🔙 Quay lại trang tải sách pdf ebook Tuyển Tập 500 Bài Toán Hình Không Gian Chọn Lọc Ebooks Nhóm Zalo 516.23076 B A N GIA O VIE N NAN G KHIE U TRUCiN G TH I T527T NJGUYE N eC/C eON G (Ch u bien) PHAN LOAI VA PHl/OfNG PHAP GIAITHEO CHUYEN DE • BOI Dl/dNG HQC SINH GIOI • CHUAN B! THI TU TAI, DAI HOC VA CAO DAN G BOG H a NOI NHA XUAT BAN OAI HOC QUOC GIA HA NOI BAN GIAO VIEN NANG KHIEU TRl/CfNG THI NGUYEN DLfC D 6 N G {Chu hien) TUYEN TAP 500 BAITOAN • HDIH imm GIAN CHON LOG • • • PHAN LOAI VA PHUdNG PHAP GIAI THEO 2 3 CHUYEN • Bo i difdng hoc sinh gioi • Chuan b i th i Ti i tai, Da i hoc va Cao dang (Tdi ban idn thvt ba, c6 svCa chUa bo sung) THir ViEN TiiVH BiKH liik^m N H A XUA T BA N DA I HO C QUO C GI A H A NO I A H A GI C QUO C HO I DA N BA T XUA A NH i Np a H - g Trcfn a Bi Ha - i Chuo g Han 61 397148) (04 : ban e Ch - p ta n Bie : thoai n Die 3) (04 : tap n Bie g Ton ; 39714899 ) (04 : chinli n Hanl 9 3971489 ) (04 : Fax • * * * ban: xuat nhiem trdch Chiu TM PHA . TS tap: bien Tong - doc Gidm A HO Y THU tap: Bien N VA I THA bdi: Saa N A G HON h sac a Nh ban: Che N VA I THA bia: bay Trinh LCilNOIDAU Chung toi xi n gidi thi$u den doc gia bp sdch: Tuyen tap cdc b^ i toan dknh cho hoc sinh Idp 12, chuan b i th i vao cac trucrng Dai hoc & Cao ding . Bo sach gom 7 quyen : . TUYE N TA P 54 6 BA I TOA N TIC H PHA N . TUYE N TA P 54 0 BA I TOA N KHA O SA T HA M S O . TUYE N TA P 50 0 BA I TOA N HIN H GIA I TIC H . TUYE N TA P 50 0 BA I TOA N HIN H KHON G GIA N . TUYE N TA P 69 6 BA I TOA N DA I SO • TUYE N TA P 59 9 BA I TOA N LUON G GIA C . TUYE N TA P 67 0 BA I TOA N RCJI RA C V A Cl/C TR I NhSm phuc vu cho viec ren luyen va on th i vao Da i hoc bkng phucrng phdp ti m hieu cac de th i dai hoc da ra, de tiT nang cao va chuan b i kien thiJc can thiet. De phuc vu cho cac do'i tUcfng t\i hoc : Cac bai giai luon chi tie t va ddy du, phan nho tCrng loai toan va dua vao do cac phucfng phap hop li . Mac du chiing toi da co g^ng het siic trong qud trin h bien soan, song vSn khong tranh khoi nhiJng thieu sot. Chiing toi mong don nhan moi gop y, phe binh tii quy dong nghiep ciing doc gia de Ian xuat ban sau sach ducfc hoan thien hcfn. Cuoi Cling, chiing toi xin cam cm NIlA XUA T BA N DA I HO C QUO C GI A H A NO I da giiip da chiing toi moi mat d l bo sach dUdc r a dcfi. NGUYEN DtfC DONG 3 TRONT TA T VIE f CHL A V U HIE I K C CA E K G BAN TAT VIE / Tl C CA A v C HO N TOA U HIE I K C CA ) (i g : ] (EFG) ; [(ABC) • g dUcfn g tUcfn ) (i : > <= • l (i o the o ke ) (i : > = • ) (EFG a v ) (ABC g dilcfn g tUdn g khon : > C •V o the o ke g khon : d> • xOi do p Phe : A D • t nha g don : = • xiiTi do p Phe : o D • t nha g don g khon : i • qua p Phe : ) cp ; (0 Q • h dien.tic : ) dt(ABC = ) S(ABC = c Sv\n • . (p C AAB v p Phe : ) k; ( 0 VT • v p Phe : ) k; ( 0 VT • p cho h hin h tic e th : ) (S.ABC V = c A H Vs • C S.ABa nghl h din : N D • y l h din : L D • n pha n toa h tic n Die : p St • a qu e h : Q H • h quan g xun h tic n Die : q Sx • min g chiJn : R CM • h tic e Th : V • A a cii u chie h hin a l ' A : A i ''7(a = ' A • i c bud : i B • ) (a g phin t ma g xuon p ho g trudn : i TH • A a cu u chie h hin a l ' A : A ) ''Vfd = ' A • p ho g trudn : i TH • A a cu u chie h hin a l ' A : A ) ''Vfd = ' A • ) (d g thftn g dtfcfn g xuoni tra e v : T V • i pha e v : P V • n de M m die T ti h eac g khoan : l) (D ; d[M • ) (D g thin g ducfn tg din t ba : T BD • M m die tii h eac g khoan : I (ABC) ; d[M •i ba u ca u ye : t ycb • i ba u ca u ye : t ycb • ) (ABC g phan t ma n de ) (ABC g phan t ma n de i pha u die : m dpc • t ma a mf 2 i bcf o ta n die i nh c go : P) ; (a • ) (P a v ) (a g phant thie a gi : t g • B A h can n die i nh : ) (AB = ) D; AB ; (S •n lua t ke : L K • Chuyen de 1 : TON G QUA N V E CA C KHAI NIE M TRON G HINH HO C KHON G GIA N • Hin h hoc khong gian la mot mon hoc ve cac v$ t the trong khon g gian (hinh hin h hoc trong khong gian) ma cac diem hin h than h nen vat the do thudng thiTcrng khong ciing nftm trong mot mat phang. • Nhif vay ngoai die m v a diidng thdng khon g drfoTc din h nghi a nhiT trong hin h hoc phAng; mon hin h hoc khong gian con xay di/ng them mot doi tuong can nghien ciifu nCfa la kha i ni#m mgt phan g cun g khon g difoTc din h nghia. Kh i noi tori kha i niem nay ta lien tuang den mot mat ban bang phang, mot mat ho nildc yen lang, mot tb giay dat dinh sat tren mot mat da di/gc la m phang.... No duoc ky hieu bdi cac chCf i n La Tin h nha : (P), (Q), (R), ... hoac cac chCf thudng H y Lap nhU (a), ((5), (y), .... • Ma t phang khong ducfc dinh nghia qua mot kha i niem khac; nhifng thifc te cho thfi'y mSt ph&ng CO nhutng tin h chat cu the sau, goi la cac tien de : O TIE N DE 1: C o i t nha t bon die m trong khon g gian khon g th^n g han g (nghia la luon luon c6 it nhat 1 diem d ngoai mot mat ph^ng tiiy y). O TIE N DE 2: Ne u mpt dtfdng th&ng v a mpt ma t ph^n g c6 ha i die m chun g thi dUcTng th&ng ay se nS m trpn v^ n trong ma t phan g ne u tren . O TIE N DE 3: Ne u ha i ma t ph&ng c6 die m chun g thi chting c6 v6 so' die m chung : n e n ha i ma t phSn g do cSt nha u theo mpt dUdng th^n g di qu a v6 so' die m chun g ay. Di/cfng thang ay goi la giao tuyen cua hai mat ph^ng. O TIE N DE 4: C o mpt v a ch i mpt m$ t phan g duy nha t di qua b a die m pha n bi#t khon g th^ng hang . O TIE N DE 5: Trenmp t m§ t phan g tuy y trong khon g gian ca c din h ly hin h hoc ph&ng scf cap (da hoc tCr Idp 6 den Idp 10 va cac dinh l y nang cao) de u diing. O TIE N DE 6: Moi doan th&ng trong khong gian deu c6 dp dai xa c dinh : tien de neu len sU bao toan ve dp dai, goc va cac tin h chat lien thuoc da biet trong hin h hoc phing . • TiT do chung ta c6 mot so each xac dinh ma t ph4ng nhi / sau : O H E QU A 1: C o mpt v a ch i mpt mfit phSn g duy nha t di qua mpt dUdng thSng v a mpt die m nS m ngoai dt^dng thang do. O H E QUA 2: Co mpt v a chi mpt m^t phdng duy nhat di qua hai di^cAig th^ng cSt nhau. O H E QU A 3: C o mpt v a ch i mpt m^t phan g duy nha t di qu a ha i di^c/ng thdng song song. • Dong thdi ta phai hieu them rkng mot mat phang se rong khong bien gidi va dUcmg th^ng c6 do dai v6 tan mac du ta se bieu dien no mpt each hin h thiifc hflu han va khiem ton nhU sau: • De thuc hien dirge phep ve chinh xdc m6t hin h hin h hoc trong khong gian ngoai cac dudng thay ve lien net, ta can phai nam chac di/pc kha i niem di/dng khuat ve bkng net dijft doan: Mpt dtfdng bi khua t toan bp ha y ch i khua t mpt doan cue bp na o do kh i v a ch i k h i ton ta i i t nha t mpt ma t phSn g du'ng phi a trvC6c ho^ c phi a tre n che no mpt each toan bp hoa c cue bp ti^cAig uTng. 5 n cho n co a t n gia g khon g tron g ph^n t ma t mo h nh^n h din c xa n Muo • geg (khon g ph&n c gia a d c hoa c gia " tii , giac m ta h hin t Mp c ca i gp a T . gian g khon g tron g phSn t m^ t mp h din c xa n lud txictng . .. , (C) , (ABCD) , (ABC) u hip y k c ca i vd thvCc h hin g phSn pt m^ u nhie y ha t mp 6 c u ne t khua i h c thu^ h hin g phdn t Ma • t ma a mc thd' h hin g ph&n t m^ g tron m n^ g thdn dvictng t Mp • t ma a cu n bie a l g khon o d g th^n g dUcTn i kh a v p b n toa y h a n toa y ha p b e cu t khua vlng g tii'oTn g cun o d g th&n g di^dn i t h i th t khua i b c thuT h hin g ph&n t m$ t mp g tron nhm m die t Mp • dUct mp c dUp i th t khua m die t mp 6 c t nha t i a m m die i ha i No • g phAn t m^ c ca a cu n bie a l g khon o d g diicta i ha u ne : p b n toa A HQ H MIN H AN H HIN C CA • ) \(d ) (d o d , bo e cu t khua e ch ) (a i b ) (d • . (a) i dud m nk n