bo ls, msofni toping.. Koeffitsiyenti k= bo lgn gomotetiyd F ko pburchk F ko pburchkk lmshdi. gr F ko pburchkning perimetri sm v yuzi,5 sm bo ls, F ko pburchkning perimetri v yuzini toping. 5. o yi 80 sm bo lgn odm soysining uzunligi, m bo lgn pytd blndligi m bo lgn simyog och soysining uzunligi nech metr bo ldi? 8 9

26 6. Xritd Toshkent v Urgnch shhrlri orsidgi msof 8,67 sm. gr rit msshtbi : bo ls, Toshkent v Urgnch shhrlri orsidgi msofni toping. III. zingizni sinb ko ring (nmunviy nzort ishi). -rsmd berilgn m lumotlr sosid drt blndligini toping.. uchburchkning tomonlri = 5 sm, = 6 sm, = 7 sm. u uchburchkning 6 tomonig prllel to g ri chiziq tomonini P nuqtd, tomonini es K nuqtd kesdi. gr PK = sm bo ls, PK uchburchk perimetrini M toping. N. -rsmd MN. gr = 6 sm, = 0 sm bo ls, MN kesmni toping.. (Qo shimch). Romb tomonlrining o rtlri to g ri to rtburchkning uchlri bo lishini isbotlng. ) b) Qiziqrli msllr. mrt kttlshtirib ko rstilgn ko zgu-lup biln qrlgnd li burchk kttligi qnchg o zgrdi?. ) Uchburchkli chizg ich rsmid tsvirlngn ichki v tshqi uchburchklr o shshmi (- rsm)? b) -b rsmdgi romning ichki v tshqi qirrlrini tsvirlovchi to rtburchklr o shshmi?. Quyidgi rus tilid berilgn mslni yechib ko ring. u biln hm rus tilidn, hm geometriydn nimg qodirligingizni bilib olsiz. Íà -ðèñóíêå èæîáðàæåíà ðóññêàÿ èãðóøêà ìàòð øêà. Âûïîëíèâ ñîîòâåòñòâóþùèå èçìåðåíèÿ, íàéòè E F êîýôôèöèåíò ïîäîáèÿ èãðóøåê: ) è ; b) è ; d) è F; e) è E. II UHURHK TMNLRI V URHKLRI RSIGI MUNSTLR Ushbu bobni o rgnish ntijsid siz quyidgi bilim, ko nikm v mlkg eg bo lsiz: ilimlr: itiyoriy burchkning sinusi,kosinusi,tngensi v kotngensi t riflrini bilish; burchkning rdin o lchovini bilish; sosiy trigonometrik yniytlrni bilish; uchburchkning yuzini burchk sinusi yordmid hisoblsh formulsini bilish; sinuslr v kosinuslr teoremsini bilish. mliy ko nikmlr: b zi burchklrning sinusi,kosinusi,tngensi v kotngensini hisobly olish; sosiy trigonometrik yniytlrni misollr yechishd qo lly olish; uchburchk yuzini uning ikki tomoni v ulr orsidgi burchgi bo yich hisobly olish; sinuslr,kosinuslr teoremsidn foydlnib hisoblshg v isbotlshg doir msllrni yechish. 50 5

27 TKIR URHKNING SINUSI, KSINUSI, TNGENSI V KTNGENSI To g ri burchkli uchburchkd =90 bo ls, tomon gipotenuz, tomon burchk qrshisidgi ktet, tomon es burchkk yopishgn ktet deyildi (-rsm). To g ri burchkli uchburchk o tkir burchgining sinusi deb, shu burchk qrshisidgi ktetning gipotenuzg nisbtig ytildi. To g ri burchkli uchburchk o tkir burchgining kosinusi deb, shu burchkk yopishgn ktetning gipotenuzg nisbtig ytildi. To g ri burchkli uchburchk o tkir burchgining tngensi deb, shu burchk qrshisidgi ktetning yopishgn ktetg nisbtig ytildi. To g ri burchkli uchburchk o tkir burchgining kotngensi deb, shu burchkk yopishgn ktetning qrshisidgi ktetg nisbtig ytildi. α burchkning sinusi, kosinusi, tngensi v kotngensi mos rvishd sinα, cosα, tgα v ctgα shklid belgilndi (o qilishi: «sinus lf», «kosinus lf», «tngens lf», «kotngens lf»). Yuqoridgi t riflrdn quyidgi formullr kelib chiqdi:. = = ; tg=. tg =.. tg. ctg =. = tg. ctg =.. = = ; ctg =. ctg =. Teorem. ir to g ri burchkli uchburchkning o tkir burchgi ikkinchi to g ri burchkli uchburchkning o tkir burchgig teng bo ls, bu o tkir burchklrning sinuslri (kosinusi, tngensi v kotngensi) hm teng bo ldi. Isbot. To g ri burchkli v uchburchklrd ( = = 90 ) = bo lsin (- rsm). U hold, v uchburchklr lomtg ko r o shsh bo ldi. Shuning uchun,. u tengliklrdn yoki sin = sin eknligini topmiz. u o tkir buchklrning kosinusi, tngensi v kotngenslri hm teng bo lishi yuqoridgig o shsh isbotlndi. Teorem isbotlndi. Msl. uchburchkd =90, =8 sm, =5 sm bo ls, uning burchgi sinusi, kosinusi, tngensi v kotngensini toping. Yechilishi. Pifgor teoremsidn foydlnib, uchburchkning gipotenuzsini topmiz: ) = + =8 +5 = 89, = 7 (sm). Uchburchkning burchgi qrshisidgi ktet, burchgig yopishgn ktet es (-rsm). Und, t riflrg ko r, Yoki tg = = = ; Jvob: ctg = = =. b) Svol, msl v topshiriqlr 6. To g ri burchkli uchburchkning o tkir burchgi sinusi, kosinusi, tngensi v kotngensi deb nimg ytildi?. tkir burchkning sinusi, kosinusi, tngensi v kotngensi nimg bog liq, nimg bog liq ems?. -rsmdgi m lumotlr sosid sin, cos, sin, cos ni toping.. To g ri burchkli uchburchkning gipotenuzsi sm g, kteti es sm g teng. Uchburchkning burchgi sinusi, kosinusi, tngensi v kotngensini toping. 5. gr to g ri burchkli ( = 90 ) uchburchkd ) =5, = 7 ; b) = 5, =; d) =, =0; e) =, =5 bo ls, v burchklrning sinusi, kosinusi, tngensi v kotngenslrini toping. 6. gr uchburchkd = 90, cos= v = sm bo ls, uchburchkning qolgn tomonlrini toping. 7. gr uchburchkd =90, sin = v =6 sm bo ls, uchburchkning qolgn tomonlrini toping. 7 d)

28 MSLLR YEHISH tgα= =. =, ctg α= tgα =. -msl. uchburchkd = 90 v sin = 0,6. gr uchburchkning blndligi,8 sm bo ls, uning ktetini v bu ktetning gipotenuzdgi proyeksiysini toping. Yechilishi. To g ri burchkli uchburchkni qrymiz (-rsm). Und, sinusning t rifig ko r, undn, (sm).,8 5 Msllr yechishd jud sqotdign yn bir muhim tenglikning to g riligini ko rstylik: to g ri burchkli uchburchkd (-rsm) Pifgor teoremsig ko r: = +. U hold, sin + cos = + = = =. sin + cos = tenglik trigonometriyning sosiy yniyti deb tldi ( trigonometriy so zi yunonch uchburchklrni o lchymn degn m noni ngltdi). -msl. gr cos = bo ls, sinα, tgα v ctgα ni toping. Yechilishi. sosiy trigonometrik yniytg ko r: sin α = cos α sinα= cos α = =. Und, Pifgor teoremsidn foydlnib ktetning gipotenuzdgi proyeksiysi ni topmiz: = = 8,8 =6, (sm). Jvob: 8 sm; 6, sm. -msl. gr uchburchkd = 90 v cos = bo ls, uchburchk tomonlri qndy nisbtd bo ldi (-rsm). Yechilishi. urchk kosinusining t rifig ko r, cos = emk, =. gr =5 desk, und = =. Pifgor teoremsig ko r, = = 69 5 =. Shundy qilib, : : = 5 : :. Jvob: 5 : : kbi. Svol, msl v topshiriqlr. -rsmdgi m lumotlr sosid quyidgilrni: ) sin, cos, tg, ctg; b) sin, cos, tg, ctg ni toping.. gr sin=0,5 bo ls, cos, tg v ctg ni toping.. gr cos=0,6 bo ls, sin, tg v ctg ni toping.. To g ri burchkli ( =90 ) uchburchkd =7 sm v sin = bo ls: ) uchburchkning blndligini; b) ktetning gipotenuzdgi proyeksiysini; d) gipotenuzni; e) ikkinchi ktetni toping gr uchburchkd =90, sin= v = =5 sm bo ls, uchburchkning gipotenuzsig tushirilgn blndligini toping. 6*. gr ) sinα= ; b) cosα=α; d) tgα= ; e) ctgα= bo ls, burchkni ysng. 7. uchburchkd = sm, =0 sm, sin=0,7 bo ls, uchburcчhk yuzini toping (5-rsm). 8. uchburchkd blndlik, =7 sm, = sm v tg = bo ls, uchburchk yuzini toping 6 (5-rsm). E 9. ( ) trpetsiyd sin=0,5; =8, =6, =0 bo ls, trpetsiy yuzini toping (6-rsm). 0. rombd sin=0,8 v =5 sm bo ls, romb yuzini toping. *. Teng yonli uchburchkning sosig tushirilgn blndligi 5 sm, sosi es 0 sm bo ls, uchburchkning ) burchklrini; b) yon tomonini; d) yuzini toping.. To g ri burchkli uchburchkd sin= v sin= bo lishi mumkinmi? ) b) d)

29 ZI URHKLRNING SINUSI, KSINUSI, TNGENSI V KTNGENSINI HISLSH. 5 grdusli burchkning sinusi, kosinusi, tngensi v kotngensini hisoblsh. Teng yonli to g ri burchkli uchburchkni qrymiz (-rsm). u uchburchkd =, = =5 bo lsin. Und Pifgor teoremsig ko r, = + = 5 yoki =. undn ni hosil qilmiz. 5 Shundy qilib, -msl. To g ri burchkli ( = 90 ) uchburchkd =5 v = 6 sm. Uchburchkning qolgn tomonlrini toping (-rsm). Yechilishi. yoki = = 6 (sm); yoki = =6 (sm). Jvob: 6 sm; 6 sm.. 0 v 60 burchklrning sinusi, kosinusi, tngensi v kotngensini hisoblsh. urchklri =0, =60 v = 90 bo lgn uchburchkni qrymiz (-rsm). 0 grdusli burchk qrshisid yotgn ktet gipotenuzning yrmig teng bo lgni uchun yoki. undn tengliklrni topmiz. sosiy trigonometrik yniytg ko r: ;. α ning 0, 5, 60 g teng qiymtlrid topilgn sinα, cosα, tgα v ctgα uchun qiymtlrni jdvl ko rinishid jmlymiz: -msl. To g ri burchkli uchburchkning gipotenuzsi 0 sm v burchklridn biri 60. Uning tgα qolgn tomonlrini toping. ctgα Yechilishi. -rsmdn foydlnmiz. Und (sm), ) (sm). Jvob: 5 sm; 5 sm. 0 Svol, msl v topshiriqlr. α burchk 0,5, 60 g teng bo ls, sinα, cosα, b) 6 tgα v ctgα qiymti nimg teng? Jvobingizni 60 soslng. d). -rsmdgi uchburchklr perimetrlrini toping.. -rsmdgi uchburchklr burchklrini toping. 8. α ning 0, 5, 60 g teng qiymtlrid sinα, 5 cosα, tgα v ctgα uchun qiymtlr jdvlini yod oling. 5. To g ri burchkli uchburchkning bir o tkir burchgi 0, ung yopishgn ktet 6 dm. Uning qolgn tomonlrini toping. 6. Teng yonli uchburchkning sosi 0 sm g, bir burchgi es 0 g teng. Uning yuzini toping. 7. uchburchkd =90, =5 sm, sin=.uchburchkning qolgn tomonlrini v cos, tg hmd ctg ni toping. 8. igonllri 5 sm v 5 sm bo lgn rombning burchklrini toping. ) b) d) sinα cosα Topilgnlrg ko r, ;

30 5 MSLLR YEHISH Follshtiruvchi mshq Jdvlning bo sh ktklrini to ldiring. sinα cosα tgα ctgα -msl. gr rombd =0 v =6 sm bo ls, rombning blndligi v yuzini toping (-rsm). Yechilishi. ) Rombning bir tomonig yopishgn burchklri yig indisi 80 g teng bo lgni uchun =80 =60. Rombning E blndligini o tkzib (-rsm), to g ri burchkli E uchburchk hosil qilmiz. Und, 60 E yoki (sm). 6 sm ) Endi rombning yuzini topmiz: S =.E=6. =8 (sm ). Jvob: h= sm; S =8 sm. -msl. teng yonli trpetsiyning kichik sosi 5 sm. gr = 60, = 6 sm bo ls, trpetsiyning yuzini toping. Yechilishi. Trpetsiyning E v F blndliklrini o tkzmiz (-rsm). Und, to g ri burchkli E uchburchkdn (sm), 5 (sm). 6 undn tshqri E =F, EF = bo lgni uchun, 60 E F = E + EF + F = = (sm). Trpetsiyning yuzini topish formulsig ko r, (sm ). Jvob: sm. Svol, msl v topshiriqlr. Teng yonli to g ri burchkli uchburchkning gipotenuzsi sm. Uning yuzini hisoblng.. lndligi sm bo lgn teng tomonli uchburchk perimetrini toping.. -rsmd berilgnlrg ko r teng yonli trpetsiylr yuzini toping.. To g ri burchkli trpetsiyning o tkir burchgi 0 g, blndligi sm g v kichik sosi 6 sm g teng. Trpetsiyning perimetri v yuzini toping. 5. yln vtri 0 grdusli yoyni tortib turdi. gr yln rdiusi 0 sm bo ls, vtr uzunligini toping. 6*. Teng yonli uchburchkning uchidgi burchgi ) 0 ; b) 90 ; d) 60. Uchburchk blndligining sosig nisbtini hisoblng. 7*. -rsmd tsvirlngn pt irmonining yon yoqlri teng yonli trpetsiy, usti es kvdrt shklid. Rsmd berilgnlrdn foydlnib, irmonni to liq yopish uchun qnch mto zrurligini niqlng. 8. Yengil mshin dovonning yuqorig ko trilish qismid 0 m yo l bosdi. gr yo lning gorizontg nisbtn ko trilish burchgi 5 bo ls, yengil mshin nech metr blndlikk ko trilgn (5-rsm)? Msus klkultord burchkning sinusi, kosinusi v tngensini hisoblsh sin v cos tugmchlri bor msus klkultord trigonometrik funksiylrning qiymtlri quyidgich hisoblndi: urchk grduslrd berilgn bo lsin, msln, sin0 :. Klkultor yoqilib, EG (grdus) tugmchsi bosildi.. So ng tugmchlr 0 Sin trtibd bosildi v tegishli jvob: 0,5 olindi. sin0 = 0,5. gr msus klkultor bo lms, drslik oiridgi ilovd keltirilgn trigonometrik funksiylrning qiymtlri jdvlidn foydlnishingiz mumkin. ) 0 b) 6 5 d) m 9 m 0 m 85 0 m 5 0 m

31 6 ) b) 0 N 80 GH LGN URHKNING SINUSI, KSINUSI, TNGENSI V KTNGENSI P M(, y) y y y y P M(, y) To g ri burchkli y koordintlr sistemsining I v II chorklrid joylshgn, rdiusi birlik kesmg teng, mrkzi koordintlr boshid bo lgn yrim yln ysymiz (-rsm). ylnni M (;y) nuqtd kesuvchi P nurni o tkzmiz. u nurning nur biln hosil qilgn burchgini biln belgilymiz. P nurning nur biln ustm-ust tushgn holdgi burchkni 0 li burchk siftid qbul qilmiz. M lumki, o tkir burchk bo lgnd (- rsm), bu burchkning sinusi, kosinusi, tngensi v kotngensi to g ri burchkli M uchburchkdn sinα= cosα= tgα= ctgα = tengliklr yordmid niqlndi. gr M=, M = y, = eknligini hisobg olsk, sinα=y, cosα=, tgα=, ctgα= () tengliklrg eg bo lmiz. Umumiy hold, 0 dn 80 gch bo lgn burchkning sinusi, kosinusi, tngensi v kotngensini hm () formul orqli niqlymiz: Istlgn α (0 α 80 ) burchkning sinusi deb M nuqtning ordintsi y g ytildi. Istlgn α (0 α 80 ) burchkning kosinusi deb M nuqtning bsisssi g ytildi. Istlgn α (0 α 80, α 90 ) burchkning tngensi deb M nuqt ordintsining bsisssig nisbtig ytildi. Istlgn α (0 <α<80 ) burchkning kotngensi deb M nuqt bsisssining ordintsig nisbtig ytildi. M uchburchkd +M =M yoki +y =. sinα=y v cosα= eknligini hisobg olsk, istlgn α (0 α 80 ) burchk uchun sin α + cos α = () tenglikni hosil qilmiz. u tenglik sosiy trigonometrik yniyt deb tlib, u oldingi drslrd o tkir burchklr uchun isbotlngn edi. mliy topshiriq. irlik kesmni 5 sm g teng deb olib, to g ri burchkli koordintlr sistemsini chizing.. Koordintlr sistemsining I v II chorgid joylshgn, mrkzi koordintlr boshid v rdiusi birlik kesmg teng yrim yln chizing.. Yrim ylnni M nuqtd kesdign v nur biln ) α= 67 ; b) α= 8 ; d) α=50 g teng burchk tshkil qildign M nur ysng.. lchshlr yordmid M nuqtning koordintlrini hmd sinα, cosα, tgα v ctgα qiymtlrini toping. Msl. 0, 90 v 80 li burchklrning sinusini toping. Yechilishi. 0 li burchk, 90 li burchk ( ;0) 0 (;0), 80 li burchk nur yordmid niqlndi (-rsm). T rifg ko r, sin0 (;0) nuqtning b) y (0;) ordintsi siftid 0 g, sin90 (0;) nuqtning M(;y) ordintsi siftid g, sin80 es ( ;0) nuqtning ordintsi siftid 0 g teng bo ldi. Jvob: sin0 =0, sin90 =, sin80 =0. Svol, msl v topshiriqlr ( ;0) 0 (;0). 0 dn 80 gch bo lgn burchkning sinusi v y d) kosinusi degnd nim tushunilishini ytib bering. (0;). α burchkning tngensi v kotngensi nim? α burchkning tngensi v kotngensi α ning qndy M(;y) qiymtlrid niqlnmgn?. gr 90 <α<80 bo ls, sinα, cosα, tgα v ctgα qiymtlrining ishorsini niqlng. ( ;0) 0 (;0). gr 0 α 80 bo ls, 0 sinα v cosα tengsizliklr o rinli bo lishini tushuntiring. 5. -rsmdgi α burchkni o lchng v uning sinusi, kosinusi, tngensi v kotngensini tegishli o lchshlr yordmid niqlng. 6*. - rsmd tsvirlngn yrim ylnni chizing. nur biln 5 v 5 li burchk hosil qiluvchi nurlrni ysng. hizilgn rsmdn foydlnib, sin5 ni sin5 biln v cos5 ni cos5 biln o zro solishtiring. 7*. lndligi sm v o tkir burchgi 0 bo lgn rombning perimetri v yuzini hisoblng y ( ;0) 0 ) y (0;) (0;) (;0) M(;y)