doa t dijf t ne t khua e ch ) (SAC g ph^n t ma i b ) (d • S u sa m nk n doa t duf e v n doa t mp OC (S, (SAB) t ma c ca u sa d g cun ) (d n nhie ) (SBC g phan t ma i ha i b C A h Can • nh m xe C A n doa a c o d , bo n toa t khua (S, (SAB) g ph^n t ma i ha i thd g don u sa . -AA ^ 1—^VFJL c./— d ^fH N CA U HIE Y K C CA • H A] n doa a c o d o b n toa e ch i b H A] • trU Ha i mat phln g (a), (P) thuf tif chiJa hai difdng thin g (di), (da) ma (dj) n (da) = I => SI la giao tuyen can tim . > Ha i mat phln g (a), (P) thuf tif chtifa hai difdng thin g (di), (da) ma (di) // (da). S _ Difng xSy song song vdi (dj) hay (da) => xSy la giao tuyen can tim . M BA O C N TOA I BA C CA . m 1 i Ba S m die a v g son g son g khon i do h can c ca 6 c D ABC i lo c gia f tii o Ch : a cii n tuye c gia (SAD ; (SDC) a v ) (SAB hi . (SBD) a v ) (SAC /a i Gia : 6 c a t , (SBD) a v ) (SAC g phan t ma i ha t Xe /a ) (1 . nhat f thu g chun m die a l S • C A o che g ducm i ha , ABCD i lo c gia T tu g Tron • ^ . (2) i nh f thij g chun m die : O = D B n : a r y su ) (2 a v ) (1 Ti/ ) (ycbt O S = ) (SBD o ) (SAC : 6 c g cun ) (SDC a v ) (SAB g phan t ma i ha t Xe hi . chung m die t mo a l S • D ABC c gia l ti a cu D C a v B A n be h can i Ha • . song g son g khon t thie a gi o the . hai t thu g chun m die a l : E = D C ^ B A ^ ) (ycbt E S = ) (SDC n ) (SAB : o d o D B/ AD/ o d ; BC ^ D A = F i vd ; (ycbt) F S = ) (SBC n ) (SAD : tif g Tucfn 2 i Ba Aa v D BC c gia m ta i ha n tar g trpn a l a G , Gj i Go . ABCD n die l ti o Ch : n tuye c gia c ca m CD.Ti , AD , BD a cii m die g trun a l K , J ) (ACD n ) (G1G2B hi ) (ADB o ) (G1G2C aJ J I = ) (ABD n ) (G1G2C /a K A c hoS K Gg = ) (ACD n ) (GiGaB hi 2 G,G = ) (CIJ ^ ) (ABK d 3 i Ba . O m ta h han h bin h hin a l D ABC y da 6 c D S.ABC p cho h hin o Ch . (SBC) a v ) (SAD g phSn t ma i ha a cu n tuye o gia m Ti aJ . (SCD) a v ) (SAB g phin t mS i ha a cu n tuye o gia m Ti hi ^ => (SAB) r^ (SCD) = uSv (ycbt) c/ Goi O = AC n BD, tiTcrng t a bai 1 => (SAC) n (SBD) = SO (ycbt). B a i 4 Cho hin h chop S.ABCD c6 day la hin h thang ABCD vdi AB la day Idtn. Gpi M la mot diem bat ky tren SD va EF la difang trung binh cua hin h thang. a/ Ti m giao tuyen ciia hai mSt phin g (SAB) va (SCD). b/ Ti m giao tuyen cua hai ma t phSng (SAD) va (SBC), c/ Ti m giao tuyen cua hai mSt phang (MEF) va (MAB). Doc gia t u giai tUcfng t u nh u cac bai tren. B a i 5 Cho hin h chop S.ABCD c6 ABCD la hin h binh hanh. Goi G,, G2 la trong ta m cac tam giac SAD; SBC. Ti m giao tuyen cua cac cSp mSt phang : a/ (SGiG^) va (ABCD) b/ (CDGiGz) va (SAB) 0/ (ADG2) va (SBC). UvCdng d§Ln Goi I , J , E, F thur ta Ik trung diem cac doan thin g AD, BC, SA, SB theo thur tvt d6. Thifc hien cac lap luan nhtf cac bai toan tren ; a/ (SG1G2) n (ABCD) = I J (ycbt) b/ (CDGiGa) n (SAB) = EF (ycbt) c/ (ADG2) ^ (SBC) = xG2y (ycbt) Trong do xGay // A D hoSc BC. Loa i 2 : TlM GIA O DI£ M CU A DUdN G THAN G 1fA MA T PHAN G L PHirONG PHA P Ca sd cua phaang phap ti m giao diem O cua dudng thang (a) va mat phSng (a) la xet 2 hai kha nSng xay ra : n Trirdng hop (a) chiJa dudng thSng (b) va (b) lai c&t diicrng thdng (a) ta i O. T i m O = (a) n (b) => O la diem can tim . n Trtfdng hap (a) khong chiifa dUcmg thin g nao cat (a). Tim ( P )^(a)va(a )n(P ) = (d) > Ti m O = (a) o (d) => O la diem can tim . n. CAC BA I TOA M G O BA N B a i 6 Cho tuf dien ABCD. Goi M , N Ian lugt la trung diem cua AC va BC. Lay diem K e BD sao cho KB > KD. Ti m giao diem ciia hai dudng thin g CD va A D vdi (MNK). 9 i Gia D C g son g son g khon N K > = D K > B K n de y e D • . I = D C o N K ) (BCD g tron o d o D ) (ycbt I = ) (MNK D C ) (MNK c N K a M ) (ADC g tron , (MNK) c M I t xe a t g Taon • E = M I n D A : O C a T ) (ycbt E = ) (MNK n D A >= 7 i Ba f thu K a v N m die i ha a v C A n tre M m die y La . ABCD n die J ti o Ch . (MNK) \di D A a v D C a cu m die o gia g Dun . ACD a v D BC c gia n di g HtfdTn . ben h hin m xe , giai u t a gi c Do ) (ycbt P = ) (MNK D C / a ) (ycbt Q = ) (MNK n D A / b 8 i Ba m die a b C B a v B S , SA n tre y La . S.ABCD c gia f tu p cho h hin o Ch (Mi vd C A a v C S a cu m die o gia m Ti . CD y ha B A c&t e th g khon P M o ch i Gia m die o gia m ti t ycb o d g thudn g ThUdn K = C S o P N ) (MNP c P N a m) (ycbt K = ) (MNP n CS 1 H = C A o K M ) mp(SAC g Tron | ) (MNP c K Ma m ) (ycbt H = ) (MNP > r C A >= 9 i Ba ta a chil g phin t ma i ngoa d S m die t mo a v C AB c gia m ta t mo o Ch . O m die t mo y la a t ) (ABC g phin t ma g tron a v N, M m die i ha y la at Loal 3 : Cfll/NG MWfl B A DI£ M TRON G KHON G GIA N THAN G HAN G I. pmroNG PHA P Co so cua phiiong phap can phai chufng minh ba diem trong yeu cau b^ i toan la diem chung cua 2 mSt phSng nao do (chfing ban A, B, C nSm tren giao tuyen (d) cua hai mat phSng do nen A, B, C thang hang). O day khong loai triJ kha n&ng chiJng min h difoc difdng thang AB qua C => A, B, C thin g hang. n. CAC BA I TOA N C O BAM B ai 10 Xet ba diem A, B, C khong thuoc mat phin g (u). Goi D, E, F Ian luot la giao diem ciia AB, EC, CA va (g). ChCifng min h D, E, F thang hang. Gia i De y thay D, E, F viTa a trong mp(ABC) vifa d trong mp(a). Do A, B, C g (a), nen (a) va (ABC) phan biet nhau. => (a) n (ABC) = A (A chuTa D, E, F) D, E, F thin g hang tren A (dpcm). B a i 11 Hai tam giac ABC, AB C khong dong phin g c6 A B n AB ' = I , AC n A C = J , BC n B C = K ChiJfng min h I , J , K thin g hang. Gia i De y I , J , K Ian luot d tren hai mat phin g phan (P) ^ (ABC) va (Q) = (A'B'C). Nen no 1^ diim chung cua ha i mat phin g do I , J , K e (A) = (ABC) n (A'BC ) => 1, J , K thin g hang (dpcm). B a i 12 Cho A, B la hai diem d hai phia khac nhau doi vdi mat phin g a va A B ci t a ta i O. Ditog hai dUdng thin g x'Ax, y'By song song nhau theo thuf tiT ci t a ta i M va N. ChuTng min h M , N , O thin g hang. Kvldng dS n TifOng t\l: fM, O, NG(S ) ^ • [8 = (Ax; By) n (a) => M , N, O thin g hang tren (8) 11 GG KHON G TRON G THAN G DtfCiN T MQ l MWf CmiUG : 4 l toa H DIN O C M DI£ T MO A Q U , PHAP G PHirON . I ) (d g thin g diXcrn h min g chuTn p pha g phucfn a cu sd a C : h din o c m die t mo a qu h min g chuTn a v B ; A y y tu m die i ha ) (d n tre m ti n ca a T g tron n sS 6 c h din o c I m die t mo i vd g han g thin o d m die 2 . gian g khon . (dpcm) h din O C I a qu ) (d >= , PHAP G PHtfON LI : n ba f cc c bifd a b n hie c thu n ca p pha g phiTcfn a cu d s oC o c g ph5n t mS i ngoa d h din o c a g thin g dUctn m Ti : i B n . dong) i (liO d a chil ) (a a m) (a h din d ^ a = I m die o gia m Ti : 2 B • a qu i d d a m h din o c m die a l I >= N BA O C N TOA I BA C CA . m 3 1 i Ba c kha a phi i ha e v d n gia g khon g tron h din o c m die i ha a l B , A o Ch I = a n A M o ch o sa n gia g khon g tron g don u lu M m die t Xe . a h din . dinh o c m die t mo a qu i d n luo J I g thin g difdn h min i Gia vk h din o c B , Ai (v h din o c O > = ) (a n B A = O i Go ) (a) a cu a phi 2 a J I = ) (a n ) MB ; (MA = ) (P mp : O C a T . hang g thin J, I , O > = J I e O : y tha y e D ) (dpcm h din o c O a qu i d J I g thin g dacfn a l a Nghi 4 1 i Ba (ABC e S m die t Xe . CD) > B A a v D C/ / B (A D ABC g than h hin o Ch thig difdn h min g ChuTn . N = D S n a , M = B S ^ '- a i vd C A h quan g don . dinh o c m die t mo Bai 15 Cho ha i difdn g thftn g don g qu y Ox , O y v a ha i die m A , B khoh g nS m tron g ma t phin g (xOy). Mo t ma t phan g lif u don g (a ) qua A B luo n luo n ca t Ox , O y ta i M , N . Chiifng t o M N qua m ot die m co dinh . Giai De y tha'y