32 7 SSIY TRIGNMETRIK YNIYTLR M(;y) y 0 T riflrg ko r, hr bir o tkir burchkk bu burchk sinusining (kosinusi, tngensi v kotngensining) bitt qiymti mos qo yilypti. u mosliklr o tkir burchkning trigonometrik funksiylri: sinus, kosinus, tngens v kotngens funksiylrini niqlydi. u funksiylr ko pinch uchburchklrni yechishd qo llnishi sbbli, ulr trigonometrik funksiylr deb tldi. Trigonometriy so zi yunonch uchburchklrni yechish degn m noni ngltdi. Endi α (0 α 80 ) burchkning sinusi, kosinusi, tngensi v kotngensi orsidgi munosbtlrni niqlylik.. Trigonometriyning sosiy yniyti deb tluvchi, α ning 0 α 80 qiymtlri uchun o rinli bo lgn ushbu sin α + cos α = () formul biln oldingi drslrd tnishgn edik.. T rifg ko r, tgα=, ctgα=, = cosα, y = sinα bo lgni uchun, tgα= (α 90 ), ctgα= (α 0, α 80 ), tgα.ctgα= (α 0, α 90, α 80 ) yniytlr o rinlidir.. () tenglikning hr ikki qismini oldin cos g, keyin es sin α g bo lib, +tg α= (α 90 ), +ctg α =, (α 0, α 80 ) () yniytlrni hosil qilmiz. y α Follshtiruvchi mshq -rsmdn foydlnib nuqtlr o rnini to ldiring: sinα =...; cosα =...; tgα =... ; ctgα =.... Msl. gr sinα= 0,6 v 90 α 80 bo ls, cosα, tgα v ctgα qiymtini toping. () Yechilishi. sosiy trigonometrik yniytdn foydlnib cos ni hisoblymiz: cosα= sin α= 0,6 = 0,6 = 0,6 = 0,8. 90 α 80, y ni α II chorkd bo lgnd, cosα 0. Shu bois ildiz ishor biln olindi. () formullrg sosn, tgα= = = ctgα= = Jvob: cosα= 0,8; tgα= ctgα= Svol, msl v topshiriqlr. tgα=, ctgα=, tgα.ctgα= yniytlr α ning qndy qiymtlri uchun o rinli?. Ifodlrni soddlshtiring: ) cos α; ) sin α sin α cos α; ) ( sinα)(+ sinα); 5) ctg α(sin α+cos α ); ) sin α+sin α cos α+ cos α; 6) tg α sin α tg α.. gr ) sinα= v 90 <α<80 bo ls, cosα nimg tengligini toping; b) cosβ= v 90 <β<80 bo ls, sinβ nimg teng; d) cosα= bo ls, sinα ning qiymtini hisoblng.. tkir burchgi 60 g, blndligi es sm g teng rombning yuzini toping. 5. Teng yonli uchburchkning sosi,8 sm, sosidgi burchgi es 0. Uchburchkning blndligi v yon tomonini toping. 6. gr ) cosα= ; b) cosα= ; d) cosα= bo ls, sinα nimg teng? 7. ) sin= ; b) cos= ; d) cosα= eknligi m lum, burchkni ysng. 8*. α v β burchklr 0 <α<β<90 shrtni qnotlntirdi. -rsmdn foydlnib isbotlng: ) sinα<sinβ; b) cosα>cosβ; d) tgα<tgβ; e) ctgα>ctgβ. 9*. nur biln nur orsidgi burchk α g t e n g. g r ) =, α=5 ; b) =,5, α=90 ; d) =5, α =50 ; e) =, α=80 ; f) =, α=0 bo ls, nuqtning koordintlrini toping. b 6 6

33 8 SSIY TRIGNMETRIK YNIYTLR (dvomi) -msl. α = 0 bo ls, sinα, cos α, tgα v ctgα qiymtlrni hisoblng. Yechilishi. ) () formulg ko r, -teorem. Hr qndy o tkir burchk uchun: sin(90 )=cos, cos(90 )=sin. () 90 Isbot. uchidgi o tkir burchgi g teng bo lgn to g ri burchkli uchburchkni qrymiz (-rsm). U hold uning uchidgi o tkir burchgi β=90 g teng. T rifg ko r, sin0 = sin(80 60 ) = sin60 = ; cos0 = cos(80 60 ) = cos60 =. Und tg0 = = ; ctg0 = = =. Jvob: sin0 = ; cos0 = ; tg0 = ; ctg0 =. sin(90 )=sinβ= =cos, Svol, msl v topshiriqlr cos(90 )= cosβ= Teorem isbotlndi. = sin. -msl. Quyidgi sonlr ichid o zro tenglrini toping: sin0, cos0, sin80, cos 80. Yechilishi. 80 = 90 0 (α=0 ) v50 = 90 0 (α= 0 ) b o l g n i u c h u n -teoremg ko r, sin80 = sin(90 0 ) = cos0, Jvob: sin80 =cos0, cos 80 = sin0. ( ;y ) cos 80 = cos(90 0 ) = sin0. -teorem. Hr qndy (0 80 ) burchk uchun: sin(80 ) =sin, cos(80 ) = cos. () y 80 (;y) y Isbot. To g ri burchkli y koordintlr sistemsid mrkzi nuqtd, rdiusi g teng yrim ylnni ysymiz (-rsm). ylnning rdiusi biln nur orsidgi burchk α bo lsin. nur biln 80 α g teng burchk tshkil qiluvchi rdiusni o tkzmiz. v to g ri burchkli uchburchklr teng. Xususn, = v = yoki = v y =y tengliklrg egmiz. Shundy qilib,. tg(90 α)=ctgα (α 0 ) v ctg(90 α) = tgα (α 0 ) yniytlrni isbotlng.. tg(80 α)= tgα (α 90 ) v ctg(80 α)= ctgα (α 0 v α 80 ) yniytlrni isbotlng.. Jdvlni to ldiring. α sinα cosα tgα ctgα. gr 90 < α <80 v ) sinα= ; b) cosα= ; d) tgα= ; e) ctgα= bo ls, α burchk kttligini toping. 5. Hisoblng: ) sin80 +cos90 ; b) sin50 + tg50 ; d) cos0 +cos50 sin0 sin50 ; e) cos0 ctg Soddlshtiring: ) cos (80 α)+cos (90 α); b) sin (80 α)+sin (90 α); d) tgα tg(90 α); e) ctgα ctg(90 α). 7. uchburchkd =50 v =7 sm bo ls, uchburchkning uchidn tushirilgn blndligini toping. 8. To g ri to rtburchkning sm g teng digonli bir tomoni biln 0 g teng burchk hosil qildi. To g ri to rtburchk yuzini toping. sin(80 α) =y =y =sinα; cos(80 α) = = = cosα. 9. gr ) sinα= ; b) sinα= ; d) sinα= bo ls, cosα ni toping. Teorem isbotlndi. () v () formullr keltirish formullri deyildi. 0*. gr ) sinα= ; b) tgα= ; d) cosα= bo ls, α ni toping. 6 65

34 9 ILIMINGIZNI SIN K RING I. hp ustund berilgn tmlrg o ng ustund berilgn t riflrdn to g risini mos qo ying.. α burchk sinusi ) α burchk qrshisidgi ktetning gipotenuzg nisbti;. α burchk kosinusi b) α burchkk yopishgn ktetning gipotenuzg nisbti;. α burchk tngensi. α burchk kotngensi d) α burchk qrshisidgi ktetning ikkinchi ktetg nisbti; e) α burchkk yopishgn ktetning ikkinchi ktetg nisbti. II. Testlr. Noto g ri formulni toping:. sin(90 α) = cosα;. cos(90 α) = sinα;. sin(80 α)= sinα; E. cos (80 α)= cosα.. gr 90 <α<80 bo ls, quyidgilrdn qysi biri musbt?. sinα;. cosα;. tgα; E. ctgα.. To g ri tenglikni toping:. sin α =+cos α;. tg α=+cos α;. =+tg α (α 90 ); E. sin.cos =. cos α. sin70 nimg teng?:. sin0 ;. sin0 ;. cos70 ; E. cos sinα= bo lgn o tkir burchkni ko rsting:. 0 ;. 5 ;. 90 ; E cosα= bo ls, α o tkir burchkni toping:. 0 ;. 5 ;. 90 ; E tgα= bo ls, α o tkir burchkni toping:. 0 ;. 5 ;. 90 ; E ctgα= bo ls, α o tkir burchkni toping:. 0 ;. 5 ;. 90 ; E Qysi o tkir α burchk uchun sinα=cosα tenglik o rinli?. 0 ;. 5 ;. 90 ; E gr sin= bo ls, cos ni toping. 5. ;. 9;. ; E gr cos=0, bo ls, tg ni toping.. 96;. 6;. 5; E. 6.. To g ri to rtburchkning digonli uning bir tomonidn mrt uzun. To g ri to rtburchkning digonllri orsidgi burchkni toping.. 0 ;. 60 ;. 90 ; E Teng yonli uchburchkning sosig tushirilgn blndligi sm, sosi es 8 sm. Uchburchkning sosig yopishgn burchgi sinusini toping.. 5 ;. ;. 7 7 ; E. 5. III. Msllr. Rsmd tsvirlngn burchklrning sinusi, kosinusi, tngensi v kotngensini toping. ) b) d) e). shrfjon uyidn shrq tomong qrb 800 m, so ng shimol tomong qrb 600 m yo l yurdi. U uyidn nech metr uzoqlikk keldi? Endi u uyig to g ri chiziq bo ylb yetib olishi uchun g rbg nisbtn qndy burchk ostid yurishi kerk?. Poyezd hr 0 m yo l yurgnd m tepg ko trildi. Temir yo lning gorizontg nisbtn ko trilish burchgini toping.. gr blndligi 0 m bo lgn bino soysining uzunligi 5 m bo ls, quyosh nurining shu bino joylshgn mydong tushish burchgini toping. 5. To g ri burchkli uchburchkning bir burchgi 60 g, ktt kteti es 6 g teng. Uning kichik kteti v gipotenuzsini toping. 6. mrkzli ylnning nuqtsidn o tkzilgn urinmd nuqt olingn. gr =9 sm, =0 bo ls, yln rdiusini v kesm uzunligini toping. 7. m to g ri chiziq v uni kesib o tmydign kesm berilgn. und =0, v m to g ri chiziqlr orsidgi burchk 60. kesm uchlridn m to g ri chiziqq v perpendikulrlr tushirilgn. kesmni toping. 8. Rombning o tkir burchgi 60 g, blndligi es 6 g teng. Rombning ktt digonli uzunligini v yuzini toping

35 . 9. Rdiusi 5 sm bo lgn ylng teng yonli trpetsiy tshqi chizilgn. gr trpetsiyning o tkir burchgi 0 bo ls, uning yon tomoni v yuzini toping. 0. gr to g ri to rtburchkd =, =0 bo ls, ung tshqi chizilgn yln rdiusini v to g ri to rtburchk yuzini hisoblng.. To g ri to rtburchkning tomonlri sm v sm. Uning bir digonli biln tomonlri hosil qilgn burchklrini toping.. gr ) sin = ; b) cos = ; d) cos = bo ls, burchkni ysng.. To g ri burchkli uchburchkning bir burchgi 0, gipotenuzsig tushirilgn blndligi 6 sm. Uchburchk tomonlrini toping.. tkir burchgi 0 g, blndligi es sm g teng bo lgn rombning yuzini hisoblng. 5. gr sin = v 90 <α<80 bo ls, cosα, tgα v ctgα qiymtini toping. 6. To g ri burchkli uchburchkning gipotenuzsig blndlik tushirilgn. gr = 60 v = bo ls, ktetni toping. 7. uchburchkd = 0, = 5. gr uchburchkning blndligi sm bo ls, uning tomonini v yuzini toping. IV. zingizni sinb ko ring (nmunviy nzort ishi). gr cos α= v 90 < α< 80 bo ls, sinα, tgα, ctgα nimg teng?. To g ri burchkli uchburchkning gipotenuzsi c =8 sm v kteti = sm bo ls, uning ikkinchi kteti v o tkir burchklrini toping.. Teng tomonli uchburchkning medinsi uning tomonidn kichik bo lishini isbotlng.. (Qo shimch). To rtburchkning hr bir tomoni qolgn tomonlrining yig indisidn kichik eknini isbotlng. Triiy lvhlr. ltin uchburchk Yunonlr burchklri 6, 7 v 7 bo lgn teng yonli uchburchkni oltin uchburchk deb tshgn. Sbbi u mn bundy joyib ossg eg ekn: sosidgi burchk bissektrissi uni ikkit teng yonli uchburchkk bo ldi (-rsm). Hqiqtn, bissektris bo lgni uchun, v burchklr hm 6 dn. emk, uchburchk teng yonli. uchburchkd burchk =7 bo lib, burchkk teng. emk, uchburchk hm teng yonli. Ntij. uchburchk uchburchkk o shsh v =. () gr uchburchkning yon tomonlri = = deb olsk, uning sosi quyidgich topildi (-rsm): = bo lsin.u hold,. = bo ldi, chunki teng yonli.. = bo ldi, chunki teng yonli.. = =. () tenglikk ko r: = 5 undn + =0. u kvdrt tenglmni yechib, = topmiz. Triiy lvhlr Ulug bek (9 9) buyuk o zbek olimi v dvlt rbobi. sl ismi Muhmmd Trg y. U sohibqiron mir Temurning nbirsi. Ulug bekning otsi Shohruh hm dvlt rbobi bo lgn. Ulug bek tminn 5 8-yillri Smrqnd yqinidgi bi Rhmt tepligid o zining mshhur rsdonsini qurdi. Rsdonning binosi uch qvtli bo lib, uning sosiy sbobi kvdrntning blndligi 50 metr edi. Ulug bekning eng mshhur sri Ziji ko rgoniy deb tluvchi stronomik jdvldir. U 08 t yulduzni o z ichig olgn. eknligini Msl. sin8, cos8, sin7, cos7 qiymtlrni hisoblng. 5 Yechilishi: Yon tomoni = = v sosi = = g teng bo lgn oltin uchburchk ni qrymiz (-rsm). Uning E blndligini o tkzmiz. To g ri burchkli E uchburchkdn E sin8 = = = 5 undn foydlnib, topilishi tlb qilingn boshq qiymtlrni 8 hisoblymiz: 5 + cos8 = sin 8 = ; sin7 = sin(90 8 ) = cos8 = ; cos7 = sin8 = Jvob: sin8 = cos7 = ; cos8 = sin7 =. E Ulug bek (9 9) 68 69

36 0 UHURHK YUZINI URHK SINUSI YRMI HISLSH -teorem. Uchburchk yuzi uning ikki tomoni biln shu ikki tomon orsidgi burchk sinusi ko pytmsining yrmig teng. Isbot. uchburchkning blndligini tushirmiz. U hold -rsmd ko rstilgn uch hol bo lishi mumkin. irinchi holni qrymiz. uchburchkd sin =. undn =. sin =.sin. Shundy qilib, ) b) d), =, = b, (-rsm) E b b b S = b sin S =.. =. b..sin= bsin. Ikkinchi v uchunchi hollrning isbotini mustqil bjring. Teorem isbotlndi. -teoremg ko r, uchburchk yuzi uchun S = bcsin v S = c sin formullr hm o rinli bo ldi. -msl. uchburchkning yuzi sm. gr =8 sm v = 0 bo ls, tomonni toping. Yechilishi. Uchburchk yuzini burchk sinusi orqli topish formulsig ko r, S =... sin undn, = =. = = (sm). 8.0,5 Jvob: sm. -msl. Prllelogrmm yuzi uning ikkit qo shni tomoni v shu tomonlr orsidgi burchgi sinusining ko pytmsig teng eknligini isbotlng. prllelogrmm, =, =b, = α (-rsm) S =bsin E Yechilishi. E blndlik tushirmiz. E uchburchkd sin = yoki E = sin = sinα. U hold, S =.E = b sinα. -teorem. To rtburchk yuzi uning digonllri biln digonllr orsidgi burchk sinusi ko pytmsining yrmig teng. Isbot. igonllr kesishishidn hosil bo lgn burchklrni qrymiz (-rsm): = α shrtg ko r, = α g vertikl bo lgni uchun, α =80 α g qo shni bo lgni uchun, =80 α g vertikl bo lgni uchun. Uchburchk yuzini burchk sinusi yordmid hisoblsh formulsig ko r: S =. sinα; S =. sin(80 α)=. sinα; S =. sinα; S =. sin(80 α) =. sinα. Yuzning osssig ko r: S =S +S +S +S = =. sinα+. sinα+. sinα+. sinα= = ( )sinα= {(. (+ )+ +. (+)}sinα = (. +. )sinα=. sinα. Teorem isbotlndi. Svol, msl v topshiriqlr. -teoremni -b v -d rsmd tsvirlngn hollr uchun isbotlng.. gr ) = 6 sm, = sm, =0 ; b) = sm, =7 sm, =60 ; d) = sm, = sm, =5 bo ls, uchburchk yuzini toping.. igonli sm v digonllri orsidgi burchgi 0 bo lgn to g ri to rtburchk yuzini toping.. Tomoni 7 sm v o tms burchgi 5 bo lgn romb yuzini toping. 5. Rombning ktt digonli 8 sm v bir burchgi 0. Romb yuzini toping. 6. Yuzi 6 sm g teng bo lgn uchburchkd = 9 sm, = 5. Uchburchkning tomonini v shu tomong tushirilgn blndligini toping. 7*. uchburchkd =, uning v uchlridn tushirilgn blndliklri es mos rvishd h b v h c bo ls, uchburchk yuzini toping. 8*. uchburchkd = 8 sm, = sm v = 60 bo ls, uning bissektrissini toping (ko rstm: S =S +S ). 70 7

37 ) b) d) SINUSLR TEREMSI Teorem. (Sinuslr teoremsi). Uchburchkning tomonlri qrshisidgi burchklrning sinuslrig proporsionl., = c, =, =b (-rsm) c b sin = b sin = c sin Isbot. Uchburchk yuzini burchk sinusi orqli topish formulsig ko r, S= b sin, S= bc sin, S= c sin. ( ) u tengliklrning dstlbki ikkitsig ko r, c b sin= bc sin, demk = sin sin. Shuningdek, ( ) tengliklrning birinchi v c b uchinchidn tenglikni hosil qilmiz. sin = sin Shundy qilib, b c = =. sin sin sin Teorem isbotlndi. -msl. uchburchkd = dm, = 0, = 65 (-rsm). tomonni toping. Yechilishi: Sinuslr teoremsig ko r, =. Undn, sin sin.sin = sin =.sin0.0,5 7,78 (dm). sin 65 0,9 Esltm: Trigonometrik funksiylrning qiymtlri msus klkultor yoki jdvllr yordmid topildi. u yerd sin65 0,9 eknligini drslikning 5-betidgi jdvldn niqldik. Jvob: 7,78 dm. -msl. Uchburchk tomonining shu tomon qrshisidgi burchgi sinusig nisbti uchburchkk tshqi chizilgn yln dimetrig teng, y ni b c = = sin = R sin sin eknligini isbotlng (-rsm). Yechilishi: Rvshnki, sinuslr teoremsig ko r, tenglikni isbotlsh kifoy. Uch hol bo lishi mumkin: sin = R -hol: o tkir burchk (- rsm); -hol: o tms burchk (-b rsm); -hol: to g ri burchk (-d rsm). -holni qrymiz: v nuqtlrni tutshtirmiz. to g ri burchkli uchburchk, chunki burchk dimetrg tirlgn. d: =. sin = Rsin. Lekin, =, chunki ulr bitt yoyg tirlgn ichki chizilgn burchklr. Und, = R sin yoki sin = R. Qolgn hollrni mustqil isbotlng (ko rstm: -hold =80 eknligidn, -hold =R eknligidn foydlning). Svol, msl v topshiriqlr ). Uchburchk istlgn tomonining shu tomon qrshisidgi 6 burchk sinusig nisbti uchburchkk tshqi chizilgn yln dimetrig teng eknligini 0 5 -msld keltirilgn - v -hollr uchun =? isbotlng.. -rsmd berilgnlrg ko r, so rlgn kesmlrni b) toping. 8. gr uchburchkd: ) sin= 0,; = 6 sm v = 5 sm bo ls, sin ni; 60 5 b) sin= ; =8 dm v = 7 dm bo ls, sin ni; =? d) d) sin= ; = 6 m v = 8 m bo ls, sin ni toping.. Uchburchkning bir burchgi 0 g teng. Uning 7 qrshisidgi tomon,8 dm. Uchburchkk tshqi 05 0 chizilgn yln rdiusini hisoblng. =? 5. Uchburchkning bir tomoni uchburchkk tshqi chizilgn yln rdiusig teng. Uchburchkning shu tomoni qrshisidgi burchgini toping. und, ikki holni qrshg to g ri kelishig e tibor qiling. 6. uchburchk uchun : := sin : sin: sin tenglik o rinli bo lishini soslng. sin:sin: sin= :5:7 tenglik to g ri bo lishi mumkinmi? 7. gr uchburchkd = 0 m, = m v = 67 bo ls, uchburchkning tomonini, v burchklrini toping. 8*. gr uchburchkd =8 dm, =, = 6 bo ls, uchburchkning burchgini, v tomonlrini toping. 7 7

38 KSINUSLR TEREMSI To g ri burchkli uchburchkd to g ri burchk qrshisidgi tomon (gipotenuz) kvdrti qolgn tomonlr (ktetlr) kvdrtlri yig indisig teng. Xo sh, to g ri bo lmgn burchk uchun-chi? Quyidgi teorem shu ususd. Teorem. (Kosinuslr teoremsi). Uchburchk istlgn tomonining kvdrti qolgn ikki tomoni kvdrtlri yig indisi shu ikki tomon biln ulr orsidgi burchk kosinusi ko pytmsining ikkilngni yirmsig teng. d), = c, =, = b (-rsm) ) b) c c c 60 b b b c b = b + c bccos Isbot. uchburchkning blndligini o tkzmiz. nuqt tomond (- rsm) yoki uning dvomid (-b v -d rsmlr) bo lishi mumkin. irinchi holni qrymiz. To g ri burchkli uchburchkd Pifgor teoremsig ko r, = +. = bo lgni uchun: = +( ) = To g ri burchkli uchburchkd + = v =cos eknligini hisobg olib, oirgi tenglikdn = +..cos, y ni = b +c bccos tenglikk eg bo lmiz. Teorem isbotlndi. -b rsmd tsvirlngn hold =, -d rsmd tsvirlngn hold =+ v cos(80 )= cos tengliklrdn foydlnib, kosinuslr teoremsini mustqil isbotlng. Esltm. Kosinuslr teoremsi Pifgor teoremsining umumlshgnidir. =90 bo lgnd (cos90 =0 bo lgni uchun) kosinuslr teoremsidn Pifgor teoremsi kelib chiqdi. -msl. uchburchkd =6 sm, = 7 sm, = 60 (-rsm). tomonni toping. Yechilishi. Kosinuslr teoremsig ko r, = b + c bccos yoki = +..cos bo lgni uchun = cos60 = =, y ni = sm. Jvob: sm. Kosinuslr teoremsidn foydlnib, tomonlri m lum bo lgn uchburchkning burchklrini topish mumkin: cos = b +c. () bc -msl. uchburchkning tomonlri =5 m, b =6 m v c = m. Kichik tomonning ktt tomondgi proyeksiysini toping (-rsm). Yechilishi. () formul sosid cos ni topmiz: cos = b +c bc = = 9 6. c 5 To g ri burchkli uchburchkd = 80 =. 9 cos bo lgni uchun =. =,5 (m). 6 Jvob:,5 m. b/ b/ b Svol, msl v topshiriqlr. Kosinuslr teoremsini -b v -d rsmd tsvirlngn hollrd isbotlng.. uchburchkd ) = sm, = sm v =60 bo ls, ni; b) = m, = m v =5 bo ls, ni; d) =7 dm, =6 dm v =50 bo ls, ni toping.. Tomonlri 5 sm, 6 sm, 7 sm bo lgn uchburchk burchklri kosinuslrini toping.. uchburchkd =0 sm, = m v sin=0,6 bo ls, tomonni toping. 5. Prllelogrmmning digonllri 0 sm v sm, ulr orsidgi burchgi 60 g teng. Prllelogrmm tomonlrini toping. 6. Tomonlri 5 sm v 7 sm bo lgn prllelogrmmning bir burchgi 0 g teng. Uning digonllrini toping. 7*. Tomonlri, b, c bo lgn uchburchkning medinsi = +c b formul biln hisoblnishini isbotlng (-rsm). 8*. Tomonlri 6 m, 7 m v 8 m bo lgn uchburchk medinlrini toping. 9. -msldgi uchburchk bissektrislrini toping. 0. -msldgi uchburchk blndliklrini toping. 7 75

39 SINUSLR V KSINUSLR TEREMLRINING ZI TTIQLRI ldingi drslrd isbotlngn sinuslr v kosinuslr teoremlridn uchburchklrg oid turli-tumn msllrni yechishd smrli foydlnish mumkin. u drsd bu teoremlrning b zi bir ttbiqlrig to tlmiz.. Kosinuslr teoremsi uchburchk burchklrini topmsdn, uning burchklr bo yich turini (o tkir, o tms yoki to g ri burchkli eknligini) niqlshg imkon berdi. Hqiqtn, cos = b +c bc formuld ) gr b +c > bo ls, cos>0. emk, o tkir burchk; ) gr b +c = bo ls, cos=0. emk, to g ri burchk; ) gr b +c < bo ls, cos<0. emk, o tms burchk. b +c = tenglik yoki b +c < tengsizlik uchburchkning eng ktt tomoni bo lgn holdgin bjrildi. emk, uchburchkning to g ri yoki o tms burchgi uning eng ktt tomoni qrshisid yotdi. Uchburchkning eng ktt tomoni qrshisidgi burchkning kttligig qrb, bu uchburchkning qndy (o tkir, o tms, to g ri burchkli) uchburchk eknligi hqid ulosg kelish mumkin. -msl. Tomonlri 5 m, 6 m v 7 m bo lgn uchburchk burchklrini topmsdn uning turini niqlng. Yechilishi. Eng ktt burchk qrshisid eng ktt tomon yotdi. Shuning uchun, gr =7, b =6, c=5 bo ls, eng ktt burchk bo ldi. cos = b +c bc = emk, o tkir burchk, berilgn uchburchk es o tkir burchkli.. Uchburchk yuzini uning ikki tomoni v ulr orsidgi burchgi orqli hisoblsh formulsi S = bcsin v sin = formullrdn uchburchk yuzini hisoblsh uchun R bc S = R formulni v uchburchkk tshqi chizilgn yln rdiusini hisoblsh uchun bc R = S formulni hosil qilmiz. = 60 = > msl. Tomonlri =5, b=6, c =0 bo lgn uchburchkk tshqi chizilgn yln rdiusini toping. Yechilishi. Geron formulsidn foydlnib, uchburchk yuzini topmiz: + b + c p = = =, S= p(p )(p b)(p c)= ( 5)( 7)( 0)=.6. = 6 6,. Und, bc R= ,. Jvob: 5,. S.6, Svol, msl v topshiriqlr. gr =7 sm, =8 sm, =9 sm bo ls, uchburchkning eng ktt v eng kichik burchgini toping.. gr uchburchkd = 7, = 58 bo ls, uchburchkning eng ktt v eng kichik tomonlrini niqlng.. Uchburchkning ucht tomoni berilgn: ) = 5, b=, c= ; b) =7, b= 8, c=5; d) = 9, b=5, c= 6. Uchburchk o tkir burchkli, to g ri burchkli yoki o tms burchkli eknligini niqlng.. Tomonlri ),, 5; b) 5,, ; d) 5, 9, 8; e), 5, 7 bo lgn uchburchkk tshqi chizilgn yln rdiusini toping. 5. uchburchkning tomonid nuqt bergilngn. kesm v kesmlrning kmid bittsidn kichik eknligini isbotlng. 6. Uchburchkning ktt burchgi qrshisid ktt tomoni yotishini isbotlng. 7. Uchburchkning ktt tomoni qrshisid ktt burchgi yotishini isbotlng. 8*. uchburchkning medinsi o tkzilgn. gr > bo ls, burchk burchkdn kichik bo lishini isbotlng. 9. Quyidgi rsmg mos msl tuzing. M 76 77

40 IKKI VEKTR RSIGI URHKNI HISLSH Vektorlrning sklyr ko pytmsi tushunchsi v osslri biln 8-sinfd tnishgn edingiz. Ikki vektorning sklyr ko pytmsi ulrning koordintlri orqli ifodlngn edi. Quyid kosinuslr teoremsi yordmid vektorlrning sklyr ko pytmsi uchun yn bir muhim formul chiqrildi. und sklyr ko pytm vektorlrning uzunligi v ulr orsidgi burchk orqli ifodlndi. b Nol vektordn frqli v b vektorlr berilgn bo lsin. Itiyoriy nuqtdn = v =b vektorlrni qo ymiz. v b vektorlr orsidgi burchk deb burchkk ytildi (-rsm). ir il yo nlgn vektorlr orsidgi burchk 0 g teng deb hisoblndi. gr ikkit vektor orsidgi burchk 90 g teng bo ls, ulr perpendikulr deyildi. Esltib o tmiz: α. ( ; ) vektorning uzunligi = +. b. ( ; ) v b(b ;b ) vektorlrning sklyr ko pytmsi b = b + b formullr biln niqlnr edi. b Nokolliner v b vektorlrni qrymiz. Itiyoriy nuqtdn = v =b vektorlrni qo ymiz (- α rsm). =α bo lsin. Und, bir tomondn kosinuslr teoremsig ko r, = +.. cosα. () Ikkinchi tomondn = = =( ) = +.. () emk, () v () g ko r. =.cosα yoki b =. b cosα. Ntij. Nol vektordn frqli ( ; ) v b(b ;b ) vektorlr orsidgi burchk uchun cos α= b. b yoki cosα = Msl. (;) v b(; ) vektorlr orsidgi burchkni toping. Yechilishi. erilgn vektorlr orsidgi burchkni deb belgilsk, formulg ko r,.+. ( ) cosα= +. = +( ) 5. =0. 0 emk, α=90. Jvob: 90. Svol, msl v topshiriqlr. gr v b vektorlr uchun ) =, b =5, α=0 ; b) =8, b=7, α= 5 ; d) =,, b =0, α=60 ; e) =0,8, b =, α= 0 bo ls, bu vektorlrning sklyr ko pytmsini toping (bu yerd α v b vektorlr orsidgi burchk).. ) ( ; ) v b(;); b) ( 5;6) v b(6;5); d) (,5;) v b( ; ) v e k - torlrning sklyr ko pytmsini hisoblng v ulr orsidgi burchkni toping.. rombning digonllri nuqtd kesishdi v bund = = sm. ) v ; b) v ; d) v ; e) v vektorlrning sklyr ko pytmsini v bu vektorlr orsidgi burchkni toping.. Nol vektordn frqli v b vektorlr berilgn bo lsin..b=0 bo lgnd bu vektorlr perpendikulr bo lishini v ksinch, v b vektorlr perpendikulr bo ls,.b =0 bo lishini isbotlng. 5*. ning qndy qiymtid ) (;5) v b(;6); b) (;) v b(;); d) (0; ) v b(5;) vektorlr o zro perpendikulr bo ldi? 6. (;), b(; ), c( ; ) v d( ;) vektorlr orsidn o zro perpendikulr juftlrini toping. 7. = tenglikni isbotlng. 8*. ilyrd o yinid nuqtd turgn shr zrbdn keyin bilyrd stoli tomonig nuqtd urildi v yo nlishini o zgrtirib nuqtdgi svtchg tushdi (-rsm). gr =0 sm, =50 sm v =0 bo ls,. sklyr ko pytmni toping. 9. F(-, ) kuch t siri ostid nuqt (5,-)holtdn (, ) holtg o tdi. u jryond qndy ish bjrildi? formul o rinli

41 5 UHURHKLRNI YEHISH Uchburchkning tomonlrini, b, c biln, bu tomonlr qrshisidgi burchklrni mos rvishd α, β, γ b c biln belgilymiz (-rsm). Uchburchkning tomonlri v burchklrini bitt nom biln uning elementlri g b deb tshdi. Uchburchkni niqlovchi berilgn elementlrig ko r, uning qolgn elementlrini topish uchburchkni yechish deb yuritildi. -msl. (Uchburchkni berilgn bir tomoni v ung yopishgn burchklri bo yich yechish). gr uchburchkd =6, β=60 v γ=5 bo ls, uning uchinchi burchgi v qolgn ikki tomonini toping. Yechilishi.. Uchburchk burchklri yig indisi 80 bo lgni uchun α=80 β γ = =75. Sinuslr teoremsidn foydlnib, qolgn ikki tomonni topmiz:. sinα = b tenglikdn b =. sinβ sinβ sinα = 6. sin60 sin ,8660 5,79 5,. 0,9659 (sin60 v sin75 qiymtlri mikroklkultord topib qo yildi, ulrni drslikning 5-betidgi jdvldn topishingiz mumkin).. sinα = c tenglikdn c =. sin γ sinγ sinα = 6. sin5 sin ,707,9,. 0,9659 Jvob: α=75 ; β 5,; c,. -msl. (Uchburchkni berilgn ikki tomoni v ulr orsidgi burchgi bo yich yechish). gr uchburchkd =6, b = v γ=0 bo ls, uning uchinchi tomoni v qolgn burchklrini toping. Yechilishi.. Kosinuslr teoremsidn foydlnib, uchburchkning uchinchi c tomonini topmiz. c = +b bcosγ= ( 0,5)= 76 8,7.. Endi, uchburchkning ucht tomonini bilgn hold, kosinuslr teoremsidn foydlnib, uchburchkning qolgn burchklrini topmiz: cos α b +c = = 0,806. bc.. 76 cosα 0,806 tenglik sosid α burchkning qiymtini 5-betdgi jdvldn niqlymiz (α o tkir burchk): α 6.. β=80 α γ 80 (6 +0 )=. Jvob: c 8,7; α 6, β. -msl. (Uchburchkni berilgn uch tomoni bo yich yechish). gr uchburchkd =0, b =6 v c = bo ls, uning burchklrini toping. Yechilishi:. Uchburchk o tms burchkli bo lishi yoki bo lmsligini ktt tomon qrshisidgi burchk kosinusining ishorsig qrb niqlymiz: +b c cos = = = 0,75 < 0. b emk, o tms burchk ekn. uni 5-betdgi jdvldn burchkning kttligini niqlshd hisobg olmiz. Jdvldn kosinusi 0,75 g teng burchk =7 eknligini topmiz. Und cos(80 α)= cosα formulg ko r, =80 =80 7 =06.. Sinuslr teoremsig ko r, sin = c sin. undn, sin =.sin c = 0.sin06 = 0.sin7 0.0,965 0,796. o tkir burchk bo lgni uchun 5-betdgi jdvldn 7 eknligini niqlymiz.. 80 (06 +7 )=6. Jvob: 7, 6, 06. Svol, msl v topshiriqlr. Uchburchkning bir tomoni v ung yopishgn ikkit burchgi berilgn: ) =5 sm, β=5, γ= 5 ; b) = 0 sm, α=75, β=60 ; d) =5 sm, β= 0, γ=0 ; e) b= sm, α=6, β=5. Uchburchkning uchinchi burchgi v qolgn ikki tomonini toping.. Uchburchkning ikki tomoni v ulr orsidgi burchgi berilgn: ) = 6, b=, γ= 60 ; b) =, b=, γ=0 ; d) b=7, c= 9, α= 85 ; e) b=, c=0, α=5. Uchburchkning qolgn burchklrini v uchinchi tomonini toping.. Uchburchkning ucht tomoni berilgn: ) =, b=, c= ; b) = 7, b=, c= 8; d) =, b=5, c= 7; e) =5, b=, c=8. Uchburchkning burchklrini toping.. Uchburchkning ikki tomoni v bu tomonlrdn birining qrshisidgi burchgi berilgn. Uchburchkning qolgn tomoni v burchklrini toping: ) =, b= 5, α= 0 ; b) =7, b= 9, α=8 ; d) b=, c=, α= 60 ; e) b=6, c=8, α= rsmd berilgn m lumotlr sosid uchburchkni yeching. b) d) ) 6 e)

42 6 MSLLR YEHISH -msl. Prllelogrmm digonllri kvdrtlrining yig indisi tomonlri kvdrtlrining yig indisig teng eknligini isbotlng. prllelogrmm, =, = b, = d, =d (-rsm). d + d =( +b ) b Yechilishi. prllelogrmmning d burchgi α g teng bo lsin. Und =80 α. v uchburchklrg kosinuslr teoremsini qo llymiz (-rsm): d d = +b bcosα, () α b d = +b b cos(80 α). cos(80 α)= cosα tenglikni hisobg olsk, d = +b +bcos. () () v () tengliklrning mos qismlrini qo shib, d + d =( +b ) tenglikni hosil qilmiz. -msl. uchburchkd =0, =, = b o l s, u c h b u r c h k - ning uchidn tushirilgn blndligini toping (-rsm). Yechilishi. ) Kosinuslr teoremsidn foydlnib, uchburchkning tomonini 0 topmiz: = +..cos= = +( )... cos0 =7, = 7. ) Endi uchburchkning yuzini topmiz: S =... sin =... sin0 =. ) Topilgnlrdn foydlnib, uchburchkning blndligini topmiz: S =.. S formuldn = Jvob: = 7 = msl. Hydovchi yo l hrkti qoidsini buzib, sot 00 d shohko chning nuqtsidn lmzor ko chsi tomon burildi v 0 km/sot tezlikd hrktini dvom ettirdi (-rsm). Sot 00 d N odimi nuqtdn toshloq yo l bo ylb 70 km/sot tezlikd qoidbuzr hydovchi yo lini kesib chiqish uchun yo lg chiqdi. N odimi chorrhd, y ni nuqtd qoidbuzr hydovchini to ttib qol oldimi? Yechilishi: uchburchkd =80 ( + )=80 (0 +50 )=80 70 =0.. lmzor ko chsidgi yo lning qismi uzunligini topmiz: sinuslr teoremsig ko r, sin = sin. u tenglikdn =.sin sin =.sin50 sin0 =.sin50 sin(90 +0 ) =.sin50 cos0.0,766,60 (km). u yo lni qoidbuzr hydovchi,60 km 0,06 0,90 =,5 0,9 0 km/sot sot =0,0.600 sekund sekundd bosib o tdi.. Endi toshloq yo lning qismi uzunligini topmiz: sinuslr teoremsig ko r, sin = sin. u tenglikdn =.sin.sin0.0, = = 0,89 (km). sin sin50 0,766 0,89 km u yo lni N odimi 0,08 sot =0, sekund 6 70 km/sot sekundd bosib o tdi. emk, chorrhg N odimi hydovchidn kechroq yetib kelr ekn. Jvob: Yo q. Svol, msl v topshiriqlr. -rsmdgi m lumotlr bo yich ning qiymtini toping.. uchburchkning blndligi m. gr = 5, = 0 bo ls, uchburchk tomonlrini toping.. ir nuqtg kttligi bir il bo lgn ikkit kuch qo yilgn. gr bu kuchlr yo nlishlri orsidgi burchk 60 v kuchlrning teng t sir etuvchisi 50 kg bo ls, bu kuchlr kttligini toping.. Uchburchkning ikki tomoni 7 dm v dm, uchinchi tomonig tushirilgn medinsi es 6 dm. Uchburchkning uchinchi tomonini toping. 0 km/sot 0 km 50 Shohko ch N posti ) S= sm 60 8 sm d) lmzor ko chsi 60 S= 75 b b) 70 km/sot S= =60sm 5. Tomonlri 6 sm v 8 sm bo lgn prllelogrmmning bir digonli sm bo ls, uning ikkinchi digonlini toping. 6. Uchburchkning 8 sm g teng tomoni qrshisidgi burchgi 60 g teng. Uchburchkk tshqi chizilgn yln rdiusini toping. 7. Teng yonli trpetsiyning kichik sosi yon tomonig teng, ktt sosi es 0 sm. gr trpetsiyning bir burchgi 0 bo ls, uning perimetrini toping. 5 sm 0 8 8

43 7 H H UHURHKLRNI YEHISHNING MLIYT Q LLNISHI α α β. lndlikni o lchsh. ytylik, nimningdir (msln, drtning) H blndligini o lchsh zrur bo lsin (-rsm). ) uning uchun nuqtni belgilymiz v H msof ni v H burchk α ni o lchymiz. Und, to g ri burchkli H uchburchkd H=H tgα= tgα. b) gr blndlikning sosi H nuqt borib bo lmydign nuqt bo ls (-rsm), yuqoridgi usul biln H blndlikni niqly olmymiz. Und quyidgich yo l tutmiz: ) H nuqt biln bir to g ri chiziqd yotgn v nuqtlrni belgilymiz; ) msofni o lchb ni topmiz; ) H v H burchklrni o lchb H= α v H= β lrni topmiz; ) uchburchkk sinuslr teoremsini qo llsk ( = α β), y ni. sinβ = = sinβ sin(α β) sin(α β) 5) to g ri burchkli H uchburchkd H blndlikni topmiz: H = sinα = sinα.sinβ sin(α β).. orib bo lmydign nuqtgch bo lα γ gn msofni hisoblsh. ytylik, nuqtdn borib bo lmydign nuqtgch bo lgn À msofni hisoblsh kerk (-rsm). u mslni uchburchklrning o shshlik lomtlridn foydlnib jvobini topgnimizni esltib o tmiz. Endi bu mslni sinuslr teoremsidn foydlnib yechmiz. ) v nuqtlrdn ko rinib turgn tekis joyd nuqtni belgilymiz. ) msofni o lchymiz: =. ) sboblr yordmid v burchklrni o lchymiz: =α, =γ. ) uchburchkd =80 α γ bo lgni uchun, sin= sin(80 α γ)=sin(α + γ). Sinuslr teoremsig ko r, yoki. sin = = sinγ sin sin(α+ γ) Svol, msl v topshiriqlr. -rsmd = m, α= bo ls, drt blndligini hisoblng.. -rsmd =8 m, α=, β= bo ls, drt blndligini hisoblng.. -rsmd = 60 m, α = 6, γ = α bo ls, msofni toping.. Futbol o yinid metrlik jrim to pini drvozg yo nltirish burchgi α ni toping (-rsm). rvozning kengligi 7 m rsmd Xiv shhridgi Kltminor tsvirlngn. gr β=5, γ =, =50 m bo ls, Kltminor blndligini toping Syohtchi Shhrisbz shhridgi qsroyni undn 7 m msofd tomosh β γ qilypti (6-rsm). gr u qsroy sosini gorizontg nisbtn,5 g teng burchk ostid, tep qismini es 5 g 6 teng burchk ostid ko ryotgn bo ls, qsroy blndligini toping. 7. Ucht yo l uchburchkni tshkil qildi. u uchburchkd = 0, =50. nuqtdn yo lg chiqqn hydovchi nuqtg imkon borich tezroq yetib bormoqchi. v yo llr toshloq, sflt yo l bo lib, sflt yo ld toshloq yo lg qrgnd brvr tezroq 5,5 7 m,5 hrktlnish mumkin. Hydovchig qysi yo ldn yurishni mslht bersiz? Qiziqrli msl Pifgor teoremsining yn bir isboti To g ri burchkli uchburchkd = c sinα, b = c cosα. u ikki tenglikni kvdrtg oshirib, hdm-hd qo shsk v sin α+ cos α= eknligini hisobgolsk, + b =c sin α+ c cos α= c (sin α+ cos α)=c. emk, +b = c. u isbot mntiqn noto g ri eknligini isbotlng. m 7 m 8 85