kh i (a ) qua y quan h A B co din h nhtfn g vAn co : ( a ) n [(Ox ; Oy ) ^ (P)] = A (qua M , N ) Nhung -j 1 AB CO din h ABo(p ) = I e A [P CO din h Nghi a l a dudn g than g M N = A lo u dQng nhOn g v an qua I co dinh . (dpcm ) toal S : CHON G MWfl BA OUdN G THA« G TRON G KHONG GIAN DON G QU Y L PHUON G PHAP , Co so cua phifan g pha p l a t a ca n chiifng min h dUdn g thiif nha t qua giao die m ciia 2 difdn g co n la i ban g 2 budrc co ba n : • B i : Ti m (d,) o (d^) = O • B2 : Chufng min h (d;j) qu a O. => (di), (d2), (d.i) don g qu y ta i O (dpcm ) Q. PHUON G PHAP , Co sd cua phaan g pha p l a t a ca n chijfng min h chun g d o i mo t ca t nha u v a do i mo t d tron g 3 ma t phan g pha n bie t qua 2 bifdc ca ba n : d ] , c: a; d j 0 da = I i i d 3 \ • B i : Xa c din h < da, d;j cz P; da ^ d3 = I2 A \ dg, d , e Y; dg n d j = I3 di \ a, p, y phan bi§t • B2 : Ke t lua n (d,); (da); (d;,) don g qu y ta i 0 = I i = I2 = I3 m. cA c BA I TOA N C O BA M Bai 16 C h o ti l die n ABCD . Go i E , F , G l a b a die m tre n ba can h AB , AC , B D sao cho E F n B C = I , E G o A D = J (vd i I ^ C wk J ^ B). Chijfng min h CD , I G v a J F don g quy . Giai X e t b a difcfng thAn g CD ; I G v a JF , t a tha'y : CD, I G e (BDC) va CD I G / 0 • IG, J F c (EFG) va I G n J F * 0 J F, CD e (ACD) va J F r> CD * 0 Va ba ma t phan g (BCD) , (EFG) , (ACD ) luo n pha n bie t ( v i I ?t c v a J ?t D ) => CD , IG , J F don g quy ta i O (dpcm). 13 c kha h Cac O don F J a v G I ; CD > = D C n G I = O a qu F J g rSn h min g churn a gi c D o 7 1 i B a d C At cd C A, E a ' A'B t ca B A o ch o sa C AB , ABC c gia m ta i ha o C h . hang g thSn G, F, E m die a b h min g Chufn / a . quy g don C C, BB' , AA' g than g difcfn h min g ChiJn / b Gidi t bie n pha g ph^n t ma i ha a cu g chun m die a b a l G, F, E y tha y e D / a . (AB'C) =) (P a v ) (ABC ^ ) ( a . (P) n) (a =) (A e G, F, E : o d o D . (dpcm) g han g th^n G, F, E y V a : u sa u nh t xe n Nha / b 0 #' BB o' AA ; (EAA') cr ' BB , AA' : 0 * C C r^ ' BB ; (GBB') c C C, BB' ^ 0 #' AA n C C ; (FCC) c ' AA , Ice . (dpcm) Oi ta y qu g don C C , BB' , AA' ^ G SON G SON E H N QUA : 2 de Chuyen G SON G SON G THAN G DLfCJN I HA l MWf G CHtJN : 1 i toa P PHA G PHirON .I nghi h din o ch n ba O C hxidc i ha n hie c thii n ca p pha g phaon a cu 0 S o C ) (a : c b ja, 0 = b 'a^ t ma t mo g tron g cun a g than g difdn i ha a tr m Kie : i B • a r y xa o d u die n nhie n hie g ran m nga u hife y ha g phan ) (1 . do o na g phan h hin 1 g tron g chun u n e Gia i S G, SG2 2 SE SF 3 Theo tinh chat trong tarn, ta c6 : -i , SG3 SG4 2 t [ SH SK 3 Dinh ly Thales va tin h chat diTcfng trung binh G,G2// = -EF;EF7 / = iA C ' ^ 3 2 G.G,, // = -H K ; HK// = - AC ^ ' 3 2 • G1G2 // = G;jG4 G1G2G3G4 la hin h bin h hanh (dpcm). Bai 19 Cho diem S d ngoai mat phSng hin h binh hanh ABCD. Xet mSt phdng a qua A D c^t SB va SC Ian lucft d M va N. Chiirng min h AMN D la hin h thang. Gia i S D6 y thay hai mSt phSng (a) va (P) c6 2 diem M vfl N 1^ di^m chung. '(a) 3 AD => M N = (a) n (SBC) ma^(SBC)3B C N iAD//BC va theo each dimg M N // A D (hoftc BC) => ADN M la hin h thang day lorn AD . (dpcm) Bai 20 Cho tuT dien ABCD. Goi M , N Ian li^gt la trung diem cua BC va BD . Ggi P la diem tren canh AB sao cho P ?t A va P # B. Xet 1 = PD A N va J = PC o AM . ChiJng min h rSng : I J // CD. Gia i Xet hai mat phang (AMN) va (PCD) c6 hai diem chung la I va J . IJ = (AMN ) r-. (PCD) 'CD c (PCD) Nhimg < MX CT (AMN) • va MN // CD ^ I J // M N hoac CD (dpcm). toai Z : CfltJfJG MW H DiidfiG THAN G SONG SONG TfCl MA T FHAN G L PHtrOWG PHAP, Co so ciia phuong phap mot la sii dung dinh ly phuong giao tuyen song song. De chiing min h d // a ta can thUc hien hai bade CO ban chufng min h : d y r- a = a tuy y • E l : Chufng min h d = y o p ma • B2 : Ket luan tif tren d // a. p n a = b . a//b 15 ^ PHAP G PHOON . n u d a v n ca n kie u die g dun f st a l p pha g phifcn a cii sd Ca ) (a g phan t ma i vcJ g son g son ) (d g thin g di/dn h min g chijfn : c btfdr i ha g ban u o g thin g dudn m ti t thie a gi y l n qua a v t sa n Qua : i B • . (A) / / ) (d h min g chiJn a v ) (a cz ) (A t vie . dii a v n ca n kie u die o the ) (a / / ) (d n lua t Ke : 2 B • M BA O C N TOA I BA c cA . m 1 Bai2 Gm ta g tron i ha i no' n dea g rSn h min g chufn , ABCD n die f tu g Tron . (ACD) v6[ g son g son i th D AAB i Gia : 6 c a t , do T ti t thu o the D B a v C B m die g trun a l 2 A , Ai i Go 2 2 AG ) AG 3 g AA ' , AA : 6 c a t , Thales y l h din o The 2 A, A '0,02// ) binh g trun g dUcrn t cha h (tin D //C 2 ,A A a 'm ) (ACD / / 2 G1G . =j ) (ACD : c D C/ / 2 G1G > = u ca c bS h tin o The 2 2 i B a l N , M i Go . ABCD h han h bin h hin a l y da D S.ABC p cho h hin o Ch . (MNCD) / / B A a v ) (SCD / / N M : h min g Chijfn i Gia c gia m ta g tron h bin g trun g dudn t cha h tin o The D C / / B A a m , AB / / N M >= ) (SCD cz D C / / N M >= . (ycbt) ) (SCD / / N M > = u d a v n ca n kie u die o The c kha h Cac O o d g phan t ma i ha g tron a v ) (SAB n ) (MNCD = N M y e D D B A/ / D C g thin n doa c ca T ti f thij o the a chiJ ) (ycbt D C 3 ) (SCD / / N M > = D C a v B A/ / N M . (dpcm) ) (CDMN / / B A > = ) (CDMN c N M/ / B A : tyl g TifOn 3 2 i B a Bai 24 Hin h cho p S.ABC D c6 da y l a hin h bin h han h ABCD , tar n O. Go i M , N Ia n \\iqt l a trun g die m SA, S B v a xe t he thiJc vect o : 3 SI - 2 SM = 3 SJ - 2 SN = 0*. ChuTng min h rSn g : a/ IJ//(SCD ) b/ SC//(MNO) . Hvfdrng di n a/ i I J // MN , M N // AB; A B // C D CD c (SCD) => IJ//(CSD ) (dpcm ) A M A O M N // CD b/ AS AC SC // M O c (OMN ) S C // (OMN ) (dpcm ) Bai 25 C ho Ax , B y l a ha i nijfa diTdng thSn g cheo nhau . Tre n A x la y die m M , tre n B y la y die m N sao cho A M = BN . Chijfng min h rSn g dUcfng thin g chufa doa n M N luo n luo n son g son g w6i ma t phan g CO dinh . Q ua A dun g Ax ' // By ; qua N difn g NN ' // BA ; v6i N ' e Ax'. Luc do tii giac ANN B l a hin h bin h han h ne n : AN ' = B N => A M = AN ' D e y AAMN ' ca n d A ne n ti a pha n giac ngoa i A t cua STAJ T se son g son g v6i MN ' v a ti a A t n a y co din h ha y A B v a A t xac din h ma t phSn g co din h (P). lMN'//A t N N ' // A B (MNN' ) // (P) Ta CO : < [ V a y : M N // (P) tiifc l a M N luo n luo n song song v6i m&t phan g co din h (dpcm) . toal 3 : HA I MA T PHAN G SON G SON G Dang 1 : CHQNG MINH HAI MAT PHANG SONG SONG L PmrOHG PHAP Co sd cua phuon g pha p chiJng min h ha i ma t phan g fx v a P son g son g nha u t a ca n thiTc hie n ha i bUdc CO ba n tron g kh i siJf dun g die u kie n ca n v a du n h u sau : • B i : Chufng min h "ma t phan g (a) chii a ha i dUcJng than g a, b don g qu y thijf ti f song son g vd i ha i dUoing than g a', b' don g qu y tron g ma t phan g P". • B2 : Ke t lua n (a) // (P) the o die u kie n ca n v a du . THL; VJENTifJHglNHTHUAN 17 M BA O C N TOA I BA ckc . n 6 2i Ba C, By , Ax g ph^n g don g Ichon a v g son g son , chieu g cun a ti a b n Tre (h min g ChOfn . khong c kha i da o d6 c C C =' BB =' AA : o ch o sa C, B' , A ' i Gia ) (ABC c AB II A'B' ' BB 3 = ' AA ) ( I i < : y e D ) (ABC c C A/ / C A' . =j ' CC =' AA ; C A, AB' y qu g don g th^n g dUcrn i ha 6 c a t n N e . (I) n kie u die a tho ) mp(A'B'C' g tron ) (dpcm ) (ABC / / ) (AB'C >= 7 2i Ba cug son g son y C a c x Ae k Ca v Ar Ti . ABCD h han h bin h hin o C h . Cy) ; (D / / ) Ax ; (B h min g Chiifn . (ABCD) g phSn t ma g tron i Gi& a v ) Ax ; (B g phin t ma i ha t xe u t g Tiran g thin g dudn p ca c ca a chuT a t T thu , Cy) ; ( D . quy g