44 II G IR Q SHIMH MSLLR V 8 9 M LUMTLR I. Testlr. Tomonlri, b, c, mos burchklri α, β, γ, yuzi S bo lgn uchburchk uchun qysi tenglik noto g ri?. =b +c bccosα;. b c ; = = sinα sinβ sinγ. S= bsinγ; E. S= bsinα.. Noto g ri tenglikni toping:. sin α+cos α=;. sin(80 α)=sinα;. cos(80 α)=cosα; E. sin(90 α)=cosα.. Uchburchkning ucht tomoni m lum bo ls, qysi teoremdn foydlnib uning burchklrini topish mumkin?. Sinuslr teoremsi;. Kosinuslr teoremsi;. Fles teoremsi; E. Geron formulsi.. Uchburchkning bir burchgi 7 g, ikkinchi burchgi 5 g teng. gr bu uchburchkning ktt tomoni g teng bo ls, uning kichik tomonini toping.. 8,;. 9,;.,8; E. 6,5. 5. Uchburchkning v 9 g teng bo lgn tomonlri orsidgi burchgi 6. Shu uchburchkning uchinchi tomonini toping..,;. 5,;. 6,9; E. 9,7. 6. gr ikki vektorning uzunliklri =, b =5 v ulr orsidgi burchk 5 bo ls, v b vektorlrning sklyr ko pytmsini toping.. 5;.. 0; E.. 7. (; -) v b(; ) vektorlrning sklyr ko pytmsini toping.. 5;. ;. ; E (- ; ) v b( ; ) vektorlr orsidgi burchkni toping.. 0 ;. 60 ;. 90 ; E Uchburchk burchklrining nisbti :: kbi bo ls, uning tomonlri nisbtini toping.. ::;. ::;. : :; E. : :. 0. Tomoni sm bo lgn muntzm uchburchkk tshqi chizilgn yln rdiusini toping.. ;.. ; E. II. Msllr. uchburchkk = 6 sm, = 60, = 75. tomonni hmd uchburchkk tshqi chizilgn ylnning rdiusini toping.. Tomonlri 5 sm, 6 sm v 0 sm bo lgn uchburchk burchklrining kosinuslrini toping.. uchburchkd = 60, = 6 sm, = sm. tomonni hmd uchburchkk tshqi chizilgn ylnning rdiusini toping.. Tomonlri 5 sm, 5 sm v 5 sm bo lgn uchburchkk tshqi chizilgn ylnning rdiusini toping. 5. Uchburchkning ikkit tomoni sm v sm, uchinchi tomonig o tkzilgn medinsi es sm. Uchburchkning uchinchi tomonini toping. 6. Prllelogrmmning digonllri sm, sm v ulr orsidgi burchk 5. Prllelogrmmning ) yuzini; b) perimetrini; d) blndliklrini toping. 7. Tomonlri v 5 bo lgn prllelogrmmning bir digonli g teng. Uning ikkinchi digonlini toping. 8. Tomonlri ), v,5; b), 7 v 5; d) 9, 5 v 6 bo lgn uchburchk turini niqlng. 9. Prllelogrmmning tomonlri 7 v 6 sm. gr uning o tms burchgi 0 bo ls, uning yuzini toping. 0. uchburchkning, tomonlrid N, K nuqtlr olingn. Und N= N, K = K. gr =, = 5, = 6 bo ls, NK kesmni toping.. uchburchkd =0, = 7 sm. Uchburchkk tshqi chizilgn yln rdiusini toping.. uchburchkning E bissektrissi o tkzilgn. E nuqtdn tomong EF perpendikulr tushirilgn. gr EF =, =0 bo ls, E ni toping.. to g ri to rtburchk tomonining o rtsi N nuqtd. gr =, = 6 bo ls, N.N sklyr ko pytmni toping.. (;), b( ;) bo lib, +b v b vektorlr perpendikulr. ni toping. 5. m(7;) v n( ; 5) vektorlr orsidgi burchkni toping. 6. Rsmd berilgnlrdn foydlnib, rsmdgi eng ktt kesmni niqlng. 7. to g ri to rtburchkning digonllri nuqtd kesishdi. gr = sm, =0 bo ls, to rtburchk perimetrini toping. 8. To g ri burchkli uchburchk bissektrislri nuqtd kesishdi ( =90 ). gr = 0, = 5 bo ls, gipotenuzni toping

45 III. zingizni sinb ko ring (nmunviy nzort ishi). Tomonlri =5, b=70, c=95 bo lgn uchburchkning eng ktt burchgini toping.. Uchburchkd b=5, α=0, β=50 bo ls, uchburchkni yeching.. PKH uchburchkd PK=6, KH=5, PKH=00. HF medin uzunligini v PFH uchburchk yuzini toping.. (Qo shimch). Uchburchkd =, b=, α=5 bo ls, β burchkni toping. Triiy lvhlr. Sinus hqid Sinus hqidgi m lumot dstlb IV V srlrdgi hind stronomlrining srlrid uchrydi. rt siyolik olimlr l-xorzmiy, eruniy, Ibn Sino, bdurhmon l- Hziniy (XII sr) sinus uchun «l-jyb» tmsini ishltishgn. Hozirgi sinus belgisini Simpson, Eyler, lmber, Lgrnj (XVII sr) v boshqlr qo llgn. «Kosinus» tmsi lotinch «komplimenti sinus» tmsining qisqrtirilgni, u «qo shimch sinus», niqrog i «qo shimch yoyning sinusi» demkdir. Kosinuslr teoremsini yunonlr hm bilgn, uning isboti Yevklidning Negizlr srid keltirilgn. Sinuslr teoremsining o zig os isbotini bu Ryhon eruniy byon qilgn. Triiy lvhlr. eruniy (to liq ismi bu Ryhon Muhmmd ibn hmd) (97 08) o rt srning buyuk qomusiy olimi. U Xorzm o lksining Qiyot shhrid tug ilgn. Qiyot mudryoning o ng qirg og i hozirgi eruniy shhrining o rnid bo lgn, u yqin vqtlrgch Shbboz deb tlgn. eruniyning mtemtik v fnning boshq sohlrig qo shgn hisssini yozib qoldirgn 50 dn ortiq sridn hm ko rish mumkin. Ulrdn eng yiriklri Hindiston, Yodgorliklr, Ms ud qonunlri, Geodeziy, Minerlogiy v stronomiy. eruniyning shoh sri Ms ud qonunlri, sosn, stronomiyg oid bo ls hm, uning mtemtikg oid nchgin kshfiyotlri shu srd byon etilgn. eruniy (97 08) u srd eruniy ikki burchk yig indisi v yirmsining sinuslri, ikkilngn v yrim burchkning sinuslri hqidgi teoremlr biln teng kuchli bo lgn vtrlr hqidgi teoremlrni isbotlgn, ikki grdusli yoyning vtrlrini hisoblb topgn, sinuslr v tngenslr jdvllrini tuzgn, sinuslr teoremsini o zig os usuld isbotlgn. III YLN UZUNLIGI V IR YUZI Ushbu bobni o rgnish ntijsid siz quyidgi bilim v mliy ko nikmlrg eg bo lsiz: ilimlr: ko pburchkk tshqi v ichki chizilgn ylnlr ning osslrini bilish; muntzm ko pburchklrning osslrini bilish; muntzm ko pburchkning yuzini hisoblsh formullrini bilish; yln v uning yoyi uzunligini hisoblsh formullrini bilish; doir v uning bo lklri yuzini topish formullrini bilish; burchkning rdin o lchovini bilish. mliy ko nikmlr: muntzm ko pburchklrni tsvirly olish; muntzm ko pburchkk tshqi v ichki chizilgn ylnlrning rdiuslrini top olish; yln v yoy uzunligini hisobly olish; doir v uning bo lklri yuzini hisobly olish

46 0 YLNG IHKI HIZILGN K PURHK T rif. gr ko pburchkning brch uchlri ylnd yots, bu ko pburchk ylng ichki chizilgn, yln es ko pburchkk tshqi chizilgn deyildi (-rsm). Istlgn uchburchkk tshqi yln chizish mumkinligi v bu yln mrkzi uchburchk E tomonlrining o rt perpendikulrlri kesishgn nuqtd yotishini 8-sinfd o rgngnsiz. ylng ichki chizilgn gr ko pburchk burchklri soni uchtdn ortiq beshburchk yoki beshburchkk tshqi chizilgn bo ls, ko pburchkk hr doim hm tshqi yln chizib yln. bo lvermydi. Msln, to g ri to rtburchkdn frqli prllelogrmm uchun tshqi chizilgn yln mvjud ems (-rsm). 8-sinf geometriy kursidn m lumki, to rtburchkk qrm-qrshi burchklri yig indisi 80 g teng bo lgnd v fqt shu hold ung tshqi yln chizish mumkin (-rsm). -msl. tkir burchkli uchburchkning v blndliklri H nuqtd kesishdi. H to rtburchk ylng ichki chizilgn eknligini isbotlng. Yechilishi. v bo lgni uchun (-rsm) H = H =90. Und H + H =80. To rtburchk ichki burchklri yig indisi 60 bo lgni uchun: + =80 + =80 H + H =80. emk, H to rtburchkk tshqi yln chizish mumkin. ylng ichki chizilgn ko pburchk uchlri yln mrkzidn teng uzoqlikd yotgni uchun yln mrkzi ko pburchk tomonlrining o rt perpendikulrlrid yotdi (5-rsm). emk, ylng ichki chizilgn ko pburchk tomonlrining o rt perpendikulrlri bir nuqtd kesishishi shrt. -msl. sosig tushirilgn blndligi 6 sm bo lgn teng yonli uchburchk rdiusi 0 sm bo lgn ylng ichki chizilgn. Uchburchk tomonlrini toping. Yechilishi. uchburchkk tshqi chizilgn yln mrkzi nuqt tomonning o rt perpendikulri bo lgn blndlikd yotdi (6-rsm). Und, = =6 0= 6 (sm) bo ldi v Pifgor teoremsig ko r, = = 0 6 =8 (sm), ==6(sm). Shuningdek, to g ri burchkli uchburchkd = + = 8 +6 = 8 5 (sm). Jvob: 8 5 sm, 8 5 sm, 6 sm. Svol, msl v topshiriqlr. gr ko pburchk ylng ichki chizilgn bo ls, uning tomonlri o rt perpendikulrlri bir nuqtd kesishishini isbotlng.. Qndy uchburchk ylng ichki chizilgn bo lishi mumkin? To rtburchk-chi?. E beshburchk ylng ichki chizilgn bo ls, = E bo lishini isbotlng.. Ktetlri 6 sm v sm bo lgn to g ri burchkli uchburchkk tshqi chizilgn yln rdiusini toping. 5. Rdiusi 5 sm bo lgn ylng bir tomoni sm bo lgn to g ri to rtburchk ichki chizilgn. To g ri to rtburchk yuzini toping. 6. Rdiusi 0 sm bo lgn ylng ichki chizilgn ) teng tomonli uchburchk; b) kvdrt; d) teng yonli to g ri burchkli uchburchk tomonlrini toping. 7. Tomonlri 6 sm, 0 sm v 0 sm bo lgn uchburchkk tshqi chizilgn yln rdiusini toping. 8. ylng ichki chizilgn EF oltiburchkd F+ F =90 bo ls, yln mrkzi F tomond yotishini isbotlng. 9. Istlgn teng yonli trpetsiy ylng ichki chizilishi mumkinligini isbotlng. 0. Teng yonli trpetsiy chizing. Ung tshqi chizilgn yln ysng. Qiziqrli msl n olti yoshli Glu (E.Glu frnsuz mtemtigi, 8 8) kollejd o qib yurgn chog lrid, ung o qituvchisi bir sot ichid ucht mslni yechib berishni so rgn. Glu yechimi unch oson bo lmgn bu msllrni 5 dqiqd yechib, hmmni hyron qoldirgn. Mn, shu msllrdn biri. Uni siz hm yechib ko ring-chi! Msl. ylng ichki chizilgn to rtburchkning to rtt tomoni, b, c v d g teng. Uning digonllrini toping

47 E YLNG TSHQI HIZILGN K PURHK ylng tshqi chizilgn E beshburchk yoki E beshburchkk ichki chizilgn yln. + = + T rif. gr ko pburchkning brch tomonlri ylng urins, u hold ko pburchk ylng tshqi chizilgn, yln es ko pburchkk ichki chizilgn deyildi (-rsm). Istlgn uchburchkk ichki yln chizish mumkinligi v bu yln mrkzi uchburchk bissektrislri kesishgn nuqtd eknligi biln 8-sinfd tnishgnsiz. gr ko pburchk burchklri soni uchtdn ortiq bo ls, bu ko pburchkk hr doim hm ichki yln chizib bo lvermydi. Msln, kvdrtdn frqli to g ri to rtburchkk ichki yln chizib bo lmydi (-rsm). Yn 8-sinf geometriy kursidn m lumki, to rtburchkk fqt v fqt qrm-qrshi tomonlri yig indisi teng bo lgnd ichki yln chizish mumkin (-rsm). ylng tshqi chizilgn ko pburchk tomonlri ylng uringni uchun yln mrkzi shu ko pburchk burchklri bissektrissid yotdi (-rsm). emk, ylng tshqi chizilgn ko pburchk burchklrining bissektrislri bir nuqtd kesishdi. Teorem. gr r rdiusli ylng tshqi chizilgn ko pburchkning yuzi S, yrim perimetri p bo ls, S =pr bo ldi. Isbot. Teorem isbotini ylng tshqi chizilgn EF oltiburchk uchun keltirmiz. yln mrkzi nuqtni ko pburchk uchlri biln tutshtirib, ko pburchkni uchburchklrg jrtmiz. u uchburchklrning blndliklri r g teng (5-rsm). Und, S =S +S +...+S F = r + r F + F r = r =pr. Teorem isbotlndi. Msl. ylng tshqi chizilgn to rtburchkning yuzi sm g, perimetri es 7 sm g teng. yln rdiusini toping. Yechilishi. S= pr formulg ko r, S r= = p = 6 (sm). Jvob: 6 sm.,5 Svol, msl v topshiriqlr. Tomoni 6 sm bo lgn ) teng tomonli uchburchkk; b) kvdrtg ichki chizilgn yln rdiusini toping.. Rdiusi 5 sm bo lgn ylng tshqi chizilgn ko pburchk yuzi 8 sm. Ko pburchk perimetrini toping.. 6-rsmdgi to rtburchklrning perimetrini toping.. 7-rsmdgi m lumotlr sosid so rlgn kesmni toping. 5. ylng tshqi chizilgn prllelogrmm romb bo lishini isbotlng. 6. To g ri burchkli uchburchkk ichki chizilgn yln rdiusi ktetlr yig indisi biln gipotenuz yirmsining yrmig tengligini isbotlng. 7. ylng tshqi chizilgn teng yonli trpetsiyning o rt chizig i uning yon tomonig teng eknligini isbotlng. 8. soslri 9 sm v 6 sm bo lgn teng yonli trpetsiy ylng tshqi chizilgn. yln rdiusini toping. 9*. to rtburchk mrkzli ylng tshqi chizilgn. v uchburchklr yuzlrining yig indisi to rtburchk yuzining yrmig tengligini isbotlng. 0*. ylng tshqi chizilgn trpetsiyning soslri v b bo ls, uning blndligi b g teng eknligini isbotlng. *. Uchlri to rtburchk bissektrislrining kesishgn nuqtlrd bo lgn EFPQ to rtburchkk tshqi yln chizish mumkinligini isbotlng. F ) b) d) b) ) r, d) =? r r r r, E r =? =?

48 MUNTZM K PURHKLR Ntij. Muntzm beshburchkning brch digonllri o zro teng. muntzm beshburchk muntzm oltiburchk muntzm skkizburchk Follshtiruvchi mshq. Qndy shkllr ko pburchk deyildi?. Ko pburchk burchklri, qo shni tomonlri, digonllri deb nimg ytildi?. Qvriq ko pburchk deb qndy ko pburchkk ytildi?. Qvriq ko pburchk ichki burchklri yig indisi hqidgi teoremni yting. T rif. Hmm tomonlri teng v hmm burchklri teng bo lgn qvriq ko pburchk muntzm ko pburchk deyildi. Teng tomonli uchburchk, kvdrt muntzm ko pburchkk misol bo ldi. -rsmd muntzm beshburchk, oltiburchk v skkizburchklr tsvirlngn. Teorem. Muntzm n burchkning hr bir burchgi 80 g teng. Isbot. Muntzm n burchkning burchklri yig indisi (n ).80 g teng (8-sinf). emk, uning hr bir burchgi. 80 g teng. Teorem isbotlndi. Msl. Muntzm 5 beshburchkd v digonllri teng eknligini ko rsting (-rsm). 5 muntzm beshburchk = Yechilishi. Uchburchklr tengligining TT lomtig ko r, v 5 uchburchklr o zro teng. Hqiqtn hm, muntzm ko pburchkning tomonlri teng v burchklri teng bo lgni uchun, = 5, = 5 v = 5. emk, = 5. undn = eknligi kelib chiqdi. Svol, msl v topshiriqlr. Muntzm bo lmgn ko pburchklrg misollr yting v nim uchun muntzm emsligini tushuntiring.. Quyidgi tsdiqlrdn to g rilrini toping: ) brch tomonlri teng bo lgn uchburchk muntzm bo ldi; b) brch tomonlri teng to rtburchk muntzm bo ldi; d) brch burchklri teng to rtburchk muntzm bo ldi; e) brch burchklri teng romb muntzm bo ldi; f) brch tomonlri teng to g ri to rtburchk muntzm bo ldi.. gr ) n=; b) n= 5; d) n= 6; e) n= 0; f) n = 8 bo ls, muntzm n burchk burchklrini toping.. Muntzm n burchkning tshqi burchgi nimg teng bo ldi? gr ) n = ; b) n= 5; d) n= 6; e) n=0; f) n = bo ls, muntzm n burchkning tshqi burchgini toping. 5. Muntzm n burchkning hr uchidn bittdn olingn tshqi burchklri yig indisi 60 g teng eknligini isbotlng. 6. gr muntzm ko pburchkning hr bir burchgi ) 60 ; b) 90 ; d) 5 ; e) 50 bo ls, bu ko pburchk tomonlri sonini toping. 7. Muntzm EF oltiburchk berilgn. ) v digonllr tengligini isbotlng. b) E muntzm uchburchk bo lishini isbotlng. d), E v F digonllr o zro tengligini isbotlng. 8. Tomoni 0 sm bo lgn muntzm ) beshburchkning; b) oltiburchkning; d) skkizburchkning; e) o n ikkiburchkning; f) o n skkizburchkning kichik digonlini hisoblng. 9. Muntzm to rtburchkning kvdrt bo lishini isbotlng. 0*. Kvdrtning tomoni g teng. Uning tomonlrig hr bir uchidn boshlb digonlining yrmig teng kesmlr qo yildi. Ntijd, -rsmd tsvirlngn skkizburchk hosil bo ldi. Uning turini niqlng v yuzini toping

49 MUNTZM K PURHKK IHKI V TSHQI HIZILGN YLNLR Follshtiruvchi mshq. Qndy ko pburchk ylng ichki chizilgn ko pburchk deyildi?. Qndy ko pburchk ylng tshqi chizilgn ko pburchk deyildi?. Istlgn ko pburchk ylng ichki (tshqi) chizilgn bo lishi mumkinmi? H α α H α α α α n n n α α H Teorem. Hr qndy muntzm ko pburchkk ichki yln hm, tshqi yln hm chizish mumkin. Isbot. ytylik,... muntzm n ko pburchk, v burchklri bissektrislrining kesishish nuqtsi bo lsin. u muntzm ko pburchkning burchgini α biln belgilylik.. = =...= n eknligini isbotlymiz (-rsm). urchk bissektrissining t rifig ko r, α = =. emk, teng yonli uchburchk. undn, = kelib chiqdi. v uchburchklr tengligining TT lomtig ko r teng, chunki =, tomon umumiy hmd α = =. Shuning uchun =. Xuddi shundy yo l tutib, =, 5 = v hokzo tengliklr o rinli bo lishi ko rstildi. Shundy qilib, = =... = n, y ni mrkzi v rdiusi bo lgn yln ko pburchkk tshqi chizilgn ylndn ibort bo ldi (-rsm).. Yuqorid ytilgnlrg ko r, teng yonli,,... n uchburchklr teng. Shuning uchun bu uchburchklrning uchidn tushirilgn blndliklri hm teng bo ldi (-rsm): H =H =... =H n. emk, mrkzli v rdiusi H kesmg teng bo lgn yln ko pburchkning brch tomonlrig urindi. Y ni, bu yln ko pburchkk ichki chizilgn yln bo ldi. Teorem isbotlndi. Ntij. Muntzm ko pburchkk ichki chizilgn v tshqi chizilgn ylnlrning mrkzlri bitt nuqtd bo ldi. u nuqt muntzm ko pburchkning mrkzi deyildi. Muntzm ko pburchk mrkzini uning ikki qo shni uchlri biln tutshtiruvchi nurlrdn ibort burchk (-rsmdgi,... burchklr) uning mrkziy burchgi deyildi. Muntzm ko pburchkning mrkzidn tomonlrig tushirilgn perpendikulrlr (- rsmdgi H, H,... kesmlr) uning pofemsi deyildi. Msl. gr muntzm n burchkning tomoni, ung ichki chizilgn ylnning rdiusi r bo ls, uning S yuzini S= nr formul biln hisoblsh mumkinligini isbotlng (-rsm). Yechilishi. Ko pburchkning yrim perimetri p= n bo lgni uchun, ylngtshqi chizilgn ko pburchk yuzini topish formulsi S =pr g ko r, S = nr bo ldi. Svol, msl v topshiriqlr. Yuzi 6 sm bo lgn kvdrtg ichki v tshqi chizilgn ylnlr rdiuslrini toping.. Perimetri 8 sm bo lgn muntzm uchburchkk ichki v tshqi chizilgn ylnlr rdiuslrini hisoblng.. Muntzm oltiburchkk tshqi chizilgn yln rdiusi uning tomonig teng bo lishini isbotlng.. Muntzm ko pburchk tomonlrining o rtlri boshq bir muntzm ko pburchk uchlri bo lishini isbotlng (5-rsm). 5. Muntzm uchburchkk ichki chizilgn yln rdiusi tshqi chizilgn yln rdiusidn ikki mrt kichik eknligini isbotlng. 6*. Muntzm ko pburchkning istlgn ikkit tomonining o rt perpendikulrlri bir nuqtd kesishishi yoki bir to g ri chiziqd yotishini isbotlng. 7. ylng ichki chizilgn muntzm ko pburchkning bir tomoni ylndn ) 60 ; b) 0 ; d) 6 ; e) 8 ; f) 7 g teng yoy jrtdi. Ko pburchkning necht tomoni bor? 8. Qog ozdn oltit teng muntzm uchburchk qirqib oling. Ulrdn foydlnib, muntzm oltiburchk ysng. Tomonlri teng bo lgn muntzm oltiburchk v uchburchk yuzlri nisbtini toping r

50 MUNTZM K PURHKNING TMNI ILN TSHQI V IHKI HIZILGN YLNLR RIUSLRI RSIGI G LNISH Follshtiruvchi mshq To g ri burchkli uchburchk o tkir burchgining ) sinusi; b) kosinusi; d) tngensi deb nimg ytildi? Tomoni n g teng bo lgn muntzm n burchkk tshqi chizilgn ylnning R rdiusi v ichki chizilgn ylnning r rdiusini hisoblsh uchun formullr topmiz. uning uchun to g ri burchkli uchburchkdn foydlnmiz. u yerd ko pburchkning mrkzi, ko pburchkning tomoni o rtsi (-rsm). Und, β= = =. = ; Svol, msl v topshiriqlr. Tomoni 5 sm bo lgn ) muntzm uchburchkk; b) muntzm to rtburchkk; d) muntzm oltiburchkk ichki v tshqi chizilgn ylnlr rdiuslrini hisoblng.. -rsmd R rdiusli ylng ichki chizilgn kvdrt, muntzm uchburchk v muntzm oltiburchk tsvirlngn. ftringizg berilgn jdvllrni ko chirib, uning bo sh ktklrini to ldiring (n ko pburchk tomoni, P ko pburchk perimetri, S uning yuzi, r ung ichki chizilgn yln rdiusi). ) b) d) R== = ; r = = = ; R r R R r R R r β β R r = =.cosβ=rcos. r u formullrdn foydlnib, yrim muntzm ko pburchklr tomoni, ichki v tshqi chizilgn ylnlr rdiuslri orsidgi bog lnishlrni topmiz.. Muntzm uchburchk uchun (n=): β= =60 ; R= = ; r = = ; R=r.. Kvdrt uchun (n= ): β= =5 ; R= = ; r = = ; R=r.. Muntzm oltiburchk uchun (n= 6): β= =60 ; R= = 6 ; r = = ; R=. Msl. Muntzm n burchkning n tomonini shu ko pburchkk tshqi chizilgn ylnning R rdiusi v ichki chizilgn ylnning r rdiusi orqli ifodlng. Yechilishi. R= v r= formullrdn n=rsin v n=r tg formullrni hosil qilmiz. Xususn, n= bo ls, =R =r R r P S R r P S R r 6 P S Rdiusi 8 sm bo lgn ylng ichki chizilgn muntzm o n ikkiburchkning bir uchidn chiqqn digonllrini toping.. ylng ichki chizilgn muntzm uchburchk perimetri sm. Shu ylng ichki chizilgn kvdrt tomonini toping. 5. Silindr shklidgi yog ochdn sosining tomoni 0 sm bo lgn: ) kvdrt; b) muntzm oltiburchk bo lgn prizm shklidgi ustun tyyorlsh kerk. Yog och ko ndlng kesimining dimetri kmid qnch bo lishi zrur? 6. - rsmd tsvirlngn, rng-brng nqshlrni tomosh qils bo ldign Kleydoskop deb ) b) nomlngn o yinchoq sizg tnish bo ls kerk. yinchoq quvur v t oyn bo lklridn ibort. 5 sm -b rsmd uning ko ndlng kesimi tsvirlngn v o lchmlri berilgn. Kleydoskop ko ndlng kesimining rdiusini toping sm 5 sm 98 99

51 5 ILIMINGIZNI SIN K RING I. Testlr. Quyidgi ko pburchklrning qysi birig ichki chizilgn yln mvjud ems? ) Uchburchkk; ) Kvdrtdn frqli rombg; ) Kvdrtg; E) Rombdn frqli to g ri to rtburchkk.. Quyidgi ko pburchklrning qysi birig tshqi chizilgn yln mvjud ems? ) Uchburchkd; ) Kvdrtdn frqli rombd; ) Kvdrtd; E) Rombdn frqli to g ri to rtburchkd.. ylng ichki chizilgn brch to rtburchklr uchun noto g ri tenglikni toping. ) =60 ; ) +=+; ) + =80 ; E) + =80.. ylng tshqi chizilgn brch to rtburchklr uchun noto g ri tenglikni toping. ) =60 ; ) +=+; ) + =80 ; E) -=-. 5. Tomonlri 5 sm v sm bo lgn to g ri to rtburchkk tshqi chizilgn yln rdiusini toping. ) 6 sm; ) 6,5 sm; ) 7 sm; E) 7,5 sm. 6. Muntzm burchkning ichki burchgini toping. ) 0 ; ) 5 ; ) 50 ; E) Hr bir tshqi burchgi 60 bo lgn muntzm ko pburchkning ichki burchklri yig indisini toping. ) 50 ; ) 60 ; ) 90 ; E) 70. II. Ysshg doir msllr.. Tomoni berilgn kesmg teng muntzm oltiburchk ysng. und muntzm oltiburchkk tshqi chizilgn ylnning rdiusi oltiburchkning tomonig teng eknligidn v -rsmdn foydlning rsmlrdgi m lumotlrdn foydlnib, berilgn ylng ichki chizilgn ) muntzm uchburchk; b) kvdrt; d) muntzm skkizburchk ysng.. 5-rsmdn foydlnib, berilgn ylng tshqi chizilgn muntzm oltiburchk ysng (5-rsmd tsvirlngn ylng tshqi chizilgn oltiburchk tomonlri shu ylng ichki chizilgn muntzm oltiburchk uchlridn ylng o tkzilgn urinmlrd yotdi). III. Hisoblshg doir msllr.. Muntzm uchburchk, kvdrt v muntzm oltiburchklrning tomonlri bir-birig teng. Ulrning yuzlri nisbtini toping.. itt ylng ichki chizilgn muntzm oltiburchk v tshqi chizilgn oltiburchk yuzlri nisbtini toping.. Muntzm ) oltiburchk; b) skkizburchk; d) o n ikkiburchkning prllel tomonlri orsidgi msof 0 sm g teng. Ko pburchk tomonini toping.. Rdiusi R bo lgn ylng... 8 muntzm skkizburchk ichki chizilgn. 7 8 to rtburchkning to g ri to rtburchk eknini isbotlng v uning yuzini toping. 5. ylng tshqi chizilgn to g ri burchkli uchburchkning gipotenuzsi shu ylng urinish nuqtsid sm v 6 sm uzunlikdgi kesmlrg bo lindi. Uchburchk yuzini toping. 6. Muntzm o nburchkning bir uchidn chiqqn eng ktt v eng kichik digonllri orsidgi burchkni toping. IV. zingizni sinb ko ring (nmunviy nzort ishi). 5. Ktetlri 0 sm v sm bo lgn to g ri burchkli uchburchkk ichki chizilgn v tshqi chizilgn ylnlrning rdiuslrini toping.. Rdiusi 5 sm bo lgn ylng tshqi chizilgn rombning bir burchgi 50 g teng. Rombning ) perimetrini; b) digonllrini; d) yuzini toping.. Tomoni sm bo lgn muntzm oltiburchkning bir uchidn chiqqn digonllrini toping.. (Qo shimch). Rdiusi sm bo lgn ylng ichki chizilgn muntzm oltiburchk v muntzm uchburchklr yuzlrining yirmsini toping. Triiy lvhlr. Istlgn muntzm ko pburchkni hm sirkul v chizg ich yordmid ysb bo lvermydi. uni 80-yild nemis mtemtigi Krl Guss ( ) lgebrik usuld isbotlgn. U gr n sonning m p p...p n yoyilmsidgi p, p,... p n turli tub sonlr k + ko rinishid bo lsgin muntzm n burchkni sirkul v chizg ich yordmid yssh mumkinligini isbotldi. u yerd m v k mnfiy bo lmgn butun sonlr. 00 0

52 6 YLN UZUNLIGI Follshtiruvchi mshq. dtd quvur bo lgining ko ndlng kesimi ylndn ibort bo ldi. Ingichk ipni bir uchidn boshlb, quvurg bir mrt o rng. ir mrt o rshg ketgn ip bo lgi quvur ko ndlng kesimi, y ni ylnning uzunligi bo ldi. Uni rsmd ko rstilgndek qilib chizg ich yordmid o lchng.. Yuqoridgi usul biln quvur ko ndlng kesimi dimetrini niqlng.. niqlngn yln uzunligini uning dimetrig nisbtini hisoblng.. Yuqorid keltirilgn o lchsh v hisoblsh ishlrini yn bir necht turli o lchmdgi quvur bo lklri uchun hm bjrib, yln uzunligini uning dimetrig nisbtini toping. 5. Mshq ntijsig ko r, yln uzunligining uning dimetrig nisbti hqid qndy ulos chiqrish mumkin? Teorem. yln uzunligining yln dimetrig nisbti yln rdiusig bog liq ems, y ni hr qndy yln uchun bu nisbt o zgrms sondir. Isbot. Ikkit itiyoriy yln olmiz. Ulrning rdiuslri R v R, uzunliklri es mos rvishd v bo lsin. tenglikni isbotlshimiz kerk. Hr ikki ylng ichki muntzm n-burchkni chizmiz. Ulrning perimetrlrini mos rvishd P v P deb belgilylik. Und, P =n.r sin, P =n.r sin bo lgni uchun (*) bo ldi. u tenglik istlgn n uchun to g ri. n soni kttlshib bors, berilgn ylng ichki chizilgn n-burchk perimetri P shu yln uzunligi g yqinlshib bordi. Shu singri P hm g yqinlshib bordi. Shuning uchun nisbt nisbtg teng bo ldi (buning to liq isboti mtemtikning yuqori bosqichlrid o rgnildi). Shundy qilib, (*) tenglikdn, bundn es tenglik kelib chiqdi. Teorem isbotlndi. yln uzunligini uning dimetrig nisbtini yunon lifbosining π hrfi biln belgilsh qbul qilingn ( pi deb o qildi). yln uzunligining uning dimetrig nisbtini π hrfi biln belgilshni buyuk mtemtik Leonrd Eyler (707 78) fng kiritgn. Yunonchd yln so zi shu hrf biln boshlndi. π irrtsionl son bo lib, mliyotd uning,6 g teng bo lgn tqribiy qiymtidn foydlnildi. Shundy qilib, = π. u tenglikdn rdiusi R g teng yln uzunligi uchun =πr formulni hosil qilmiz. Msl. Tomoni 6 sm bo lgn muntzm uchburchkk tshqi chizilgn yln uzunligini toping. Yechilishi. Muntzm uchburchkk tshqi chizilgn yln rdiusini topish formulsi R = g ko r, R = = (sm). Endi, yln uzunligini topish formulsidn = πr =π. =π (sm). Jvob: π sm. Svol, msl v topshiriqlr. Qndy son π biln belgilndi? Rdiusi R g teng yln uzunligini topish formulsidn foydlnib, jdvlni to ldiring (π, deb hisoblng). 8 8π 6,8 R 0,7 0,5. gr yln rdiusi ) mrt oshs; b) sm g oshs; d) mrt kmys; e) sm g kmys, yln uzunligi qnchg o zgrdi?. gr Yer shri ekvtorining 0 milliondn bir qismi m g teng bo ls, Yer shrining rdiusini toping.. ) Tomoni g teng bo lgn muntzm uchburchkk; b) ktetlri v b bo lgn to g ri burchkli uchburchkk; d) sosi v yon tomoni b bo lgn teng yonli uchburchkk tshqi chizilgn yln uzunligini toping. 5. ) Tomoni g teng kvdrtg; b) gipotenuzsi c g teng bo lgn teng yonli to g ri burchkli uchburchkk; d) gipotenuzsi c, o tkir burchgi α bo lgn to g ri burchkli uchburchkk ichki chizilgn yln uzunligini toping. 6. Teplovoz m yo l yurdi. und uning g ildirgi 00 mrt ylndi. Teplovoz g ildirgining dimetrini toping. 7. Nei vtomobili g ildirgi ylnsining rdiusi sm g teng. vtomobil 00 km yo l yurs, uning g ildirgi nech mrt ylndi (-rsm)? sm 0 0

53 7 YLN YYI UZUNLIGI. URHKNING RIN LHVI l = n. n li mrkziy burchk tirlgn yoy uzunligi. ytylik, rdiusi R g teng bo lgn ylnd n li mrkziy burchk berilgn bo lsin (-rsm). und ylnning mrkziy burchkk tirlgn yoyining grdus o lchovini n yoki n li yoy deb yuritilishini esltib o tmiz. Rdiusi R g teng butun yln, y ni o lchovi 60 bo lgn yoy uzunligi πr g teng bo lgni uchun, li yoy uzunligi g teng bo ldi. =R U hold, n li yoy uzunligi l=.n formul biln niqlndi (-rsm). α R. urchkning rdin o lchovi. urchkning grdus o lchovi biln bir qtord uning rdin o lchovi hm ishltildi. yln yoyi uzunligining rdiusg nisbtini yuqoridgi formulg sosn: =.n g teng. α= rdin emk, yln yoyi uzunligining rdiusig nisbti fqt shu yoyg tirlgn mrkziy burchk kttligig bog liq ekn. u ossdn foydlnib, burchkning rdin o lchovi siftid uddi shu nisbtni olmiz: α = =.n. dtd, rdin so zi yozilmydi. Msln: 5 rdin o rnig 5 deb yozildi. ir rdin 80 π grdusg teng: rdin= 80 π urchkning grdus o lchovidn rdin o lchovig o tish uchun α=.n formuldn foydlnildi. R.n l Shundy qilib, n li burchkning rdin o lchovini topish uchun uning grdus o lchovini g ko pytirish kifoy ekn. Xususiy hold, 80 li burchkning rdin o lchovi π g teng, 90 li, y ni to g ri burchkning rdin o lchovi g teng bo ldi. α rding teng mrkziy burchkk mos yoyining uzunligi l=αr formul biln hisoblndi. Msl. Ikkit burchgi mos rvishd 0 v 5 bo lgn uchburchk burchklrining rdin o lchovlrini toping. Yechilishi. Uchburchkning 0 li burchgi rdin o lchovi 0. =, 5 li burchgi rdin o lchovi 5. =. Uchburchk ichki burchklri yig indisi 80 g, y ni π g tengligi hqidgi teoremg sosn uchburchkning uchinchi burchgining rdin o lchovini topmiz π =. Jvob: ; v. Svol, msl v topshiriqlr. Rdiusi 6 sm bo lgn ylnning grdus o lchovi ) 0 ; b) 5 ; d) 90 ; e) 0 bo lgn yoyi uzunligini toping.. ) 0 ; b) 60 ; d) 75 g teng burchkning rdin o lchovini toping.. ),; b) ; d) rding teng burchkning grdus o lchovini toping.. gr yln rdiusi 5 sm bo ls, uning ) ; b) ; d) rding teng mrkziy burchgi tirlgn yoy uzunligini toping. 5. Rdiusi sm bo lgn ylng uchburchk ichki chizilgn. gr ) =0 ; b) =0 bo ls, nuqtni o z ichig olmgn yoy uzunligini toping. 6. ylnning teng vtrlri ylndn teng yoylr jrtishini isbotlng. 7*. Ikkit yln bir-birining mrkzidn o tdi. u ylnlrning umumiy vtri hr ikki ylndn jrtgn yoylr uzunliklri nisbtini toping. 8*. Rdiuslri teng bo lgn ucht ylnlr bir-birig tshqridn v rdiusi R g teng bo lgn ylng ichkridn urindi (-rsm): ) ylnlr rdiusini toping; b) bo ylgn shklni chegrlovchi yoylr uzunliklri yig indisini toping. Qiziqrli msl -rsmd tsvirlngn ikkit tishli g ildirklr birbirig tishltilgn. G ildirklr rdiusi R v R. irinchi g ildirk n mrt ylngnd ikkinchi g ildirk nech mrt ylndi? R R 0 05

54 8 IR YUZI T rif. Tekislikning berilgn nuqtsidn berilgn R msofdn ktt bo lmgn msofd yotuvchi brch nuqtlridn tshkil topgn shklg doir deb tldi. und nuqt doirning mrkzi, R es doirning rdiusi deb tldi. Mzkur doirning chegrsi mrkzi nuqtd, rdiusi es R g teng bo lgn ylndn ibort bo ldi. ) Follshtiruvchi mshq ir vrq qog ozg yo g on chiziq biln yln chizing v - rsmd ko rstilgndek, uning bir necht dimetrlrini o tkzib, doirni teng bo lklrg b) bo ling. So ng bu bo lklrni qiyib oling v -b rsmd ko rstilgndek terib, F shklni hosil qiling. gr doir F istlgnch ko p teng bo lklrg bo linib, bu bo lklr rsmd ko rstilgn trtibd terils, ntijd to g ri to rtburchkk jud yqin F shkl pydo bo ldi. ) F shklni to g ri to rtburchk shklig jud yqinligini hisobg olib, uning tomoni tminn nimg teng bo lishini toping (ko rstm: tomonni doir rdiusi biln tqqoslng). b) F shklning tomoni tminn nimg teng α bo ldi? (Ko rstm: v tomonlr yo g on chiziq biln chizilgnig, y ni yln yoychlridn ibort eknligig e tibor bering) d) F shklning to g ri to rtburchk shklig jud yqin eknligini hisobg olib, uning yuzini tqribn hisoblng. F shkl yuzi doir yuzig jud yqin eknligini nzrd tutib, doir yuzi hqid ulos chiqring. Teorem. Rdiusi R g teng bo lgn doirning yuzi πr g teng. Isbot. Rdiusi R v mrkzi nuqtd bo lgn ylnni qrymiz. ylng tshqi chizilgn... n v ichki chizilgn... n muntzm n I II burchklrning yuzlri mos rvishd S n v S n bo lsin (-rsm). v uchburchklr yuzlrini topmiz: S = = R; S = = cosα = Rcosα. I I II II Und, S n =n R = P n R, S n =n Rcosα= P n R cosαц ц () I II u yerd P n v P n mos rvishd... n v... n ko pburchklrning perimetrlri. α= bo lgni uchun n ning yetrlich ktt qiymtlrid cosα ning qiymti birdn, P n v P n lrning qiymtlri yln uzunligi, y ni πr dn istlgnch km frq qildi. Und, () tengliklrg ko r, n ning yetrlich ktt qiymtlrid ko pburchklrning yuzi πr g yqinlshib bordi. undn, doirning yuzi uchun S =πr formul kelib chiqdi. Teorem isbotlndi. Msl. Sirk rensi ylnsining uzunligi m. ren rdiusi v yuzini toping. Yechilishi. ) yln uzunligini topish formulsidn rdiusni topmiz (-rsm): R R = π., 6,5 (m). ) oir yuzini hisoblsh formulsidn renning yuzini topmiz: S =πr,. 6,5,8 (m ). Jvob: R 6,5 m; S,8 m. Svol, msl v topshiriqlr. oir yuzini hisoblsh formulsini soslng.. Rdiusi R g teng bo lgn doirning S yuzini topish formulsidn foydlnib jdvlni to ldiring (π =, deb oling). R 5 5, 6,5 7 S 9 9 π. gr doir rdiusi ) k mrt oshs; b) k mrt kmys, doir yuzi qndy o zgrdi?. Tomoni 5 sm bo lgn kvdrtg ichki chizilgn v tshqi chizilgn doirlrning yuzini toping. 5. Tomoni sm bo lgn muntzm uchburchkk ichki v tshqi chizilgn doirlrning yuzini toping. 6. Rdiusi R bo lgn doirdn eng ktt kvdrt qirqib olindi. oirning qolgn qismi yuzini toping. 7. Tomonlri 6 sm v 7 sm bo lgn to g ri to rtburchkk tshqi chizilgn doir yuzini toping. 8. Tomoni 0 sm v o tkir burchgi 60 bo lgn rombg ichki chizilgn doir yuzini toping. 9*. To g ri burchkli uchburchk tomonlrini dimetr qilib yrim doirlr chizilgn. Gipotenuzg chizilgn yrim doir yuzi ktetlrg chizilgn yrim doirlr yuzlri yig indisig teng bo lishini ko rsting (-rsm)