don D fAB//C y IAx//C ) (dpcm ) Cy ; (D / / ) Ax ; (B >= 8 2i Ba kg ph^n t ma i ha g tron d F ABE a v D ABC h h^n h bin h hin i ha o C h . (BCE) / / ) (ADF i Gia g dirdn p ca c ca a chuT T ti f thii ) (BCE a v ) (ADF g phlin t ma i H a / . quy g don g thdn E iAF//B C AD//B) (dpcm ) (BCE / / ) (ADF GN6 D G THAN G DldN C CA H MIN G CHUfN : 2 Dan • B2 : Ket luan d], d2, ds, ... c (a) // (P) => di, d2, d^j, ... dong phing trong (a); (a) phai chufa cac giao diem cija d,, da, ds, .... n . CA C BA I TOA N C O BA N Bai 29 Cho tiJ dien ABCD c6 AB = AC = AD. Chufng minh rSng ba diTcfng phan giac ngo^i cdc goc SAC. CAI), I5AB cung nSm trong mot mat phlng. Giai Goi Ati, At2, Ata la ba diTdng phan giac ngoai ciia goc : fiAfc, CXt), I5A6 theo thuT tu do. Do cac tam gidc can tai dinh A nen cac phan giac ngoai song song vdi canh day, nen : At, //BC c (BCD) AtaZ/CDe (BCD) ;At3//BDc(BCD) At,, At2, At3 // (BCD) => At,, At2, Ata dong ph^ng (trong (P) // (BCD) \k (P) qua A) (dpcm). Bai 30 Cho hinh chop day la luc gidc deu. Chufng minh rang giao tuyen cua mat ben doi nhau thi dong phlng. Giai De y thay : (SAB) n (SED) = t, // AB, ED (SBC) o (SEE) = ta // BC, E E ^(SCD) n (SEA) = // CD, FA => t,, ta, tg//(ABCDEF) Vay t,, t2, tg dong ph^ng trong (a) // (ABCDEF) va (a) qua S. (dpcm) Bai 31 Tren bon tia phan biet Ax, By, Cz va Dt song song cung chieu, lay cac diem A', B', C, D' sao cho AA' = BB' = CC = DD'. Chutng minh rin g AB , B'C, CD', DA', AC , B'D" cung songsong vdi mat ph^ng ABCD. Htfdng di n Doc gia tu giSi iMng t\l hai bai toan tren. 19 N GIA G KH6N G TRON S THAIJE " Li H DES trong d2 d,, y tuy thing ditang Hai : ) (thu|ln i ly h Din •k (a) nhau song song phdng in^t cdc tren chdn gian khong : le ty ling tUcmg thang doan cdc ra tao (y) II (P) II A,A. : ) (dao 2 ly h Din -k i kha i ha n de n tar n qua a t , dao y l h din t xe i kh c TrUd • : n ha g chin , so y t y da c ca n de t xe i kh u sa m nie 9 B1B A , A) (* 3 B2B 3 A2A . (*) o s y t y da a cu c go p ca a l ) B] ; (Ai • . (*) o s y t y da a cu n ngo p cSt c ca a l ) B3 ; (A3 a v ) B2 ; (A2 • y t y da a cu g than c ba ) (doan a l n ngo p cft c cd a v c go p ca i no n Doa • : gian khong trong so ty day c6 Neu : y l h Din •. A,A da, (*) song song cung A3B3 A2B2, AiB,, thang bac cdc thi (d2) (d,), thdng dinh. y l h din a cu c kha u bie t pha 6 c a T : u ch i Gh O : u sa f nhi o da s Thale hai tren ra xdy da (*) so ty day c6 kien dieu Vai bac 3 trong mot thi ) fd^ (dj), thdng dudng mot vdi song song se A^B^ A2B2, A,B,, thang lai. con thang bac hai chda phdng mat ) A3B3 ; 2 (A2B = ) //(a i "A,B ) (A ) )-(A3B3;AiBi //(P A2B2 ) A2B2 ; i (A,B S ) A3B3//(Y ) (di i A (goc) 2 A) tren n (ngp 3 A) difdi n (ngo pht (ma M G SON G SON G THAN G DJCiN H MIN G CHUfN : 3 g Dan S THALE Y L H DIN G BAN •. pmroNG PHAPj Ta chutng min h dUdng thin g (d) na m trong ma t phin g (a) // (()) => (d) // (p). m. cAc BA I TOA N C O BA N Bai 32 Cho tut dien ABCD c6 A B = CD. Goi M va N la ha i diem lUu dong tre n A B va CD sao cho AM = CN. Chutng min h M N luon song somg vdi mSt phIn g co dinh^ Giai Neu dat AB = CD = a; A M = C N = x. De y thay tren A B va CD ta co day t y AM ^ CN AB CD i(A; C) la cap go'c |(M; N) va (B; D) la hai cap ngon tUcJng ufng. Ap dung dinh l y Thales dao trong khong gian th i ba bac thang AC, M N va BD ciing song song v M N // (CDF); v i A B // CD c (CDF) => M N // (DEF) = (CDF) (dpcm) Bai 34 Cho hin h vuong ABCD va ABEF d trong ha i ma t phIn g khac nhau. Tren cac difdng cheo AC va BF, ta Ian lugt lay cac diem M , N sao cho A M = BN . Chutng min h rang M N // (CEF). Giai Do ha i hin h vuong ABCD, ABEF bang canh nen bang nhau => AC = BF. „.,,,..- ^ AM BN Gia thie t => = AC BF Ap dung dinh ly Thales cho cac doan bac thang : A B, MN , CF voti de y EF cz (CEF); AB // EF c (CEF) ^ M N // (CEF) (dpcm) 2 1 5 3 i Ba s N M m die i ha y la t lua n Ia a t , nhau o ch6 y B a v x A a ti i ha n Tre pht mS t mp i vd g son g son n luo n luo N M g rSn h min g Chufn . trade) o ch 0 dim g HUoTn : het e Trad 1 = , BN : i bd h din , N m die y la' y B ) tn/dc o ch , 0 > k i (v k = ; AM : i bd h din j M m die y la x A . dinh o c ] N \k j M m die i ha ^ 1 n nhie n Hie : o c a t h hin n tre g dan h eac tii a v t thie a gi o The i BN i AM , M A , A M N B MA N B , B N N M s Thal^ y l h din a cu o da y l h din o the n N e = ) ((5 h din o c g phin t mS i vd g son g son n luo n luo g son B a qu d g thin g dadn a v B A a cha ) Bd ; ( A ) (dpcm . N,M] i vd g son 6 3 i B a don chuye m die t mo a l M. d2 a v j d u nha o che g thin g dadn i ha o C h . MN n doa a cu I m die g trun h tie y qu m Ti . d2 n tre g don n chuye m die dan Hu&ng B , dj e A ( 2 d a v i d a cu g chun e go g vuon n doa a l B A i G o id . AB a eii m die g trun a l O ; d2) i = = : Taeo X I B O 2 ' d g phln t ma g tron nkm 1 0 i th o da s Thale y l h din o The c xa g phan t ma a l c tiif , da a v i d i vd g son g son O a qu ) ( P g son t lacr n Ia O a qu 2' d a v l' d g thin g dadn i ha i ba h din d-^. a v ] d i vd g son g ran o c g khon a v ] d n tre y cha N a v M ; han Giai . (P) n tre y y tii y cha I n ne e hug g son g thin g dacfn g dOn a t I a Qu . ) P ( G I m die t mo y La : Dao • l ' d e ' M m die y La . E i ta l' d t ci y na g thin g dadn , '2 d i vd g son . E a qu O i vdr g xOn i do' a l I y tha o ch h bin g trun g dadn y l h Din . N' d 2' d t cl I M g son g thin g dadn c ca g difn ' N a v ' M f TC . M'N' a cu m die g trun . N d 2 d a v M d ] d t el t lag n Ia g Chun . AB i vd g son Chuyen de 3 : PHlTONG PHAP TIE N D E Ta da thay di/gc kh i giai toan hin h hoc trong khong gian tif hai chuyen de trUdc mot each chi/a tiidng min h iSm viec sCf dung hai tien de 5 va tien de 6 the nao ? Den day, de" khftc phuc viec do. Chung toi diTa vac mot chuyen dfe PHUCfNG PHAP TIEN' DE vdi mot mong muon la doc gia se thiTc sa thay difoc mot each chinh xac han, tUdng minh hon : si^ can thiet cu a tien de 5 v a tien de 6. Hien nhien viec gidi thieu rong ra i nhif the doi hoi doc gia can chuan bi mot it kien thiife ve sir vuong goc va nhOmg kha i niem ve cac hin h khoi. Sau nhOrng suy nghi va tran tr d trong suot qua day hoc va viet sach chiing toi hy vpng duge doc gia dong cam vcii vi$e dat chuyen de 3 a v i tr i nay trong quyfin sach chi mot each lidc le cung la du. L nnxova PHA P Co so ciia phifong phap la sii dung sii c^n thie t cua hai tien de 5 va tien de 6 d4 xay diTng va chufng min h mot so bai toan co ban trong khong gian kh i hin h thanh nen cac vat the (hien nhien 4 tien de d trirdc da duoc ngam hieu la luon luon di/gc sOf dung). n. CA C BA I TOA M C O BA M B a i 37 Cho a, b, c la ba difdng thin g khong ciing nkm trong mpt mat phAng va doi mot cSt nhau. Chufng minh rSng : a, b, c dong guy. Gia i That vay : gia sijf a, b, c khong dong quy, th i cac giao diem ciia chiing lap than h ba diem khong thang hang va ba difcfng thftng cung nam trong mot ma t phang. Tra i vdi gia thiet. Theo phep chufng min h phan chiifng ycbt dUcrc chijfng min h xong. B a i 38 Cho 3 tia Ox, Oy, Oz doi mot vuong goc. a/ Chufng minh rkng ba ti a do khong cung nk m trong mot mat phang. b/ Ijay tren ba tia Ox, Oy, Oz Ian lifgt cac diem A, B, C (khac goc O). Chijfng min h rang : (AB + BC + CAf ^ eiOA' + OB^ + OC^) c/ Ky hieu a, p, y la ba goc tarn giac ABC, a, b, c la do dai OA, OB, OC. Tin h cosa, cosp, cosy va chufng to rang a, [3, y nhon. Gia i a/ That vay : gia sCf ba tia cijng thuge mot mat phang, v i Ox va Oy ciing vuong goc v6i Oz, nen Ox va Oy cung nam tren mot du'dng thang. Dieu do tra i vdi gia thiet. Do do ycbt di/gc chufng min h bang phep chufng minh phan chufng. b/ Ap dung bat dang thufc Bunhiacovky, ta eo : (AB +BC +CA)^ < 3(AB ' + BC^ + CA^) = 3(0A^ + 0B^+ OB^ + OC^ + OC^ +OA^) (AB + BC + CAf <= = i cos( : 6 c a t f ti g Tiran ^ b^Vb^+c + ^ V a ) (dpcm n nho g cun Y , P : o d o D 9 3 i B a = y cos ; 0 > 0 > b2.Va^+c + 2 V c t mo u nha i vdr o ta t mo i do z O , Oy , Ox a ti a b n gia g khon g tron o Ch . phSng g don i pha z O , Oy , Ox a ti a b g rSn i Gia y O ; Ox n tre sAn n cho a t a v g phSn g don g khon z O , Oy , Ox f sii a Gi ) (dvcd 1 = B O = A O : o ch o sa o d f ti A 1 C O o ch o sa C m die n cho a t , Oz a ti a cu ' Oz i do a ti n tre i thd g Don — = C A " OAsin60 = CA 2 i - = C O " OAcos60 = CO 2 : a t o ch C BO A g tron n cosi m ha y l h Din ° 2OB.OCcos60 - C O + ' OB = ^ BC - = i 2.1.