55 9 S = S = IR LKLRI YUZI K T rif. oirning yoyi v bu yoy oirlrini doir mrkzi biln tutshtiruvchi ikkit rdiusi biln chegrlngn qismi sektor deyildi. Sektorning chegrsi bo lgn yoy sektor yoyi deyildi. -rsmd K v L yoyli ikkit sektor tsvirlngn (ulrdn birinchisi bo ylgn). L Rdiusi R g v yoyining grdus o lchovi n g teng K bo lgn sektorning S yuzini topish uchun formul keltirib chiqrmiz. Yoyi g teng sektorning yuzi doir (y ni yoyi 60 g teng sektor) yuzining qismig teng bo lgni uchun, yoyi n grdus bo lgn sektorning yuzi S =.n yoki S= Rl formul orqli topildi. u yerd l n li sektor yoyining L uzunligi. n.n S n.n+s T rif. oirning yoyi v bu yoy oirlrini tutshtiruvchi vtri biln chegrlngn qismi segment deyildi. -rsmd K v L yoyli ikkit segment tsvirlngn (ulrdn birinchisi bo ylgn). Yrim doirdn frqli segmentning S yuzi S = S sektor ± S =. n ±S formul bo yich hisoblndi (- v - rsmlrg qrng). Msl. Yoyning grdus o lchovi 7 bo lgn sektorning yuzi 5π g teng. Sektor rdiusini toping. Yechilishi. Sektor yuzini topish formulsig ko r,. 7 = 5π. undn, R = 5π.60 7π =5, demk, R = 5. Jvob: 5. Svol, msl v topshiriqlr. Sektor yuzini topish formulsini keltirib chiqring.. Segment yuzini topish formulsini keltirib chiqring.. Yoyining grdus o lchovi ) 0 ; b) 5 ; d) 0 ; e) 90 v rdiusi 7 sm bo lgn sektor v segment yuzlrini toping.. 5-rsmd tomoni g teng bo lgn muntzm uchburchk, kvdrt v muntzm oltiburchk tsvirlngn. o ylgn shkllr yuzini toping. und sektorlrning rdiuslri ko pburchk tomonining yrmig teng. 5. Nishond rdiuslri,,, g teng bo lgn to rtt yln bor. Eng kichik doir yuzini v hr bir hlq yuzini toping (6-rsm). 6. Rdiusi 0 sm g teng bo lgn doird rdiusg teng vtr o tkzilgn. Hosil bo lgn segmentlr yuzini hisoblng. 7. Rdiuslri 5 sm dn bo lgn ikkit doir mrkzlri orsidgi msof 5 sm. oirlr umumiy qismining yuzini toping. 8. Rdiusi 0 sm bo lgn doirg ichki v tshqi chizilgn muntzm o n ikkiburchklr yuzini hisoblng. Ntijlrni doir yuzi biln solishtiring. Qiziqrli msl 7-rsmd tsvirlngn guldon rsmini ucht to g ri chiziq biln: ) shundy to rt bo lkk bo lingki, ulrdn to g ri to rtburchk yig ish mumkin bo lsin; b) ikkit to g ri chiziq biln shundy uch qismg bo lingki, ulrdn kvdrt yig ish mumkin bo lsin. Triiy lvhlr. Uzoq vqtlr mobynid dunyoning ko plb mtemtiklri doir kvdrtursi deb nom olgn quyidgi mslni yechishg hrkt qilgnlr: sirkul v chizg ich yordmid yuzi berilgn doir yuzig teng bo lgn kvdrt yssh. Fqt XIX srning oirid bu msl yechimg eg emsligi isbotlngn. 5 ) b) d) sm 08 09

56 50 S MSLLR YEHISH -msl. v nuqtlr ylnning dimetrini ucht, v kesmlrg jrtdi., v dimetrli ylnlr uzunliklrining yig indisi dimetrli yln uzunligig teng eknligini isbotlng (-rsm). Yechilishi. yln uzunligini topish formulsidn foydlnib,, v dimetrli ylnlrning,, uzunliklri yig indisini topmiz: S + + = π+ π+ π = π( + +). ++= v dimetrli ylnning uzunligi π g teng bo lgni uchun + + =. Shu tenglikni isbotlsh tlb qilingn edi. S -msl. to rtburchkning uchlrini mrkz qilib bir il rdiusli sektorlr yslgn (-rsm). u sektorlrdn itiyoriy ikkitsi umumiy nuqtg eg ems hmd brchsining rdiusi sm. Sektorlr yuzlrining yig indisini toping. Yechilishi. ) To rtburchkning,,, burchklri mos rvishd α, α, α, α bo lsin. Und, ko pburchk ichki burchklrining yig indisi hqidgi S teoremg ko r, α + α + α + α =60. ) Sektor yuzini topish formulsig ko r (R = sm), S =.α, S =.α, S =.α, S =.α. () ) () tengliklrning mos qismlrini qo shmiz. Und, S + S + S + S = (α + α + α + α )=. 6 0 = =π (sm ). Jvob: π sm. Svol, msl v topshiriqlr. Perimetri m bo lgn kvdrt v uzunligi m bo lgn yln berilgn. u yln biln chegrlngn doir yuzi biln kvdrt yuzini tqqoslng.. Rdiusi 8 sm bo lgn doirdn 60 li sektor qirqib olingn. oirning qolgn qismi yuzini toping.. igonllri 6 sm v 8 sm bo lgn rombg ichki chizilgn doir yuzini hisoblng.. -rsmd bo yb ko rstilgn shkl yuzini toping. Und kvdrt, = sm. 5*. -rsmd rimed pichog i deb tluvchi π. shkl bo yb ko rstilgn. Uning yuzi formul biln hisoblnishini isbotlng (bund, =90 v = eknligidn foydlning). 6. gr = 6 sm, = sm bo ls, -rsmd bo yb ko rstilgn shklning yuzi v perimetrini (uni o rb turgn yoylr uzunligi yig indisini) toping. 5 ) S S =S +S b) Triiy lvhlr. Gippokrt oychlri. Gippokrt oychsi ikki yln yoylri biln chegrlngn v quyidgi ossg eg bo lgn shkldir: gr ylnlr rdiuslri v oych yoylri S S =S +S 5 tirlgn vtr berilgn bo ls, oychg tengdosh kvdrt yssh mumkin. Pifgor teoremsi qo llnils, 5- rsmd tsvirlngn S 5 gipotenuzg qurilgn yrim doir yuzi ktetlrg qurilgn yrim doirlr yuzlri yig indisig teng bo ldi (07-betdgi 9*-mslg qrng). Shuning uchun 5-b rsmdgi oychlr yuzlrining yig indisi uchburchk yuzig teng (mushohd qilib ko ring!). gr rsmdgi uchburchk o rnig teng yonli to g ri burchkli uchburchk olsk, hosil bo lgn ikki oychdn hr birining yuzi uchburchk yuzining yrmig teng bo ldi. oir kvdrtursi hqidgi mslni yechishg urinib, yunon mtemtigi Gippokrt (miloddn vvlgi V sr) ko pburchk biln tengdosh bir nech il oychlrni itiro qilgn. Gippokrt oychlrining to l jdvli fqt XIX XX srlrd tuzilgn. S 0

57 III G IR Q SHIMH MSLLR V 5 M LUMTLR I. Testlr. 5 grdusli burchkning rdin o lchovi nimg teng? π π. g teng;. g teng;. g teng; E. g teng.. Rdiusi sm bo lgn ylnning grdus o lchovi 50 bo lgn mrkziy burchgi tirlgn yoy uzunligini toping.. 5π sm;. 5π sm;. 0π sm; E. 5π sm.. Rdiusi 6 sm bo lgn ylnd 5π rding teng mrkziy burchk tirlgn yoy uzunligini toping.. 5π sm;. 5π 6 sm;. π 5π sm; E. sm.. Tomoni 5 sm g teng bo lgn kvdrtg tshqi chizilgn yln uzunligini toping.. 5 π;. π;. π; E. 5π. 5. imetri 6 g teng doir yuzini toping.. 9π;. 6π;. π; E. π. 6. Yoyining grdus o lchovi 50, rdiusi 6 sm bo lgn doirviy sektorning yuzini toping.. 5π sm ;. 6π sm ;. 0 π sm ; E. π sm. 7. Yoyining uzunligi sm v rdiusi 6 sm bo lgn doirviy sektorning yuzini toping.. 5π sm ;. 6π sm ;. 0 π sm ; E. π sm. 8. Yoyining grdus o lchovi 0, rdiusi g teng bo lgn doirviy segmentning yuzini toping.. 6π - ;. 6π + ;. π - ; E. π +. II. Msllr. EFKL muntzm skkizburchkning tomoni 6 sm. Uning digonlini toping.. Kvdrt rdiusi dm bo lgn ylng ichki chizilgn. Kvdrt qo shni tomonlrining o rtlridn o tuvchi vtrni ylndn jrtgn yoylrning uzunligini toping.. ylnning 90 li yoyi uzunligi 5π sm. yln rdiusini toping.. Rdiusi 0 g teng ylndn uzunligi 0π g teng yoy jrtildi. u yoyg mos mrkziy burchkni toping. 5. Ikkit doirning umumiy vtri bu doirlrni chegrlovchi ylnlrdn 60 li v 0 li yoylrni jrtdi. oirlr yuzlrining nisbtini toping. 6. Tomonlri,, 5 bo lgn uchburchkk ichki v tshqi chizilgn doirlr yuzlrini toping. 7. oir vtri 60 li yoyni tortib turdi. u vtr jrtgn segmentlr yuzlri nisbtini toping. 8. Muntzm oltiburchk yuzining ung ichki chizilgn doir yuzig nisbtini toping. 9. Tomoni g teng bo lgn EF muntzm oltiburchk berilgn. Mrkzi nuqtd v rdiusi bo lgn yln bu oltiburchkni ikki qismg jrtdi. Hr bir qism yuzini toping. 0. To g ri burchkli uchburchkd =7, = 90, =5 sm. dimetrli ylnning uchburchk ichid yotgn yoyi uzunligini toping.. oirg ichki chizilgn muntzm skkizburchk berilgn. Uning ikki qo shni uchlrig o tkzilgn rdiuslr doirni ikkit sektorg jrtdi. u sektorlr yuzlrining nisbtini toping.. To g ri burchkli uchburchkd =0, =90, =8 sm. kesm uchburchkk tshqi chizilgn doirni ikki segmentg jrtdi. o yb ko rstilgn segment yuzini toping (-rsm).. Kichik yln ktt ylng hmd uning 0 dimetrig urindi. gr dimetrg urinish nuqtsi yln mrkzi v = bo ls, rsmd bo ylgn shkl yuzini toping (-rsm).. Muntzm EF oltiburchkning tomoni 6 g teng v mrkzi nuqtd. Mrkzlri v nuqtd v rdiuslri teng bo lgn ylnlr nuqtd urindi. o ylgn soh yuzini toping (-rsm). 5. To g ri burchkli uchburchkd =90, =, =. Mrkzi gipotenuzd bo lgn yln uchburchk ktetlrig urindi. u yln uzunligini toping. III. zingizni sinb ko ring (nmunviy nzort ishi). Tomoni 6 sm bo lgn kvdrtg tshqi chizilgn yln uzunligini v ichki chizilgn doir yuzini toping.. Tomoni sm bo lgn muntzm ko pburchkk ichki chizilgn yln rdiusi sm g teng bo ls, ung tshqi chizilgn yln rdiusini toping. F E

58 . 0 li yln yoyining uzunligi sm bo ls, ) yln rdiusini; b) yoyi 0 bo lgn sektor yuzini; d) yoyi 0 bo lgn segmentning yuzini toping. Qiziqrli msl In v Yn Rsmd tbitdgi qrm-qrshiliklrni ifodlovchi In v Yn deb nomlngn itoy rmzi tsvirlngn. ) In v Yn rmzlri yuzlri tengligini ko rsting; b) bitt to g ri chiziq biln bu rmzlrning hr birini yuzlri teng bo lgn ikki bo lkk bo ling. d) In v Yn rmzlr perimetrlrini (ulrni o rb turgn yoylr uzunliklri yig indisini) toping. Triiy lvhlr. yln uzunligini hisoblsh jud qdimdn dolzrb mummo bo lgn. yln uzunligini ung ichki chizilgn ko pburchk perimetrig lmshtirish usuli keng trqlgn. rt siyolik mtemtiklr hm doirg ichki chizilgn muntzm ko pburchklrni yssh, ulrning tomonlrini doirning rdiusi orqli ifodlsh msllri biln shug ullngnlr. bu Ryhon eruniy Qonuni Ms udiy srid doirg ichki chizilgn ko pburchklrning tomonini niqlsh biln shug ullnib, ichki chizilgn beshburchk, oltiburchk, yettiburchk,..., o nburchk tomonlrini niqlsh usulini ko rstdi. u hisoblsh ntijsid u π, qiymtg eg eknligini niqlydi. Qdimgi obil v Misr qo lyozmlri v mitlrid π uchg teng deb olingn. u o sh dvr niqlik tlbi uchun yetrli bo lgn. Keyinchlik rimliklr π uchun, ni ishltishgn. π soni uchun rimed bergn qiymt, bo lib, bu mliy msllrni hl qilishd jud m qul edi. Xitoy mtemtiklrid π,55... v /7. Hindlrning Sulv Sutr ( rqon qoidsi ) srid π uchun,008 v,6... v 0,6... qiymtlr uchrydi. Mirzo Ulug bekning stronomiy mktbi nmoyndlridn biri Jmshid G iyosiddin l-koshiy -yild yozgn yln uzunligi hqid kitob nomli risolsid ylng ichki v tshqi chizilgn muntzm ko pburchk tomonlri sonini ikkilntirish yo li biln 8 = tomonli muntzm ko pburchklr perimetrini hisoblb, π uchun π =, qiymtni hosil qilgn. u 6 t o nli rqmgch niqdir. mmo l-koshiyning sri uzoq vqtgch Yevropd nom lum bo lgn. Yevropliklrdn belgiylik Vn Romen 597- yild 0 tomonli muntzm ko pburchkk rimed usulini ttbiq etib, π uchun 7 t o nli rqmlri niq bo lgn qiymt topgn. Gollndiylik Rudolf vn Seylon (50 60) bu niqlikni 5 t o nli rqmlrgch olib borgn. Hozirgi dvrd elektron hisoblsh mshinlri yordmid π uchun milliondn ortiq o nli rqmlri niq bo lgn qiymtlr topilgn. Kundlik hisoblshlr uchun, qiymt, mtemtik hisoblshlr uchun,6 qiymt, htto stronomiy v kosmonvtik uchun,586 qiymt kifoydir. IV UHURHK V YLNGI METRIK MUNSTLR Ushbu bobni o rgnish ntijsid siz quyidgi bilim v mliy ko nikmlrg eg bo lsiz: ilimlr: proporsionl kesmlrning osslrini bilish; to g ri burchkli uchburchkd gipotenuzg tushirilgn blndlikning osslrini bilish; o zro kesishuvchi vtrlr kesmlri to g risidgi hmd ylnni kesuvchi to g ri chiziq kesmlri to g risidgi osslrni bilish. Ko nikmlr: kesmlrning nisbti v proporsionl kesmlrg doir msllrni yech olish; to g ri burchkli uchburchkd gipotenuzg tushirilgn blndlikning osslridn foydlnib,msllr yech olish; kesuvchi vtrlr kesmlrining v kesuvchi to g ri chiziq kesmlrining osslridn foydlnib,msllr yechish. 5

59 5 ) KESMLR PRYEKSIYSI V PRPRSINLLIK m nuqtning, nuqtning, kesmning m to g ri chiziqdgi proyeksiysi α b) α E P α F R α α E F m b m Q m Follshtiruvchi mshq. Kesmlr nisbti nimni ngltdi?. Qndy kesmlr proporsionl deyildi?. Fles teoremsini yting. Tekislikd m to g ri chiziq v kesm berilgn bo lsin. v nuqtlrdn m to g ri chiziqq v perpendikulrlr tushirmiz (-rsm). kesm kesmning m to g ri chiziqdgi proyeksiysi (soysi) deyildi. kesmning m to g ri chiziqdgi proyeksiysini qurish mli kesmni m to g ri chiziqq proyeksiylsh deyildi. Teorem. ir to g ri chiziqd yoki prllel to g ri chiziqlrd yotdign kesmlr berilgn bo lsin. Ulrning yni bir to g ri chiziqq proyeksiylri berilgn kesmlrg proporsionl bo ldi. b, ning, ning, E F EF ning m to g ri chiziqdgi proyeksiylri (-rsm) = E F = EF () Isbot. ) gr v b to g ri chiziqlr m to g ri chiziqq prllel bo ls, =, =, EF=E F bo lishi hmd () tengliklr o rinli eknligi rvshn. b) ordi-yu v b to g ri chiziqlr m to g ri chiziqq perpendikulr bo ls, v, v, E v F nuqtlr ustm-ust tushdi. Shuning uchun,, E F kesmlrning uzunligi nolg teng bo ldi v () tengliklr bjrildi. d) Endi boshq holni qrymiz. -rsmd tsvirlngnidek, to g ri burchkli P, Q, EFR uchburchklrni qurmiz. Und b bo lgni uchun, P= Q= FER. emk, P, Q v EFR to g ri burchkli uchburchklr o shsh. undn tengliklrni hosil qilmiz. Teorem isbotlndi. = = E F EF Msl. v kesmlr prllel to g ri chiziqlrd yotdi. gr = sm, =5 sm v kesmning biror m to g ri chiziqdgi proyeksiysi 8 sm bo ls, kesmning shu m to g ri chiziqdgi proyeksiysini toping. Yechilishi. kesmning m to g ri chiziqdgi proyeksiysi bo lsin. Und, isbotlngn teorem v msl shrtidn foydlnib, proporsiy tuzmiz: 8 =. 5 u tenglikdn =0 bo lishini topmiz. Jvob: 0 sm. Svol, msl v topshiriqlr. Kesmning berilgn to g ri chiziqdgi proyeksiysi nim?. ir to g ri chiziqd yoki prllel to g ri m chiziqlrd yotgn kesmlrning yni boshq bir to g ri chiziqq proyeksiylri berilgn kesmlrg proporsionl eknligini isbotlng.. v b to g ri chiziqlr orsidgi burchk 5 g teng. to g ri chiziqd uzunligi 0 sm bo lgn kesm olingn. kesmning b to g ri chiziqdgi proyeksiysini toping.. kesmning uchlri l to g ri chiziqdn 9 sm v sm uzoqlikd yotdi. gr kesm l to g ri chiziqni kesib o tms v = sm bo ls, kesmning l to g ri chiziqdgi proyeksiysini toping v -rsmlrdgi m lumotlr sosid binolrning blndliklrini toping. 6. To g ri chiziq v ung prllel bo lmgn kesm chizing. Kesmning to g ri chiziqdgi proyeksiysini ysng. 7. Koordintlr tekisligid (;) v (;-) nuqtlr belgilngn. kesmning koordint o qlridgi proyeksiylrining uzunliklrini toping. 8. v b to g ri chiziqlr orsidgi burchk α eknligi m lum. to g ri chiziqd kesm olingn. kesmning b to g ri chiziqdgi proyeksiysini toping. 9*. v kesmlrning l to g ri chiziqdgi proyeksiylri o zro teng. v kesmlrning uzunliklri hqid nim deyish mumkin? Misollr keltiring m 0 m 5 m 75 m m -? -?

60 5 PRPRSINL KESMLRNING XSSLRI Fles teoremsining umumlshmsi bo lgn muhim ossni isbotlymiz. Teorem. urchkning hr ikkl tomonini kesib o tgn prllel to g ri chiziqlr uning tomonlridn proporsionl kesmlr jrtdi., (-rsm) = = Isbot. v nuqtlrdn g prllel v to g ri chiziqlrni o tkzmiz. U hold, birinchidn, = = bo ldi, chunki ulr o zro prllel bo lgn, v to g ri chiziqlrni to g ri chiziq kesgnd hosil bo lgn mos burchklrdir. Ikkinchidn, = 5= 6, chunki ulr tomonlri prllel bo lgn burchklrdir. emk, uchburchklr o shshligining lomtig ko r, bo ldi. U hold, = = () 5 tengliklrni hosil qilmiz. 6 undn tshqri, v to rtburchklr prllelogrmm, chunki shrtg ko r; ysshg ko r. Shuning uchun, bu prllelogrmmlrning qrm-qrshi tomonlri o zro teng bo ldi: = v =. () () v() tengliklrdn = = bo lishi kelib chiqdi. Teorem isbotlndi. mliy mshq. Kesmni berilgn nisbtd bo lish. erilgn kesmni to rt bo lkk shundy bo lingki, bo lklrning o zro nisbti m:n:k:l kbi bo lsin. uning uchun quyidgilrni qdm-bqdm bjrmiz: -qdm. Itiyoriy o tkir burchk chizib, uning bir tomonig uzunliklri = m, =n, = l v =k g teng bo lgn kesmlrni -rsmd ko rstilgndek qilib, ketm-ket qo yib chiqmiz. -qdm. urchkning ikkinchi tomonig berilgn kesmg teng kesmni qo ymiz. -qdm. v nuqtlrni tutshtirmiz. -qdm.,, nuqtlr orqli g prllel, v kesmlrni o tkzmiz. Yuqoridgi teoremgko r, berilgn = kesm,, v nuqtlr biln m:n:l :k nisbtd bo lingn bo ldi. Topshiriq: u tsdiqni mustqil rvishd soslng. m n l k mliy topshiriq. To rtinchi proporsionl kesmni yssh., b v c kesmlr berilgn. v b kesmlr c v d kesmlrg proporsionl, y ni :b=c:d eknligi m lum. d kesmni ysng (-rsm). l k -qdm. Itiyoriy o tkir burchk chizib, uning bir tomonig = v =b kesmlrni -rsmd ko rstilgndek qo ymiz. b -qdm. Ikkinchi tomonig es = c kesmni qo ymiz. c -qdm. v nuqtlrni tutshtirmiz. d=? -qdm. nuqtdn g prllel to g ri chiziq o tkzmiz. b Topshiriq: izlnyotgn d kesm bo lishini c soslng. Svol, msl v topshiriqlr. Uzunligi sm bo lgn kesm berilgn. Uni ) 5:; b) ::7; d) :5::7 nisbtdgi bo lkchlrg bo ling.. Rsmd hr bir bo lk birlik kesmdn ibort bo ls, v, EF v MN, v F, N v E, EN v M kesmlrning nisbtlrini toping. E F M N. m, n kesmlr l v k kesmlrg proporsionl. gr ) m= sm, n= sm v l=8 sm; b) m= sm, n= sm v l=7 sm bo ls, k to rtinchi kesmni quring v uzunligini toping.. To rtburchkning perimetri 5 sm v tomonlri ::5:6 kbi nisbtd bo ls, uning hr bir tomonini niqlng. 5. To rtburchkning burchklri o zro ::5:6 kbi nisbtd bo ls, uning kichik burchgi nimg tengligini toping. 6. Uzunligi, 5 v 6 bo lgn kesmlr berilgn. Uzunligi,8 g teng kesm ysng. 7*. Perimetri 60 sm bo lgn to rtburchkning bir tomoni 5 sm, qolgn tomonlri es :: nisbtd eknligi m lum. Uning ktt tomonini toping. m n 8 9

61 5 T G RI URHKLI UHURHKGI PRPRSINL KESMLR Xoss. To g ri burchkli uchburchkning to g ri burchgi uchidn tushirilgn blndligi uni o zig o shsh ikkit uchburchkk jrtdi., =90, blndlik (-rsm), Isbot. v uchburchklr to g ri burchkli bo lib, burchk es ulr uchun umumiy. emk,. Shu singri, v to g ri burchkli bo lib, ulr uchun umumiy. emk,. -rsmd tsvirlngn v kesmlr mos rvishd v ktetlrning gipotenuzdgi proeksiylri deb yuritildi. T rif. gr, b v c kesmlr uchun :b = b :c bo ls, b kesm v c kesmlr orsidgi o rt proporsionl kesm deb tldi. rt proporsionllik shrtini b =c yoki b = c ko rinishd yozish mumkin. Yuqorid isbotlngn ossg soslndign bo lsk, o rt proporsionl kesmlr hqidgi quyidgi teoremlr osonlikch isbotlndi: -teorem. To g ri burchkli uchburchkning to g ri burchgi uchidn tushirilgn blndlik ktetlrning gipotenuzdgi proyeksiylri orsid o rt proporsionl bo ldi. Hqiqtn hm, isbotlngn ossg ko r,. undn, = =. =.. -teorem. To g ri burchkli uchburchkning kteti gipotenuz biln shu ktetning gipotenuzdgi proyeksiysi orsid o rt proporsionldir (-rsm). Hqiqtn hm, isbotlngn ossg ko r,. undn, = =. =.. Xuddi shung o shsh =. eknligini isbotlsh mumkin. Msl. Ktetlri 5 sm v 0 sm bo lgn to g ri burchkli uchburchk kichik ktetining gipotenuzdgi proyeksiysini toping., = 90, blndlik, =5 sm, = 0 sm (-rsm) =? Yechilishi. ) Pifgor teoremsidn foydlnib, uchburchk gipotenuzsini topmiz: = + = =65, y ni =5 sm. ) Ikkinchi teoremdn foydlnib, ni topmiz: =. = 5 = =9 (sm). Jvob: 9 sm. 5 Ikkinchi teoremdn ntij siftid Pifgor teoremsining Pifgorning o zi yozib qoldirgn isboti kelib chiqdi (-rsm). - teoremg ko r, =. + =. +. =.( +)=. =. =. Shundy qilib, + =. Svol, msl v topshiriqlr. Isbotlng (-rsm): ) ; b) b =b c.c, = c.c; d) h c = c.b c.. To g ri burchkli uchburchkning gipotenuzsig tushirilgn blndligi gipotenuzni 9 sm v 6 sm g teng kesmlrg bo ldi. Uchburchk tomonlrini toping.. To g ri burchkli uchburchkning gipotenuzsi 5 sm g, bir kteti es 9 sm g teng. Ikkinchi ktetning gipotenuzdgi proyeksiysini toping.. -rsmdgi m lumotlr sosid uchburchkning tomonlrini toping. 5*. Ktetlrining nisbti :5 kbi bo lgn to g ri burchkli uchburchk ktetlrining gipotenuzdgi proyeksiylri nisbtini toping. 6*. Ktetlrining nisbti : kbi bo lgn to g ri burchkli uchburchk berilgn. Ktetlrning gipotenuzsidgi proyeksiylridn biri ikkinchisidn 6 sm g uzun. Uchburchk yuzini toping. 7. Ktetlrining gipotenuzsidgi proyeksiylri sm v 8 sm bo lgn to g ri burchkli uchburchk yuzini toping. 8*. uchburchkd =90, blndlik, E bissektris v E :E = :. ) :; b) S E :S E ; d) : nisbtlrni toping. ) b) 7 d) b b c,8 6 c,8 c,0 0

62 55 ERILGN IKKIT KESMG RT PRPRSINL KESMNI YSSH b 5 b b b b b To g ri burchkli uchburchkning to g ri burchgidn tushirilgn blndligi gipotenuzni v b kesmlrg bo ls, blndlik b g teng bo lishini ko rgn edik (-rsm). emk, berilgn ikki kesmg o rt proporsionl kesmni yssh uchun: ) gipotenuzsining uzunligi +b g teng (-rsm); ) to g ri burchgidn tushirilgn blndligi shu gipotenuzni v b bo lklrg bo ldign to g ri burchkli uchburchk yssh kifoy. uning uchun to g ri burchkli uchburchkk tshqi chizilgn yln mrkzi gipotenuzning o rtsid joylshgnidn foydlnmiz (-rsm). Yssh: ) To g ri chiziq chizmiz v und = v =b bo ldign qilib, v nuqtlrni belgilymiz (-rsm). ) kesmning o rtsi nuqtni topmiz. Mrkzi nuqtd bo lgn dimetrli yrim yln ysymiz (-rsm). ) nuqtdn to g ri chiziqq perpendikulr to g ri chiziq o tkzmiz (-rsm). u t o g r i c h i z i q yrim ylnni nuqtd kesib o tgn bo lsin. Und to g ri burchkli uchburchk, = b biz ysshimiz zrur bo lgn kesm bo ldi. Yssh bjrildi. rt proporsionl kesmni ysshd to g ri b burchkli uchburchkning kteti gipotenuz biln shu ktetning gipotenuzdgi proyeksiysi orsid o rt proporsionl eknligidn foydlnish hm mumkin (5-rsm). Svol, msl v topshiriqlr. Uzunliklri v b bo lgn kesmlr berilgn. Uzunligi b bo lgn kesmni ysng.. Uzunligi v b g teng kesmlr berilgn. Pifgor teoremsidn foydlnib, uzunligi ) + b ; b) b bo lgn kesmlrni ysng.. Uzunligi g teng kesm berilgn. Uzunligi ) ; b) ; d) 5; e) 6; f) 8; g) 0 bo lgn kesmlrni ysng.. 6-rsmdgi m lumotlr sosid uchburchkning yuzini toping. 5. ylndgi nuqtdn dimetrg perpendikulr tushirilgn. gr = sm, = sm bo ls, doir yuzini toping. 6. ldingi msldgi uchburchk yuzini toping. 7. To g ri burchkli uchburchk to g ri burchgining bissektrissi gipotenuzni 5: kbi nisbtd bo ldi. To g ri burchk uchidn tushirilgn blndlikning gipotenuzdn jrtgn kesmlri nisbtini toping. 8. Rdiusi 8 sm g teng doirg bir burchgi 0 bo lgn to g ri burchkli uchburchk ichki chizilgn. oirning uchburchkdn tshqridgi qismi t segmentdn ibort. n shu segmentlr yuzlrini toping. 9*. 7-rsmd =, = b, demk, = = ( yln mrkzi). Rsmdn foydlnib, b tengsizlikni isbotlng. Qiziqrli msl ylnning dimetri to rtt teng bo lkk bo lindi v 8-rsmd ko rstilgndek yrim ylnlr ysldi. gr =d bo ls, rsmd bo yb ko rstilgn hr bir shkl yuzini hisoblng. 6 ) b) 6,5 d) 8 7 b 8

63 56 YLNGI PRPRSINL KESMLR -teorem. ylnning v vtrlri nuqtd kesishs,. = =. tenglik o rinli bo ldi. Isbot. v vtrlr (-rsm) ko rstilgn trtibd joylshgn bo lsin. Uchlrini v vtrlr biln tutshtirmiz. Shund v burchklr bitt yoyg tirldi, demk, =. Yn rvshnki, =. u ikki tenglikdn, lomtg ko r, v uchburchklrning o shshligi kelib chiqdi. shsh uchburchklr mos tomonlri es proporsionl: yoki. =.. Teorem isbotlndi. -teorem. yln tshqi sohsidgi P nuqtdn ylng P urinm ( urinish nuqtsi) v ylnni v nuqtlrd kesib o tuvchi to g ri chiziq o tkzilgn bo ls, P =P.P bo ldi. Isbot. P v P uchburchklrni qrymiz (-rsm). Und, E P = = P hmd P bu uchburchklr uchun umumiy burchk. emk, P v P uchburchklr ikki burchgi bo yich o shsh. undn, yoki P = P.P. Teorem isbotlndi. Msl.,, v nuqtlr ylnni,, v yoylrg jrtdi. gr v nurlr nuqtd kesishs, u hold. =. tenglik o rinli bo lishini isbotlng. Yechilishi. Msl shrtig mos chizm chizmiz (-rsm) v nuqtdn E urinm o tkzmiz. Und, -teoremg ko r,. =E.=E. =.. Svol, msl v topshiriqlr. -rsmd biln belgilngn nom lum kesmni toping.. nuqtdn ylng urinm ( urinish nuqtsi) v ylnni v nuqtlrid kesdign kesuvchi o tkzilgn. gr ) = sm, = sm bo ls, kesmni; b) =5 sm, = 0 sm bo ls, kesmni; d) = sm, =,7 sm bo ls, kesmni toping.. ylng to rtburchk ichki chizilgn. v nurlr nuqtd kesishdi. gr ) =0 dm, =6 dm, =5 dm bo ls, kesmni; b) = 0 dm, = 8 dm, = dm bo ls, kesmni toping.. ylnning dimetri v bu dimetrg perpendikulr vtri E nuqtd kesishdi. gr E= sm, E = 8 sm bo ls, vtrni toping. 5. v kesmlr nuqtd kesishdi. gr = bo ls,,, v n u q t - lrning bir ylnd yotishini isbotlng. 6. Rdiusi dm bo lgn yln mrkzidn 5 dm uzoqlikd P nuqt olingn. P nuqtdn uzunligi 5 dm bo lgn vtr o tkzilgn. P v P kesmlrni toping. 7. -rsmd. =. tenglikni v uchburchklrning o shsh eknligidn foydlnib isbotlng. 8*. 5-rsmlrdgi m lumotlr sosid.= =. tenglikni isbotlng. 9*. Ikki yln nuqtd urindi. to g ri chiziq birinchi ylng nuqtd, ikkinchi ylng es nuqtd urindi. =90 eknligini isbotlng. ) b) d) 5 ) b) 9 5,5 6 0,5 0, 0, 5

64 57 MSLLR YEHISH ldingi drsd yln kesuvchilri v vtrlrining osslrini isbotlgn edik. Endi shu osslrning yrim ususiy hollri biln tnishmiz. -msl. R rdiusli ylnning ichki sohsidgi R-p P nuqt yln mrkzidn p msofd joylshgn P bo lsin. Und P nuqtdn o tuvchi itiyoriy p vtr uchun P.P =R p () R tenglik o rinli bo lishini isbotlng. Yechilishi. P nuqt orqli ylnning dimetrini o tkzmiz. Und, P =R p, P =R+p (-rsm). Kesuvchi vtrlr hqidgi teoremg ko r, P.P =P.P = (R p)(r + p) =R p. () tenglik isbotlndi. -msl. Rdiusi 6 sm bo lgn ylnning mrkzidn sm uzoqlikd P nuqt olindi. P nuqt orqli vtr o tkzildi. gr P = sm bo ls, P kesmni toping. Yechilishi. Msl shrtig ko r, R =6 sm, d = sm, P = sm. U hold () tenglikk ko r,.p =6 = 6 6 = 0. undn, P =0 sm. Jvob: P =0 sm. P p R -msl. R rdiusli ylnning tshqi sohsidgi P nuqt yln mrkzidn p msofd joylshgn bo lsin. Und P nuqt orqli o tuvchi v ylnni v nuqtlrd kesuvchi itiyoriy to g ri chiziq uchun P.P =p R () tenglik o rinli bo lishini isbotlng. Yechilishi. ylnning mrkzi orqli o tuvchi P to g ri chiziq yln biln v nuqtlrd kesishsin (-rsm). Und, shrtg ko r, P =p R, P = p +R. yln tshqi sohsidgi nuqtdn o tkzilgn kesuvchilr hqidgi teoremg ko r, P.P =P.P = (p R)(p + R) = p R. Shundy qilib () tenglik isbotlndi. -msl. Rdiusi 7 sm bo lgn ylnning mrkzidn sm uzoqlikdgi P nuqtdn o tuvchi to g ri chiziq ylnni v nuqtlrd kesdi. gr P =0 sm bo ls, vtrni toping. Yechilishi. Shrtg ko r, R =7 sm, p = sm. U hold () formulg ko r, P.P = p R = 7 =69 9 = undn, P = = = (sm).emk, P 0 =P P = 0 = (sm). Jvob: sm. Svol, msl v topshiriqlr. Rdiusi 5 sm bo lgn yln mrkzidn sm uzoqlikd P nuqt olingn. vtr P nuqt orqli o tdi. gr P= sm bo ls, vtr uzunligini toping.. Rdiusi 5 m bo lgn yln mrkzidn 7 m uzoqlikd P nuqt olingn. P nuqt orqli o tuvchi to g ri chiziq ylnni v nuqtd kesdi. gr P =m bo ls, vtr uzunligini toping.. -rsmdgi m lumotlr sosid biln belgilngn kesmni toping ( yln mrkzi).. -rsmdn foydlnib, mslni yeching. Und, ) P = 5 dm, =7 dm, = dm, P =? b) P = 5 dm, = dm, P = dm, =? 5. ylnning =7 sm v =5 sm vtrlri P nuqtd kesishdi. gr P :P =: bo ls, P nuqt vtrni qndy nisbtd bo ldi? 6. ylnning nuqtsidn dimetrg perpendikulr tushirilgn. gr = sm, =8 sm bo ls, kesmni toping. 7*. ylng ichki chizilgn to rtburchkning digonllri K nuqtd kesishdi. gr =, =, = v K :K =: bo ls, kesmni toping. 8*. ylng ichki chizilgn to rtburchkd : =: v : = : bo ls, : nisbtni toping. ) b) d) P P P P 6 7

65 IV G IR Q SHIMH MSLLR V 58 M LUMTLR I. Testlr. To g ri burchkli uchburchkning gipotenuzsig tushirilgn blndligi hqid noto g ri tsdiqni ko rsting:. Ktetlridn kichik;. Uchburchkni ikkit o shsh uchburchklrg jrtdi;. Ktetlrining gipotenuzdgi proyeksiylri orsid o rt proporsionl; E. Gipotenuzning yrmig teng.. v vtrlr nuqtd kesishdi. Noto g ri tsdiqni toping:. = ;. v uchburchklr o shsh;.. =.; E. =.. To g ri tsdiqni toping:. Teng kesmlrning proyeksiylri hm teng bo ldi;. Ktt kesmning proyeksiysi ktt bo ldi;. ir to g ri chiziqdgi teng kesmlrning proyeksiylri teng bo ldi; E. Proyeksiy uzunligi proyeksiylnuvchi kesm uzunligig teng bo ldi.. To g ri burchkli uchburchkning gipotenuzsig tushirilgn blndlik uni ikkit uchburchkk jrtdi. u uchburchklr:. Teng;. Tengdosh;. shsh; E. Teng yonli. 5. Uzunligi v b bo lgn kesmlrning o rt proporsionli nimg teng?. +b;. b;. ; E. : b. 6. to rtburchk mrkzli ylng ichki chizilgn. Noto g ri tsdiqni ko rsting:. ;. + = + ;.. =.; E.. =.. II. Msllr.. To g ri burchkli uchburchk ktetlrining nisbti : g teng. u uchburchkning gipotenuzsi 50 sm. Uchburchkning to g ri burchgi uchidn tushirilgn blndligi gipotenuzdn qndy uzunlikdgi kesmlr jrtdi?. ylnning v vtrlri E nuqtd kesishdi. gr E = 5 sm, E = sm v E =,5 sm bo ls, E ni toping.. Rdiusi 6 m bo lgn ylnning mrkzidn 0 m uzoqlikd K nuqt olindi v K nuqtdn ylng urinm o tkzildi. Urinmning urinish nuqtsi P biln K nuqt orsidgi msofni toping.. uchburchkd =90 v blndlik,8 dm. gr =,6 dm bo ls, tomonni toping. 5. ylnning v vtrlri nuqtd kesishdi. gr = 6, = v = bo ls, kesmni toping. 6. ylnd,,, nuqtlr belgilngn, v nurlr nuqtd kesishdi. gr =5, =, =6 bo ls, vtrni toping. 7. ylng nuqtd urinuvchi to g ri chiziq ustid nuqt olindi. gr = v nuqtdn ylngch bo lgn eng qisq msof 8 bo ls, yln rdiusini toping. 8. Yrim ylndgi nuqtdn dimetrg tushirilgn perpendikulr kesmd v 9 g teng kesmlr jrtdi. kesmni toping. 9. To g ri burchkli uchburchkning blndligi gipotenuzni dm v dm g teng kesmlrg bo ldi. Uchburchk yuzini toping. 0. Rdiusi 5 sm bo lgn mrkzli ylnning vtrid nuqt olingn. gr = sm, =,5 sm bo ls, kesmni toping.. Rdiusi 5 m bo lgn mrkzli ylnni v nuqtlrd kesuvchi to g ri chiziqd P nuqt olindi. gr P =5 m, =,8 m bo ls, P msofni toping.. To rtt prllel to g ri chiziq berilgn. Ulr burchk tomonlrini v, v, v hmd v nuqtlrd kesdi. und,,, nuqtlr burchkning bitt tomonid yotdi. gr =8, = v =9 bo ls, kesmni toping.. yln burchkk ichki chizilgn. gr burchk uchidn ylngch bo lgn msof rdiusg teng bo ls, burchk kttligini toping.. ylng dimetrning uchidn urinm v kesuvchi o tkzilgn. yln biln nuqtd kesishdi. gr = bo ls, burchkni toping. 5. To g ri burchkli uchburchkning ktetlri nisbti : kbi. Uchburchkning gipotenuzsig tushirilgn blndlik uni ikkit uchburchkk bo ldi. Ulr yuzlrining nisbtini toping. III. zingizni sinb ko ring (nmunviy nzort ishi). yln tshqrisidgi nuqtdn ylng urinm o tkzilgn. u nuqtdn ylngch bo lgn eng qisq msof sm g, urinish nuqtsigch bo lgn msof es 6 sm g teng. ylnning rdiusini toping.. to g ri burchkli, = 9 dm, =6 dm bo ls, shu uchburchkk ichki chizilgn yln rdiusini hisoblng.. Nuqtdn to g ri chiziqq ikkit og m o tkzilgn. gr og mlr : nisbtd bo lib, ulrning proyeksiylri m v 7 m bo ls, og mlrning uzunliklrini toping. 8 9