-. ~ i + 1 = ' BC « 2 2 = C B « ) (1 = C B = C A : o d o D 2.1.1(- - 1 + 1 = ' 2OA.OBcosl20' - ^ OB + ' OA = ^ AB : u t g Tifon ) (2 a V = B A « doz O ; Oy , Ox > <= B A e C > <= B A = B C + A C : c difg a t ) (2 a v ) (1 f Tt ) dau n ba f st a gi ) (dpcm . phang g don i pha z O , Oy , Ox y Va 0 4 i B a ChiJ. 90" = z (5 y a v " 45 = & x ^ y xO o ch o sa z O , Oy , Ox a ti a b o Ch => A B + A C = B C <=> A , B , C tMn g han g (do do die u gi a siif l a v 6 ly ) V a y Ox , Oy , Oz don g phan g (dpcm ) Bai41 C ho ba ti a Ox , Oy , Oz tho a die u kie n xOy = 60° , yOz = 90 ° v a z6x = 120° . 1/ Chufng min h ran g ba ti a Ox , Oy , Oz khon g ciin g nk m tron g mo t ma t ph^ng . 2/ La y b a die m A s Ox , B e Oy , C e Oz sao cho O A = O B = O C = a. Chiln g min h rSn g AAB C vuong v a ti m qu y tic h ta m ducfng tro n ngoa i tie p ta m gia c kh i a tha y doi . 1 1 1 3 • + - / La y 3 die m A , B , C nh u cau 2, nhiin g la i tho a the m die u kie n \ e . • OA OB OC min h rang ma t phan g (ABC ) kh i lif u don g luo n luo n d i qua mo t die m co dinh . Giai ^0 y / (h.l) (h.2) 1/ Gi a sd Ox ; Oy ; Oz don g ph^n g tron g mp(a ) na o do , t a co ha i kh a n&ng : 1 . Chutag T i a Oy hoa c na m tron g mie n goc xOz (xe m h.l ) hoac mie n ngoa i goc ic8z (xe m h.2 ) th i xOy = 30° * 60 ° (h.2 ) hoa c xOy = 150 ° * 60 ° (h.l ) (v6 l y vd i gi a thie t xOy = 60° ) V a y ba ti a Ox ; Oy ; Oz khon g don g phSn g (dpcm). z 2/ Din h l y ha m cosi n cho t a : • A B = VOA"* ^ OB^ - 2OAOBcos60 0 A B = + - 2a.a.— = a A C = VOA^ +OC''^ - 2OAOCcosl20 ° => A C = ja ^ -a ^ -2.a.a.f-~ l = a Vs V V 2 ; M a B C = a V2 Do do : i AB'^ ^BC ^ ^(aV2)2 = Sa^ A C ^ = (aVs)^ = 3a^ A C ^ = AB ^ + BC ^ AAB C vuon g d B (dpcm ) G o i (d) l a tru e tro n ngoa i tie p zVABC v a I l a ta m dUcfng tro n ngoa i tie p ta m gia c do , t a co : O A = O B = O C O e (d) AAB C vuon g ta i B th i : l A = I B = I C I e (d) v a I l a trun g die m A C => O I c: (d) Q u y tic h cua I l a ti a O t l a tru e ciia (ABC) . 3/ Tron g AOAC , vd i D l a cha n dUcrng pha n gia c tron g OD , t a co : , ^20 ° 1 2 cos! 1 1 _ K ^ ) 1 1 1 (1) OA OC OD OA OC OD Tifan g tif ABOD , vdfi E l a cha n dUc(ng pha n gia c tron g OE , t a co : 2cos |:^ i „ S5D 1 1 V 2 J OD OB OE 1 OB 2 cos OE 1 OD (2) 25 : thi§'t a gi a t o ch ) (2 + ) (1 , do c Lu s co 2) fliOt C O B O AO> <= 1 =D O ) ' VE O2 = E oO 1 =. ' 1 D O cun D O c gia n pha a ti n ne n gia g khon g tron h din o c C O ; OA a ti i H a c E O c gia n pha a ti n ne n gia g khon g tron h din o c D O ; OB a ti i Ha ^ rgOt) s 2co = E O J 2 I. gian g khon g tron h din o c E m die a t o ch t cons = o ch o sa g don u lo C , B , A i kh y Va h din o c E a qu n luo g nhun 2 4 i B a • + -C O B O AO ph^n t ma i th 1 = — ca n Tre . ABCD h han h bin h hin a l y da o c D S.ABC p cho h hin o Ch o ch o sa j C, Bi , A] m die c c^ g ufn g taanC S AS san) o ch k , 0 > k ( k: i SC , SA = O i (vd h din o c m die 1 i ta O S t ca ) mp(AiBiCi i th i do y tha i C. Bi , A i i Gia n co y y tu C ZiAB a cu n tuye g trun a l M A i Go ) (h.2 . AC a v B A n tre i ti f thij y y tu C, B' : c thuf e h y nga o c a t , M' = C B' o M A i K h ) minh g chufn u t a gi c (Do ) (* — 2 = —+ ^ ^ • ' AM C A ' AB : o c a t i th , O' = , A,C ^ O S i go u ne , d6 o D ) (* O S ^ C S AS ' SO i SC i SA) (h.l ) (1 > = n Ne , C S AS ] SC i SA) (2 > <=' SO , O S „ = > <= k = 2 O S ' SO . (ycbt) h din O C' O > <= 3 4 i B a 2 ) sSn o ch , 0 > k ( — k die m Ti . a g ban u de y da h can a v n be h can o c u de p cho h hin o Ch OA Trong d6 : K 4 K 4 2 (21 V2 a>/2 a ASOA vuong can tai O => OH = OA. 2 2 2 2 minS= i .aV^.^ = ^ xa xay ra khi H Ik trung diem SA. (ycbt) Bai44 Trong mat phang (P) cho hinh vuong ABCD, tren dudng thing (d) khong nam trong (P) qua A lay diem S. Tim vi tr i cua S de I = (SA^ + SB^ + SC^ + SD^) dat gia tr i nho nhat. Giai Ap dung dinh ly dUdng trung tuyen trong cac tarn gidc ASAC va ASBD, ta c6 : -,2 SA^ + SC^ -~ 2SO^ + — ^ 2 ISB^+SD ^ =2S02+: ^ I 2 1 1(2) „ Lay ^ S = 480^ + ^ + ^ 2 2 (1) (2) ^ S = 4SO^ + AC'^ (3) Be y trong (3) chi c6 SO la thay doi, do do S nho nhat khi va chi khi SO nho nhat. Trong mp(0; d) co dinh ha OH 1 d tai H. => OH = d[0; (d)] = minSO (do (d) co dinh) => minS = 40H^ + AB^ xay ra khi S = H (ycbt) Bai 45 Cho 3 diem A, B, C khong thuoc mat phang (P). Gia sd cac doan thin g AB va BC deu clt (P). Chufng minh rang doan thing AC khong clt (P). Giai Gia suf AC n (P) = I va goi : M = AB - (P); BC n (P) = N • Do M, N, I thuoc hai mat phing phan bi$t (P) va (ABC) nen thing hang. • NhUng M, N, I la ba diem nl m trong ba canh cua AABC ma thing hang thi dan den di6u v6 ly. Vay doan AC khong the nao clt mp(P) difoc. (dpcm) Bai 46 1/ Cho n diem trong do khong co 4 digm nao dong phlng. Chijfng minh rin g khong co 3 diem nao trong chung thing hang. 2/ Cho n diem trong do bat ky 4 diem nao cung d6ng phlng. ChuTng minh rin g n diem do dong phlng. 2 7 i Gia , B, Af si a Gi . cho a d m die n tU a r y la m die 4 a l D, C, B, A y y tu n Cho / 1 y na g thin g Dudn . C a v B , A a qu i d g thSn g diTcrn a l d i Go / : 6 c a T . (a) g phftn t ma t mo h din c xa D m die i vd g cun ) (1 ) (a e C; B; A ; D / o na m die 4 6 c g "khon t thie a gi l vo i tra ) (1 y tha' y e D \ " / . phang" g cun o ch a d m die n g tron . (dpcm) g han g thin y a m die n g tron o na m die 3 6 c e th g khon y Va o na y k t ba m die 4 a l t cha h tin i vd c kha m die n t xe a t d gi y Ba / 2 . phang ) 4 ^ n ( . A„ ; A3 ; A2 A^; a l y a m die n i Go . dung n nhie g difan n toa i ba i th 4 = ni Kh • . 4 > nf SL a Gi • m die t X6 . (a) g phln t ma h din c xa 3 A; A2 ; Ai m die a B j A a v 3 A; Aj ; Aj m die 4 t thie a gi o The . n) < i < 3 i (v(J j A . n ; 5 ; 4 = i i mo i vd ) (a e j A a l c tiJ g phan g don . (dpcm) ) mp(a c thup y a m die n a c t ta y va r Nhi 7 4 i B a h can i ha a cu m die g trun a l t lug n Ia J; I i Go . ABCD n die J ti o Ch C. Ni ta D B h can t ca ) (IJM g phln t Ma . AC h can n tre y tuy m die t mo . nhau g ban h tic n die n pha i ha h than N IMJ n die t thie i Gia ) (IJM g phin t ma g Tron O = J I o N M >= C J Al D J " BI 1 = a b g tron m na J I ; BD ; AC i th o da s Thale y l h din o The . nhau u de h eac a v g son g son g phan t m a . MN a cii m die g trun a l O >= a v J MI c gia m ta g tron K N a v H M o ca g difdn c ca o d o D . nhau g ban a l J N I ) (NIJ t d = ) (MIJ t d h tic n die n pha i ha h than N IMJ n die t thie a chi J I y Va ) (dpcm . nhau g ban 8 4 i B a 2 ^./c/-i 2 TT 2 AB^+CD ^ ^ AB^+CD ^ ,, 2 L i => L = 4MG^ + IJ'' + > + IJ'' = hang so 2 2 Dang thufc xay ra kh i va chi kh i M = G => E = MA^ + MB^ + MC^ + MD ^ dat gia trj nho nhat kh i va chi kh i M d G, trong tarn cua tuT dien. (ycbt) B ai 49 Cho tarn dien Oxyz vuong va mot dil m A co dinh ben trong ta m dien. Khoang cdch tCr A den ba mat (Oyz); (Ozx) va (Oxy) Ian luat la a; b; c. Mo t ma t phang luu dong (a) qua A cMt Ox d M ; Oy o N va Oz or P. b 1/ ChiJng min h : = 1 (1) OA OB OC 2/ Dinh v i tr i cua mat phin g (a) de hin h chop O.MN P c6 the tich nho nhat. Gia i 1/ De y thay hin h chop O.MN P xem nhu duac hap thanh bdi ba hin h chop A.ONP; A.OMP; A.OMN Ta CO : VQ.MNP = VAO.NP + VA.OMP + VA.OMN Khoang each tif A den ba mat phln g (Oyz); (Ozx) va (Oxy) la : A I = a; A J = b; A K = c Ta CO : - OM.ON.OP = - ON.OP.a + - OP.OM.b - OM.ON.c 6 6 6 6 Chia hai ve cua d^ng thilc nay cho —OM.ON.OP, ta diiac 6 1 = (dpcm) (1) OM ON OP 2/ Ap dung ba't d^ng thiJc Cauchy va dua no vao (1) ta c6: 1 > 3 1 0 M ON OP OM.ON.OP > 27abc <=> Vo.MNP ^ I abc Dang thilc xay ra kh i va chi kh i : a b c 1 OM " ON ~ OP " 3 0 M = 3a ON = 3b 0P = 3c Vay the tich tuT dien O.MN P nho nha t b^ng - abc xay ra kh i mp(a) = mp(MNP ) dugc din h nhu tren. (ycbt) 29 C GO G N6 VU E H N QUA : 4 de Chuyen G PHAN T^ fdJ C GO G WON G THAN G DUCIN : 1 i Loa l Vd C GO G VUON G THAN G Ol/dN H MIN G CHQN : 1 g Dan U D A V N CA N KIE U DIE G BAN P PHA o PHiroN L c go g vuon dg thftn g difcfn h min g chufn p pha g phuan a cu d s Ca g chutn i pM n ca \k u d a v n ca n ki^ u die g bSn a g phln t ma i vd g don b, a g th^n g difcfn i ha i vdr c go g vuon d g th^n g difan : h min / . a g phSn t mS g Iron y qu . (dpcm) a =>d_L 0 = b a^ ; dla;bcu N BA O C N TOA I BA C CA . n 0 5i Ba ASA i ha t Bie . loi c gid J ti a l D ABC y di D S.ABC p cho h hin o Ch . (SAD) 1 B A g rSn h min g Chiifn i Gia : O C a T) (SAD I C A S 1 B A') (SAD c A D L B ^^A . (dpcm) ) (SAD 1 B A 1 5i Ba O m ta h han h bin h hin a l D ABC y da 6 c D S.ABC p cho h hin o Ch . (ABCD) 1 O S h min g Chufn Sn tuyS' g trun 6 c D ASB ^ v C ASA : Si ta n ca c gia m ta i ha n de y e D . uTng g tuon o ca g dudn a l g cun O S >= ) (ABCD c D B L OtS ) (ABCD c C A i O is ^ . (dpcm) ) (ABCD 1 O S >= B ai 53 Cho hin h ch6p S.ABCD c6 day la hin h cha nhat \k SA J_ (ABCD). Goi AE , A F 1^ dudng cao cua cac ASAB va ASAD. ChOfng min h r^ng SC 1 (AEF). Gia i D 6 y AD±(SAB)v a BC//A D ^ BC 1 (SAB) 3 EA ^ EA 1 BC (1) (Sau nay ta c6 the chufng min h (1) bSng dinh ly 3 dadng vuong goc se nhanh hdn hoac bdng tin h chat giao tuyen cua hai mftt phin g vuong goc). Hon nOa EA 1 SB (c^ch dung) (2) Tif (1) vk (2) cho : EA _L (SBC) 3 SC SC 1 EA (3) Mot cAch tuang tif SC I A F (4) Tif (3) va (4) => SC 1 (AEF) (dpcm). B ai 54 Cho hin h chop S.ABCD c6 day ABCD la hin h chuT nhat, goi I , J la trun g diem AB , CD v^ gia sii SA = SB. ChOfng min h rSng CD 1 (SIJ). Gia i De y thay I J la dtTorng trun g binh cua hin h chO nhat ABCD. =>IJ//ADvaB C =>CDlIJc(SIJ ) (1) Taong tif ASA B can ta i S c6 SI la trun g tuyen nen n6 cung la dudng cao. A B 1 SI ; ma CD // A B ^ CD 1 SI c (SIJ) (2) TH (1) va (2) => CD 1 (SIJ) (dpcm). B a i 55 Cho tijf dien ABCD c6 H , K la true tarn cdc tarn gidc ABC va DBC. Gia sC( rkn g H K 1 (DBC). Chufng min h AH , D K va BC dong quy. Gia i Goi AI , DI i 1^ cdc dudng cao qua H vk K. ^ , fKH I (DBC) 13 B C B C 1 H K Ta CO : < [BC x DI i (each difng) BC 1 (ADIi) 3 Al l => BC 1 AI , Theo each difng BC _L A I (1) (2) Tif (1) va (2) cho : B C 1 Al l B C 1 K I Ii^ I Vfiy AH , D K v^ BC dong quy ta i I . • Cac h kha c : Xem cau a), Bai 56. B a i 56 Cho tiJ dien SABC c6 SA 1 (ABC). Goi H va K Ian luot la trifc tarn ckc tarn gidc ABC v^ SBC. Chufng min h ; a/ AH ; SK; BC dong quy b/ SC 1 (BHK) c/ H K 1 (SBC). 31 i Gia g thin ' A; K; S h min g chiin e D. A' = C B o H A o Ch / a C B ±' SA : h min g chufn a t g han : O C a T ' BC^AA ) (ABO X SA o (d A 'BCIS ' SA 1 C B > =) (SAA' 1 C B : a r y su a t o d f Ti . (dpcm) y qu g don C B a v K S; AH : i no e th O C y Va ) (1 K B 1 C S : 6 c a t t thie a gi o The / b ^ \^^^^ : o Matkhactac ) (doSAKABO A JBHXS C S 1 H B ^ ) (SAC 1 H B : n ne . (dpcm) ) (BHK ± C S : a r y su a t ) (2 a v ) (1 f Ti K H ± C B > = ) (SAA' ± C B : 6 c a t / a u ca o The d K H_ J C S > = ) (BHK L _ C S : 6 c a t / b u ca o The . (dpcm) ) (SBC ± K H : a r y su a t ) (4 a v ) (3 f Ti 7 5 i B a ) (2 ) (3 ) (4 nijf g dita A a Qu . (P) g phAn t ma g tron m na D ABC g vuon h hin o Ch gg vuon Ma qu g than Badng . Ax n tre g don u lU m die t mo a l M n Cho . S i ta ) (P t ca ) mp(MCD \6i c go g vuon Ma qu g thSn g difdn . R i ta . hang g than R; B; A : h min g Chufn / 1 dua nijf n tre g don u lif Mi kh S R n doa a cu I m die g trun h tic y qu m Ti / 2 i Gi^ C B 1 R M ) (MBC 1 R M : 6 c a t t thie a gi o The / 1 R A 1 D A > =) (MAR 1 D A ^ M A 1 D A : O C a d a M B A a v R A a m) (P p m g tron a g cun D A; AB ; AR y Va . (dpcm) g han g than R; B; A > = D Ai vd c go g vuon g cun . hang g than S, D, A > = n tre f ti g TiJon . MB ± R M ^ ) (MBC ± R Mo D / 2 : n ne A Mo ca g difcrn 6 c M d g vuon R MB c gia m Ta R AB.A = ' M A A 1 R M Dang 2 : CHJN G MINH Dl/dNG THAN G VUONG GO C Vdl MA T PHAN G BAN G TRU C Dl/CiNG TRO N L PHirONG PHA P Ca stf ciia phi/ang phap chiirng min h difdng thin g d vuong goc v6i mat phang a bftng van dung dinh nghia true dxidng tron: la dxtiing thSng vuong goc vdri ma t phan g chii'a dUdng tron ta i tam cu a no bang hai btfdp ca ban nhU sau : n B i : Ti m mot diem S a dinh each deu cac dinh da giac day ABC... nha sau : SA = SB = SC = ... Tim diem O d day each deu cac dinh da giac day ABC... OA = OB = OC =... • Ba : No'i hai diem S, O do thanh true d cua di/crng tr6n. No la dudng thftng vuong goc vdi moi mat phang ehiJa duac di/cfng tron (ABC). n. CA C BA I TOA N C O BA N B ai 58 Cho hinh vuong ABC D canh a. Ve cung ve mot phia (ABCD); cac doan AA'' C C vuong goc (ABCD) sao cho AA' = C C = a. Chufng min h : A'C 1. (BCD). Gia i De y den mp(BC'D) (a v i tr i cac dinh cua ta m giac BC'C), ta c6 : ^ • CB = C C = CD = a =* C e (d) : true di/orng tron (BCD ) ngoai tiep ABCD . I AT) VAA^TAD ^ • Tuang tU : < A' C - A C = VAB ^ - AD ^ a- ^ A-B - VA'A'^ + AB 2 = a- ^ => A' e (d) Vay : A C c (d) hay A C ± (BCD ) (dpcm). B a i 59 Cho hinh chop S.ABC c6 fisC = 120°, CsX = 60°, A ^ = 90° va SA = SB = SC = a. Goi I latrung diem BC. ChiJtng min h rang : a/ AABC vuong. b/ SI J _ (ABC). Gia i a/ Dat : SA = SB = SC = a > 0 (cho san) BC = 2CI = 2. ^ = aV s (ASIC 1^ niJra A deu) Ta CO : •iCA = a (.\ASC deu) A B = a>/2 (AASB vuong ca n ta i S) < <^ BC ' = CA^ + AB 2 [CA^ - AB^ = a^ - 2a^ - Sa^ Theo dinh ly Pythagore dao => ACAB vuong ta i A (dpcm). 3 3 g'lAc m ta a cu n huye h can i vd g Cifn n tuye g trun g dUcfn t cha h tin o The / b ) (1 C I = B I = A l >= ) (2 C S = B S = A S O C a d a nff n H a 1 I S > = C zVAB p tie i ngoa n tro g ducrn e tru a l I S : o ch ) (2 a v ) (1 ^ T i 0 Bai6 a v " 60 = C IJA 6 c i tho h hin a l D ABC y da D S.ABC p cho h hin o C h . ABC c gia m ta m ta g tron a l G i Vd . (ABCD) x G S g rSn h min i Gia i : 6 c C AAB y e D) thoi h hin h (can C B - A B. deu C AAB ) (gt " 60 C gA i . ABC u de c gia m ta p tie i ngoa C AB n tro g dudn m ta a l G >= C G = B G = A G ; o d o D ) (ABC n tro g dudn a eii e tru ) (d 6 G ^-- ) (d e S C S = B S = A S : 6 c t thie a G i . (dpcm) ) (ABCD 1 G S y ha ) (ABC 1 G S y ha ) (d c G S i la m T o 1 6 i B a a l I i Go " 90 = t AD a v D S = C S = A S 6 c D S.ABC p cho h hin o C h . (ABCD) L - I S g rkn h min i Gia ) 90° = 6 ( C AD c gia m ta f ti y e D D I = C I = A l >= . SD = C S = A S t thie a gi p hcr t K c . .