66 .* (Qo shimch msl) PQ v undn uzun ET kesmlr berilgn. Shundy to rtburchk ysngki, = =PQ; =ET bo lib, digonllri kesishdign nuqt uchun. =. tenglik o rinli bo lsin. Qiziqrli msl Uchburchk - rsmd ko rstilgnidek qilib to rtt bo lkk bo lingn v -b rsmd ko rstilgnidek qilib qyt yig ilgn. yting-chi, ortiqch kvdrt qyerdn pydo bo lib qoldi?? ) b) V Yunon ochi Ermizdn vvlgi 500-yillrd pydo bo lgn bu shklni hyotning rmzi siftid non ustig chizgnlr. u shklni qlin qog ozg chizib olib, uni rsmd ko rstilgn chiziqlr bo ylb qirqing. Hosil bo lgn bo lklrdn kvdrt yssh mumkinligig ishonch hosil qiling. PLNIMETRIY KURSI YIH TKRRLSH Ushbu bobni o rgnish ntijsid siz quyidgi bilim v mliy ko nikmlrg eg bo lsiz: geometriyning plnimetriy qismi bo yich o tilgn mvzulrni esg olish; plnimetriy kursi bo yich o zlshtirilgn bilim, ko nikm v mlklrni musthkmlsh; ykunlovchi nzort ishig tyyorgrlik ko rish. G E M E T R I Y 0

67 59 KRINTLR USULI Tekislikdgi to g ri burchkli koordintlr sistemsi biln 7-sinf lgebr kursid tnishgnsiz ( -rsmlr). Quyid shu mvzug oid geometrik msllrni qrymiz. (;b) 0 y (;0) 0 (0;b) (, b) nuqtning koordintlri: uning bssisssi; b uning ordintsi. y y N y 0 y ordintlr o qi bssisslr o qi 0 koordintlr boshi ( ;y ) M (;y) ( ;y ) -msl. Uchlri koordintlr tekisligining birinchi chorgid bo lgn kesm berilgn bo lsin: ( ;y ) v ( ;y ), >0, y >0, >0, y >0 (-rsm). kesmning o rtsi bo lgn (;y) nuqtning koordintlrini toping. Yechilishi. u holtd N kesm soslrining uzunliklri v bo lgn trpetsiyning o rt chizig i, M kesm es soslrining uzunliklri y v y bo lgn trpetsiyning o rt chizig i bo ldi. Trpetsiy o rt chizig i osssig ko r, ; () bo ldi. u formullrning to g riligini kesmning boshq holtlri uchun hm shung o shsh mushohd biln ko rstish mumkin. -msl. Uchlri ( ; ), (; 5), (; ), ( ;) nuqtlrd bo lgn to rtburchkning prllelogrmm eknligini isbotlng. Yechilishi. () formuldn foydlnib, to rtburchkning v digonllri o rtsining koordintlrini topmiz: : :,,. ; emk, to rtburchkning hr ikki digonli o rtsi bitt (0; ) nuqt bo lr ekn. oshqch qilib ytgnd, to rtburchk digonllri (0; ) nuqtd kesishdi v shu nuqtd teng ikkig bo lindi. u to rtburchkning prllelogrmm bo lishi lomtlridn biridir. Svol, msl v topshiriqlr. Ko pburchklrning yuzlrini hisoblng (-rsm).. ylnning 8 sm g teng vtri ylndn 90 g teng yoy jrtdi. yln mrkzidn vtrgch bo lgn msofni toping.. Tomonlri ) 5, 5 v 6; b) 7, 65, 80 bo lgn uchburchk yuzini toping.. Tomonlri ),, ; b) 5, 9, 8 bo lgn uchburchkk ichki chizilgn yln rdiusini toping. 5. Uchlri quyidgich bo lgn kesmlr o rtsi koordintlrini toping: ) (; ), (5;6); b) (; ), (;); d) ( ;5), (;); e) ( 0,7;), ( 0,;,). 6*. gr (;0), (;), (;) bo ls, prllelogrmmning uchi koordintlrini toping. 7*. Prllelogrmm burchklri bissektrislri kesishgn nuqtlr to g ri to rtburchk uchlri bo lishini isbotlng. 8. Ktetlri 0 sm v 0 sm bo lgn to g ri burchkli uchburchkk ichki v tshqi chizilgn ylnlrning rdiusini toping. 9. ylng ichki chizilgn to rtburchkning ucht burchgi :: kbi nisbt hosil qilishi m lum. Uning burchklrini toping. 0. Rdiusi 6 sm bo lgn ylnning 60 g teng yoyini tortib turgn vtrini toping.. Rdiuslri 6 sm bo lgn ylnlr mrkzlri orsidgi msof 6 sm g teng. ylnlrning umumiy vtri uzunligini toping. f) ) b) y y ( ;0) e) y d) y y (;6) (;) (;) (6;0) (5;) (0;) (;0) (0; ) (;0) (5;) (6;6) (0;0)

68 60 KRINTLR USULI V VEKTRLR -msl. Koordintlr tekisligid berilgn ( ;y ) v ( ;y ) nuqtlr orsidgi msof = ( ) +(y y ) formul biln hisoblnishini ko rsting. y Yechilishi. ytylik, v nuqtlr -rsmdgidek joylshgn bo lsin (, y y y y y ). v nuqtlrdn koordint o qlrig prllel to g ri chiziqlr o tkzmiz v ulrning kesishish nuqtsini biln belgilymiz. Und, = hmd = y y y. to g ri burchkli uchburchkk Pifgor teoremsini qo llsk, = + = ( ) +(y y ) bo ldi. Undn, 0 = ( ) +(y y ) formulni hosil qilmiz. u formulning = yoki y =y bo lgnd hm to g riligig ishonch hosil qiling. -msl. gr ( ; ), (; ), (; ), ( ; ) bo ls, to g ri to rtburchk eknligini isbotlng. Yechilishi. ) digonl o rtsining, y koordintlrini topmiz: ;. digonl o rtsining, y koordintlrini topmiz: ;. emk, to rtburchk digonllri bitt ( ; ) nuqtd kesishib, shu nuqtd teng ikkig bo linr ekn. u prllelogrmm eknligini ko rstdi. ) prllelogrmm digonllrining uzunligini topmiz: = ( ( )) +( ( )) = +( ) = 0; = ( ( )) +( ( )) = + = 0. emk, prllelogrmmning digonllri o zro teng ekn. u (to g ri to rtburchk lomtig ko r) to g ri to rtburchk eknligini bildirdi. Svol, msl v topshiriqlr. gr ) (;7), ( ;7); b) ( 5; ), ( 5; 7); d) ( ;0), (0;); e) (0;), ( ;0) bo ls, kesmning uzunligini hisoblng. 5 N. gr M(;0), N(; ), P(5; 9) bo ls, MNP uchburchk perimetrini toping. 0. Kolliner v y vektorlr chizing v +y vektorni ysng.. gr,, v nuqtlr bir to g ri chiziqd 5 N yotms v = 0,7 bo ls, to rtburchk turini niqlng. 5. Nokolliner v b v e k t o r l r b e r i l g n. g r ) P b=y+b bo ls, v y sonlrni toping. 6. gr, v kesmlr uchburchklrning medinlri, itiyoriy nuqt bo ls, + + = + + tenglikni isbotlng. 7. uchburchk medinlri nuqtd kesishdi., v vektorlrni = v b= vektorlr orqli ifodlng. 8. Jismg hr biri 5N bo lgn ikkit kuch t sir ko rstypti (-rsm). gr bu kuchlrning yo nlishlri orsidgi burchk 0 bo ls, ulrning teng t sir etuvchisi kttligini toping. P b) 9. Teng tomonli uchburchkk tshqi chizilgn ylnning rdiusi 6 sm. Uchburchk perimetri v yuzini toping. 0. ylng nuqtdn o tkzilgn urinmd nuqt olindi. nuqtdn ylnning eng yqin nuqtsigch bo lgn msof sm g, eng uzoq nuqtsigch bo lgn msof es 8 sm g teng. kesmni toping. *. Rdiuslri turlich bo lgn ikkit yln nuqtd P to g ri chiziqq urindi. u ylnlrg mos rvishd P dn frqli P v P urinmlr o tkzilgn. gr v bu urinmlrning ylng urinish nuqtlri bo ls, P =P tenglikni isbotlng (-rsm). 5

69 6 YLN V IR -msl. Koordintlr tekisligid mrkzi (; b) nuqtd v rdiusi R bo'lgn ylndgi itiyoriy M(; y) nuqtning v y koordintlri ( ) +(y b) =R () tenglikni qnotlntirishini isbotlng. y Yechilishi. (; b) berilgn yln mrkzi, M(; y) shu ylnning itiyoriy nuqtsi bo ls, u hold M=R bo ldi. Koordintlr tekisligid M(,y) berilgn ikki nuqt orsidgi msofni topish formulsig (-betdgi -mslg qrng) ko r, b M= ( ) +(y b). (,b) Shundy qilib, 0 ( ) +(y b) =R. irgi tenglikning hr ikkl qismini kvdrtg oshirib, () tenglikni hosil qilmiz. Esltm. () tenglm mrkzi (; b) nuqtd bo lgn R rdiusli yln tenglmsi deyildi. y -msl. Koordintlr tekisligid ushbu ( ) + (y + 6) =5 0 - tenglm biln niqlngn ylnni ordintlr o qidn jrtgn kesmning uzunligini toping (,-6) Yechilishi. erilgn yln biln ordintефlr o qi kesishgn nuqtlrning bssisslri nolg teng bo ldi. =0 bo lgnd, berilgn tenglmdn foydlnib, bu nuqtlrning ordintsini topmiz: (0 ) + (y + 6) =5, (y + 6) =9, y = 9 yoki y =. emk, yln v ordintlr o qi (0; 9) v (0; ) nuqtlrd kesishdi. u nuqtlr orsidgi msof 6 birlikk teng. Jvob: 6. -msl. Mrkzlri nuqtd joylshgn ikkit doir hlq tshkil qildi. Ktt doirning sm g teng vtri kichik doirg nuqtd urindi (- rsm). gr hlqning kengligi 8 sm bo ls, u hold bu hlqning yuzini toping. Yechilishi. Ktt doirning rdiusini R biln, kichiginikini es r biln belgilymiz. Msl shrtig ko r, =R=r+8 (sm) v = r. undn tshqri, nuqt vtrning o rtsi, y'ni =6 sm, uchburchk es to g ri burchkli bo ldi. Pifgor teoremsig ko r, + = уыbo lgni uchun, r + 6 =(r + 8) tenglmni hosil qilmiz. u tenglmni yechib, r = sm eknligini topmiz. Und R=r+8=0 (sm) bo ldi. Ktt doir yuzidn kichiginikini yirib, berilgn hlq yuzi S ni topmiz: S=πR πr =0 π π=00π π= =56π (sm ). Jvob: 56π sm. Svol, msl v topshiriqlr. Quyidgi tenglmlr biln berilgn ylnlr mrkzlrining koordintlrini v rdiusini yting. Shu ylnlrni ysng. ) ( ) + (y + ) = ; b) ( ) + (y ) = 6; d) + y = 5; e) + (y ) = 9.. ylng ichki chizilgn to rtburchkning, v uchlridgi burchklri nisbti :: kbi. To rtburchk ichki burchklrini toping.. ylnning :8 qismig mos mrkziy burchkni toping.. Mrkzi nuqt bo lgn ylnd nuqt olingn. Mrkzi nuqtd bo lgn boshq yln nuqtdn o tdi. u ikki yln nuqtd kesishdi. burchkni toping. 5. ylnning v vtrlri nuqtd kesishdi. gr = sm, =6 sm v = sm bo ls, v kesmlrni toping. 6. ylng ichki chizilgn to g ri to rtburchkning digonli bitt tomonidn ikki mrt ktt. u to rtburchk uchlrining ylndn jrtgn yoylrining grdus o lchovlrini toping. 7. ylng tshqi chizilgn trpetsiyning o rt chizig i 7 sm. Trpetsiy perimetrini toping. 8*. Rdiusi 5 sm bo lgn doir mrkzidn 7 sm uzoqlikdgi K nuqtdn 7 sm uzunlikdgi vtr o tkzilgn. K v K kesmlrni toping. 9. Muntzm skkizburchkning bir uchidn chiqqn eng ktt v eng kichik digonllri orsidgi burchkni toping. 0. Uchlri koordintlr tekisligidgi ( ; ), (; ), (; 8) nuqtlrd bo lgn uchburchk berilgn. ) =90 eknligini ko rsting; b) uchburchkk tshqi chizilgn doirning mrkzini, rdiusini v yuzini toping. R 6 6 r 6 7

70 6 TKRRLSH 6 TKRRLSH Msl. uchburchkd medin, = 6, = 7 v = 9. Uchburchk yuzini toping. Yechilishi. nurd nuqtdn ==5 6 bo ldign qilib, nuqtni tnlymiz (-rsm). Und 7 6 =, = bo lgni uchun prllelogrmm bo ldi. 7 v uchburchkning yuzlri teng. Geron 9 formulsidn foydlnib, uchburchk yuzini hisoblymiz: P = =5; S = 5.(5 9)(5 5)(5 7)=70. Jvob: 70. Svol, msl v topshiriqlr. v EFK uchburchklr o shsh: v EF, v FK ulrning mos tomonlri. gr = sm, =5 sm, =7 sm v EF : =, bo ls, EFK uchburchkning tomonlrini toping.. v uchburchklr o shsh v ulrning mos tomonlri nisbti 6:5 g teng. uchburchk yuzi uchburchk yuzidn 77 dm g ortiq. Uchburchklr yuzlrini toping.. uchburchk medinlri kesishgn nuqt bo lsin. gr uchburchk yuzi sm bo ls, uchburchk yuzini toping (-rsm).. ylnning nuqtsidn dimetrg perpendikulr tushirilgn. gr = 9, = bo ls, kesmni toping (-rsm) Tomoni 6 m, bu tomonig yopishgn burchklri 0 v 5 bo lgn uchburchkning yuzini toping. 6. soslri 8 dm v 6 dm, yon tomonlri es 5 dm v 7 dm bo lgn trpetsiyning blndligini toping. 7. Rdiusi sm bo lgn ylng yuzsi 0 sm bo lgn teng yonli trpetsiy tshqi chizilgn. Trpetsiy tomonlrining uzunliklrini toping. 8. To g ri burchkli uchburchkk ichki chizilgn ylnning gipotenuzg urinish nuqtsi gipotenuzni sm v sm bo lgn kesmlrg jrtdi. Uchburchkning ktetlrini toping. Msl. Ktetlri v bo lgn to g ri burchkli uchburchkk ichki v tshqi chizilgn ylnlrning mrkzlri orsidgi msofni toping (-rsm). Yechilishi. ) uchburchkd =90, = v = bo lsin. Und, Pifgor teoremsig ko r, = + =5. ) To g ri burchkli uchburchkk tshqi chizilgn ylnning E mrkzi gipotenuzning o rtsid bo ldi: 5 E = =. ) Uchburchkk ichki chizilgn yln rdiusi ni topmiz ( ichki chizilgn ylnning gipotenuzg urinish nuqtsi): = = =. ) v E kesmlrni topmiz: = + = 5+ =; E =E E = 5 =. 5) To g ri burchkli E uchburchkdn E kesmni topmiz: 5 E= +E = + =. 5 Jvob:. Svol, msl v topshiriqlr. Teng yonli uchburchkd = = sm v = 0 bo ls, uning E blndligini toping.. Trpetsiyning soslri 5 dm v 8 dm, yon tomonlri es,6 dm v,9 dm. Trpetsiy yon tomonlrining dvomi nuqtd kesishdi. nuqtdn trpetsiy uchlrigch bo lgn msoflrni toping.. burchkning bir tomonig =5 sm v =6 sm kesmlr, ikkinchi tomonig es = 8 sm v F =0 sm kesmlr qo yilgn. v F uchburchklr o shshmi? Jvobingizni soslng.. To g ri to rtburchkning yuzi 9 dm, digonllri hosil qilgn burchklrdn biri es 0 g teng. To g ri to rtburchk tomonlrini toping. 5. gr teng yonli uchburchkning sosi sm v yon tomoni sm bo ls, u hold uchburchkk tshqi chizilgn yln rdiusini toping. 6. Rombning blndligi sm bo lib, digonllridn biri 5 sm. Romb yuzini toping. E 8 9

71 7. gr prllelogrmmd (; ), ( ;) v ( ;) bo ls, uning uchi koordintlrini toping. 8. Ikkit kvriumg yuqori chetidn 0 sm pst qilib suv quyildi (-rsm). Qysi kvriumd suv ko p? 0 sm 50 sm 0 sm 9. Qutig nech pket mev shrbti sig di (-rsm)? 50 sm 0. litrli mev shrbti pketi to g ri to rtburchkli prllelepiped shklid (-rsm). itt qdoq uchun qnch mteril kerk bo ldi? 8 sm 0 sm 0 sm 60 sm sm 7 sm *. 5-rsmd tsvirlngn yog och bo lklrining hjmini hisoblng. 5 0 sm 0 sm ) b) 5 sm d) sm 7 sm,5 dm 5 dm dm 6 dm 5 sm dm 7 dm 6 sm 0 sm 6 sm sm sm 6 TKRRLSH Msl. Rombning o tms burchgi uchidn o tkzilgn blndlik romb tomonlridn birini o tkir burchgi uchidn boshlb hisoblgnd 5 sm v 8 sm bo lgn kesmlrg jrtdi. Romb yuzini hisoblng. Yechilishi. romb, >90, E blndlik, E = 5 sm, E = 8 sm bo lsin (-rsm). ) Romb tomonini topmiz: = E + E = = (sm). ) To g ri burchkli E uchburchkk Pifgor 5 8 teoremsini qo llb, E blndlikni topmiz: E E = E = 5 = (sm). ) Romb yuzini topmiz: S =.E =. =56 (sm ). Jvob: 56 sm. Svol, msl v topshiriqlr. gr teng tomonli uchburchkd (0;0) v (:) eknligi m lum bo ls,. sklyr ko - pytmni toping (-rsm).. soslri v bo lgn trpetsiyning digonllri nuqtd kesishdi. gr = 8 sm, (;) =0 sm v = 50 sm bo ls, kesmni toping.. gr =,7 sm, = sm, =, sm, = dm, =60 dmv = 8 dm bo ls, v uchburchklr o shshmi? (0;0). Perimetri 6 sm bo lgn prllelogrmmning digonllri kesishishidn hosil bo lgn ikkit uchburchkdn birining perimetri ikkinchisinikidn 8 sm ortiq bo ls, prllelogrmmning tomonlrini toping g teng burchkk bir-birig tshqridn urinuvchi ikkit yln ichki chizilgn. Kichik ylnning rdiusi sm bo ls, ktt yln rdiusini toping. 6. Ktt sosi bo lgn trpetsiyning digonli tomonig perpendikulr v =. gr trpetsiyning perimetri 0 sm v = 60 bo ls, tomon uzunligini toping. 7. gr yln dimetrining uchlri ylnning biror urinmsidn 8 sm v sm uzoqlikd eknligi m lum bo ls, yln uzunligini toping. 8. soslrining uzunliklri v yuzi mos rvishd 8 sm, sm v sm bo lgn teng yonli trpetsiyning yon tomonini toping. 0