\ACD p tie i ngoa ) (ACD n tro g dudn e tru a l I S > =i ) (ABCD ^ ) (ACD 1 I S ^= . (dpcm) ) (ABCD l SI . « 2 6 i B a SCT - ) SBt 6 c u de c gia c lu a nijf a l D ABC 6 c D S.ABC p cho h hin o C h ( 1 A S a v ) (BCD 1 1 0 g rftn h min g Chijfn . SD a v D A m die g trun a l t lifg i Gia h can \'6i g ufn n tuye g trun g dudn a cii t cha h tin n de y e D n i m n h tool I: DUdNG THAN G WON G G^ C DUdJNG THAN G Dang 1 : CHUfNG MINH HAI Ol/dNG THANG VUONG GOC NHAU BANG DjNH NGHIA OUdNG THANG VUONG GOC Vdl MAT PHANG LPBUONGPHA P Co sd cua phiTOng phap chOfng min h dudng thin g d vuong goc vdfi diTcfng thSng a kh i ta sit dung dinh nghia : d 1 a => d 1 a (tuy y trong a), qua 2 bade ca ban : • Bi : Quan sat va quan ly gia thie t ti m mp(a) chijfa dudng thdng a can chufng min h no vuong g6c vdi d. • 82 : ChOrng min h d _L (a) => d 1 a (dpcm). n. cAc BA I TOA N C O BA N Bai63 Cho ti l di?n ABC D c6 AC = A D va BC = BD. ChiJng min h A B 1 CD. Giai Goi I la trung diem canh CD va de y hai trung tuyen cung la dudng cao trong ha i tarn giac can cung ddy CD la : AADC va ABCD. I CD 1 BI c (ABI) [CD 1 A I c (ABI) CD 1 (ABI) 3 A B => CD J_ A B (dpcm). Bai6 4 Cho hinh chop S.ABCD c6 d^y ABCD la hinh thoi v ^ SA 1 (ABCD). Chijfng min h BD 1 SC. Giai De y : BD c (ABCD) ma SA 1 (ABCD) => BD 1 SA; ma SA DB 1 SC (dpcm). Bai6 5 Cho hin h chop S.ABCD c6 ABCD la niJfa hin h luc giac deu va SA 1 (ABCD). Mo t mat phang qua A vuong goc vdi SD tai D' cat SB; SC ta i B', C . ChuTng min h ti l giac AB'C'D' noi tiep difoc. Giai , _ i CD 1 AC (ti'nh chat luc giac diu) ^ ^ ' ^CD L SA (vi SA ^ (ABCD)) => CD 1 (SAC); ma (SAC) 3 A C => A C 1 CD (1) Theo each dilng SD 1 (a) EE (AB'C'D') A C => A C 1 SD (2) Tir (1) va (2) => A C J_ (SCD); ma (SCD) ZD CD ' => A C 1 CD ' <=> Actr = 90° (3) Tuong tif : BD 1 (SAB) 3 AB' => AB' 1 BD (4) 35 ) (5 D S 1 ' AB > = ) (AB'C'D' ^ ) (a g difn h eac o The ' B'D 3 ) (SBD a m ; (SBD) 1 ' AB > = ) (5 a v ) (4 f Ti ) (6 " 90 = T ABT « ' B'D ± ' AB >= . (dpcm) c dua p tie i no ' AB'C'D c gia T Tu > =) (6 a v ) (5 f ti g Cun 6 6 i B a 6 c n die f tij e d u sa 6 s I DA N KIE U DIE h min g ChiJn . ABCD n die f tii o Ch l BD - ' BC =' AD - ' AC' ^ D C 1 B A i Gia : p ho g trtfcrn i ha g b^n n kie u die h min g chiln a T ) (ABH ± D C D C 1 B Af sij a Gi : ) (=> n ki# u Die • H A ± D C > = H B o ca g dUcm n cha a l H . CD m die g trun a l Mi vdr c gia m ta g tron g luan c thiJ e h g dun p A H . M 2CD -_ ' AD - ' AC f i < >= H . M 2CD - ^ -BD iBC' . (dpcm) ' BD - ' BC =' AD - ' AC >= ) (1 ' BD - ' BC =' AD - ' AC a Gi : ) (<= n ki$ u Di^ • o ca g di/dn n cha c ca a l 2 BH , AHj , CD m die g trun a l Mi Gp ) (2 7 .MH 2Ci5 = ^ AD - ' AC I \ : O C a t , tfng g tuon) (3 2 2CD.MH = ' BD - ' ;BC : ) (3 a v ) (2 o ch ) (1 g dun f Sij ) (4 2 H = i H> <= 2 MH =, MH > = 2 2CD.MH = , .MH 2CD g phSn t ma i ha a c , (ABH2) ^ ) (ABHi 1 D C 6 c a t ) (4 \.\S a l a Nghi . (dpcm) B A 1 D C o d o d , A B a v B A i do h can i ha e d ) so i (da u d a v n ca n kie u Die : n lugi t Ke O . (dpcm) ' BD - ' BC =' AD - ' AC a l u nha c go g vuon 7 6 i B a go g vuon i th u de n die f tu a cii y k t ba i do h can i ha g rkn h min g Chiifn i Gia u de c gia m ta a cu t cha h tin o The . AB h can m die g trun a l E i Go : dien l ti t ma c ca Bai 68 Cho tiif dien ABCD c6 A B J_ (BCD) va Sct) = 90". Goi B H la dadng cao AABC. Chilng min h ABHD vuong. Giai D e y : CD 1 BC1 C D x ABi Cach difng dirdng cao Bai 69 CD 1 (ABC) 3 B H B H 1 CD c (ACD) => B H 1 AC c (ACD) B H _L (ACD) 3 H D B H 1 H D hay ABH D vuong d H (dpcm). Chufng min h rSng : Trong mot ti l di$n neu c6 2 cSp canh doi vuong goc nhau th i c&p canh doi thuT ba cung vuong goc nhau. Giai Gia sur cho tiJ dign ABC D c6 : ABIC D ACIB D Ta can chilng min h : BC 1 A D That vay dimg A H 1 (BCD). Ta c6 : C Dl AB (gt) CDiA H CD _L (ABH) => CD 1 B H Va: ;BDlAC(gt) BD ± (CAH) => BD 1 C H [BDlA H H ai ket qua tren cho thay H la trilc ta m cua ABCD luc d6 : fBCiD H ^ ^BCIA H BC 1 (ADH ) BC 1 A D (dpcm). • Ghi chii : Co thi chvCng minh bdi todn bdng phuang phdp vecta. Dang 2 : CHUfNG MINH HAI OUCiNG THANG VUONG GOC NHAU BAN G DjNH LY BA Dl/CJNG VUONG GOC L PHODN G PHA P Ca sd cua phuang phap can van dung dinh ly ba dudng vuong goc nh u sau : , », T , . > I AM la dildng xien (so vdi (a)) Gia stf A H 1 (a) => < , ; HM la hinh chieu (ciia AM xuong (a)) t h i difcrng (d) n'km trong (a) thoa : (d) 1 A M (ducrng xien) c* (d) J. H M (hinh chieu) Do do philcfng phdp gom 2 budc thuc hanh : • B i : Xac dinh dudng vuong goc vdi mot ma t phSng (a) t ^ do ti m ra dudng xien ® va hin h chieu ® . n B2 : Dirdng thin g thut @ la (d) nam trong mat phln g (c • Neu : (D 1 ® (D J_ ® => (ycbt). • Neu : (D ± ® (D 1 ® => (ycbt). 37 N COBA N TOA I BA c cA .n 0 7 i B a . nhat a ch h hin a l D ABC a v ) (ABCD 1 A S 6 c D S.ABC p cho h hin o Ch . vuong c gia n tar g nhfln Ik u de ) (SAD , (SCD) , (SBC) , (SBA) n be i Gia ) (ABCD 1 A S : O C a T ) (ABCD 1 A S : y tha y e D ) (ycbt A d g vuon B JASA ^ _ B A X A |S ) (ycbt A d g vuon D [ASA ^ D A 1 AIS n xie g dUcJn a l : BS u chie h hin a l : B A B A 1 C B a M ) g6c g vuon g dtfcrn a b y l h (din B S 1 C B >= . (ycbt) B a g vuon C ASB >= d g vuon C ASD > = ) goc g vuon g dudn 3 y l h (din D S 1 C Di ti g Tiran (dpcmg vuon c gia m ta g nhifn a l u de n be m&t 4 6 c p cho h hin i la m T o 1 7 i B a a cu u chie h hin a l Hi Go . BD 1 C A vk D C L _ B A 6 c D ABC n die T TU . BC 1 D A a v D BC c gia m ta m ta c trU a l H g rSn h min i Gi& : i da o Ke , B = D C > r H B o ch o sa HB i C = D saochoCHnB HC j D = C B n H D o ch o sa HD : O C a T) thiet a (gi B A ± DC(B 1 H Ai (v H A 1 DC H )3B CD1(ABH >= ) (1 D ABC o ca g dUcfn a l , BB y ha i B i ta H B L _ D C >= . Cj i ta H C ± D B H C. z ) (ACH ± D B a t g Ti/on ) (2 D ABC o ca g dudn a l H C >= ) (dpcm D ABC m ta c trU a l H : a t o ch ) (2 a v ) (1 Tii n xie g dudn \k : j AD ) (BCD _ J H A y tha y e Du chie h hin a l : , HD j AD _ J C B > = ) ABCD m ta c trif a l Hi (v i HD 1 C B a M . (dpcm) D A ± C B D A 3 ) (ADDi ^) (AHDi 1 C B >= 2 7 i B a t nha ? chC h hin g ph^n t ma v6i c go g vuon g dUdn i ha a l y D , Cx o Ch . nh^t Q ch h hin a l F ABE g ran h min g Chufn . F a v E i ta y D, Cx t ca B A Bai 73 Trong hinh chop S.ABCD day la hin h chC nhat ABCD. Goi SH la diTcfng cao hin h chop va SK; SL thijf tu la dirdng cao cac tam giac SAB va SCD. Chijfng min h rSng H , K, L thSng hang. Gia i Deyvd i SH I (ABCD) fSH : la dudng xien ^HK : ]k hinh chieu MaAB 1 SK (each difng) AB i H K (dinh ly ba dif&ng vuong goc) TucJng t u CD ± H L (dinh ly ba di/cing vuong goc) 'AB x HK Tom tai : H , K, L thftng hang (dpcm). HK/ B / HL Bai 74 : 1 Cho tiJ dien SABC c6 ABC la tam giac deu canh a, cac mftt (SAB); (SBC) va (SCA) hap vdi (ABC) cac goc bftng nhau va bftng a. 1/ Chiitng min h rftng : hin h chieu H cua S len (ABC) la ta m du&ng tron noi tiep AABC 2/ Tinh tong dien tich 4 mftt cua tuT dien S.ABC. Gia i S 1/ Goi 1, J, K Ian lifot la hin h chieu ciia H len BC; CA; A B Do dinh ly ba dUcfng thftng vuong goc. BC 1 SI; CA 1 SJ; A B 1 SK Do d6 goc phin g cua cac mftt ben (SBC); (SAC) v ^ (SAB) tao vdi (ABC) Ian lugt la S^ , SJi?, gS^ gli l =S3tl - SKfi = a De y thay tam giac vuong SHI, SHJ, SH K bftng nhau nen : H I = H J = H K Vay, H la tam dUdng tron HQI tiep AABC. ( H cung la trong tam , true tam , ta m difdng tr6n ngoai tiep cua AABC) (dpcm). -.S [SHAB = SsAB COS a I cos a HAB 1 2/ Theo dinh l y dien tich va hin h chieu ta c6 : •JSHBC = ^sac cos a <=> -ISSBC .S SHCA =SscACOs a SsAB + SsBC + SscA + SABC = (SHAB + S,IBC + ^HCA) + S^Bc COS a SSAB + SsBC + SscA + SABC = " " SABC + SABC a^Va ( l + cosa)a2, ^ 'SCA COS a 1 COS a HBC .s IICA I, COS a 4 COS a (ycbt). 3 9 G PHAN T MA I TfCJ G WON G PHAN T MA : 4i toa G G VUON G PHAN T MA I HA H MIN G CHLJfN : 1 g Dan , PHAP G PHtfON L e da nghi h din g dun t sij n ca p pha g phuan a cu d s r Cc : u sa U nh c go g vuon g phfln t ma i ha h min g chufn . 90" g ban o d g phan t ma i ha a cu c go h min g Chufn ° 90 = ^ (o? = e aMf =p( ) KP ) (a , PHAP G PHirCM . n . dii a v n ca n kie u die g dun sii n ca p pha g phuan a cii o s Ca c go g vuon g than g ducfn t mo a chijf t nha T thu t ma h min g Chufn : n ba f cc c bLfd i ha a qu i ha f thij t ma i v(J h tin 6 (c ) (d g thin g dudn a r m ti e dt thie a gi y l n Qua : i B • . (P) c ) (d a v) toan i ba o ch t vie i lA : c ducf n lua t ke i th ) (a L _ ) (d h min g Chifn : a B • . (ycbt) ) (P ±) (a N BA ca N TOA I BA C CA . m 5 7i B a g Chtjfn A. a g vuon C AAB a v ) (ABC 1 A S 6 c C S.AB p cho h hin o Ch i Gia A S a l n tuye o gia 6 c ) (SAB a v) (SAC y tha ye D A jAB^S a v A |AC^S . (dpcm) ) (SAC 1 ) (SAB ] (SAC) ; [(SAB) = C SA =" 90 ^ : ye D : c kha h Cac OA fABlS) (SAC 1 B A C A X A B . (dpcm) ) (SAC ± ) (SAB ) (SAB c B A a m 6 7i B a BD 1 SA (vi SA 1 (ABCD) =^ BD 1 (SAC) ma BD c (SBD) => (SBD) 1 (SAC) (dpcm). Bai 77 BD) Cho hinh chop S.ABCD c6 day la hinh chuT nhat ABC D va SA i (ABCD). Goi E va F la hinh chieu cua A len SB va SD. Chilng minh rSng : a/ (AEF) 1 (SCD). b/ (AEF) _L (SAC). Gia i , „ , AE 1 SB (do Mch dtfng) a/ Ta CO : •! AE i BC (vi BC J . (SAB)) => A E 1 (SBC ) Ma A E c (AEF ) => (AEF ) 1 (SBC ) (dpcm). hi Phan trade ((d) 1 (a)), ta da chufng minh duac : S C 1 (AEF ) Ma S C c (SAC ) ^ (SAC ) 1 (AEF ) (dpcm). Bai 78 Cho tuT dien S.AB C c6 S A = S B = S C va ZSLABC vuong can tai B . Goi I va J 1^ trung diem AC va BC . Chufng minh rkng (SAC ) ± (ABC ) va (SIJ ) 1 (SBC) . Gia i De y : [SA = SB = SC (gia thiet) [lA = IB = IC (AABC vuong can d B) ^ SI 1 (ABC); VI SI la true difdng tr6n ngoai tiep AABC. Ma SI c (SAC) => (SAC) 1 (ABC) (dpcm). Ta CO : I J la difcfng trung binh AABC =; IJ//= : -AB 2 ^ BC 1 IJ Lai CO : BC i SI (vi SIl(ABC) ID BC) B C 1 (SIJ) Mat khac B C c (SBC) => (SBC) 1 (SIJ) (dpcm). Dang 2 : CHUfNG MINH Dl/dNG THANG VUONG GOC VCI MAT PHANG BANG GIAO TUYEN CUA HAI MAT PHANG LPHirONG PHAP , Ca sd cua phi/ong phap can sijf dung dinh ly lien quan den giao tuyen cua 2 mat phing vuong g6c nhau nhu sau : De chufng minh (a) 1 (a) ta chiJng minh : (a) c (P) ± (a) vk (a) 1 (d) = (a) n (P) (tai M). / M 7 41 , PHAP G PHiroN . n n qua n lie y l h din g dun f siJ n ca p pha g phucfn a cu sd Ca n qua n lie y l h din g dun f siJ n ca p pha g phucfn a cu sd Ca t mS i vd c go g vuon g cim g ph^n t ma 2 a cu n tuye o gia n de t mS i vd c go g vuon g cim g ph^n t ma 2 a cu n tuye o gia n de : u sa u nh a br thii g phin : u sa u nh a br thii g phin / : h min g chufn a t ) (a _ J ) (a h min g chufn e D / : h min g chufn a t ) (a _ J ) (a h min g chufn e D / ) (y n) (P =) (a / ) (y n) (P =) (a (a) ± (y) w); ±) ^p av M BA ca M TOA I BA c cA . m 9 7 i Ba l chf h hin a l D ABC a v Si ta n ca B ASA 6 c D S.ABC p cho h hin o Ch : g rkn h min g Chum . (ABCD) 1 ) (SAB f su a gi a v B A . (SAB) 1 C B / b . (ABCD) 1 I S /a o ca g dUcfn Ik g cun I S n tuye g trun n ne Si ta n ca B ASA : O C a T /a ) (SAB c I S; AB 1 I S ^ B An tuye o gia o the ) (ABCD 1) (SAB a M . (dpcm) ) (ABCD ±I S ^ ) (ABCD z c C B; AB 1 C B u t g Tuan /b jy B An tuye o gia o the ) (ABCD 1) (SAB a m . (dpcm) ) (SAB L BC ^ 0 8 i Ba (a v) (SAC g phln t ma i ha ; deu C AAB 6 c C S.AB p cho h hin o Ch hm ta e tru a l t lUcf n Ia Ha v On co C B m die g trun a l I i Go . (ABC) : g rkn h min g ChOrn ) (COH 1 B S / c ) (SBC 1 ) (SAI / b ) SAl(ABC /a i Gia : Xet /a) (SAC n ) (SAB = AS) (ABC l (SAC) ; (SAB) I . (dpcm) ) (ABC 1 AS ) (ABC 1 A S : y e D /bn xie g dudn \k : fSI < >= ) (1 I A 1 C B a m ) (2 I Sl BC ^ u chie h hin a l : [AI ) deu C AAB o (d Tirong tii chufng minh difcrc (BOH) ± (SBC) => (COH) n (BOH) = OH 1 (SBC) (dpcm). Bai 81 Cho hai tam giac ACD va BCD n&m tren hai m&t phSng vuong goc nhau. Cho biet AC = AD = BC = BD = a va CD = 2x. Goi I va J Ian lifcrt la trung diem cua AB v^ CD. 1/ Chufng minh : IJ 1 AB va IJ J . CD 2/ Tinh AB va IJ theo a va x. 3/ m gia tr i nao cua x thi (ABC) 1 (ABD) ? Giai 1/ Hai tam giac can AACD = ABCD JA = JB => IJ 1 AB (tai I) Van do cic tam giac ACD val BCD can tai A va B => CD ± (AJB) => CD 1 IJ (tai J) (dpcm). 2/ Theo gia thiet (ACD) ± (BCD) theo giao tuyen CD ndn : AJ 1 CD => AJ 1 (BCD) AJ 1 JB Tam giac AJC vuong goc d J , tam giac AJB vuong d J cho nen : 'AJ 2 = AC^ - CJ2 = - 1 1 r-. V- , => IJ = -A B = -V2 (a2 -x^ ) (ycbt). AB^ - AJ2 + JB^ = 2(a2 - x^) ^ AB = 72(0^ 2 2 3/ ACAB cfin d C nen : CI 1 AB C I l ID « IJ = -CD <=> AB = CD 2 « 2(a^ - x^) = 4x^ <^ x=* = ^ « X = ^ (ycbt). 3 3 Oang 3 : Di/NG DOAN THANG QUA MOT DIEM VA VUONG GOC Vdl MOT MAT PHANG LPHVONGPHAP , Co sd cua phaang phap difng thiir nhat nhu sau; gom hai budc ca ban : • Bi : Tim mSt ph^ng (P ) 1 (a) va (P ) 3 M. • B2 : Ha MH 1 (d) = (a) n (P ) MH la doan thin g thoa yeu cAu bai toan. 43 2 PHAP G PHirON L I : c bUd i ha m go i nh l thi p pha g phifcfn a cii sa Ca . gian g khon g tron n sft 6 c g thSn g dif&n m Ti : j B • . (a) X) (a ) (a / / ) (b i thd g don Ma qu ) (b g thAn g difcfn g Difn : 2 B • ) (a o ) (b = Hi go av . dimg n ca g thftn n doa a l H M >= N BA CO N TOA I BA C CA . m 2 8 i B a O ch h hin a l D ABC a v ) (ABCD 1 A S 6 c D S.ABC p cho h hin o Ch . a n de S f ti c go g vuon n doa g dan y Ha . Mi ta A S t ca i Gia ) (SAB ± C B a m; (a) c C B y e D . Bx n tuye o gia o the ) (SAB 1 ) (a >= x B e H; Bx 1 H S a H . (ycbt) g difn n ca g thin n doa a l H S y Va 3 8 i B a th hin a l D ABC y da a v ) (ABCD _ J A S 6 c D S.ABC p cho h hin o Ch . (SBD) n tre n le O Aa cii M m die g trun a v A a cu u chie h hin h din c Xa s i Gia ) thoi h hin o ch6 g difdn i ha t cha h (tin C A 1 D fB , , . jjg D B 2 1 ) (ABCD i (v A S 1 D |B ' ^ ^ . SO n tuye o gia o the ) (SAC 1 ) (SBD ^ H i ta ) (SBD 1 H A Hi ta O S 1 H A a H ) (ycbt ) (SBD n tre H a cu u chie h hin a l H y Va O S e K; AH / / K Mg Dirn . (SBD) 1 H A O C a D M i ta ) (SBD 1 K M >= . (ycbt) ) (SBD n tre Ma cii u chie h hin a l K y Va I TB N A O T l GlA . I V ) 2000 - Y N QUA N VIE C (HO 4 8 i B a Nhifng: SAIB H (2) Tir(l)va(2) BH _L (SAC) 3 SC; HK. Do do : BH 1 SC (3) BH 1 HK (4) Ta lai c6 : BK 1 SC (5) (each ditag). CQng tCf (3); (5) => SC 1 (BHK) (ycbt). Xet rieng (4) cho ta ABHK vuong tai H Tir ABBA 1 B H^ S B^ 1 B H^ 2a 2 a' a BH^ = a^ » BH = a H K S H Xet : ASHK oo ASCA (6) CA S C _ (aV2 )v/2 _ V6i ASBA vuong can tai B => S H = 2 S C a.a lyfE (6) H K = CA. S H Vay dien tich S cua ABHK la : V(aV2f + (aVsf 1 aVs Vs S = -BH.H K = -.a. 10 a (ycbt). Chuyen de 5 : PHUONG PHAP THAI CO THE THEN MOT MAT PHANG I. PHLfdNG PHAP Co so cua phuong phap la sCt dung hai tien de 5 va tien de 6 : nghia la khi trai mot co the len tren mot mat phftng (ifu viet di/dc chon sSn c6 loi cho bai toan) thi tien de 5 va tien de 6 cho ta each qua bai toan phSng se lam cho bai toan don gian hon. Gom 2 hiidc ca ban : • B i : Lira chon mot mat phSng (a) dac biet OU viet c6 Igi cho bai toan, sau do chuyen doi cac yeu to trong khong gian quan he vdi vat the hinh hoc d gia thiet xuong mSt ph4ng (a) do. O B2 : Ap dung tien de 6, ta thay trong (a) cac yeu to difOc chuyen doi luon bao dam cac tinh chat ve goc va do dai ... Ap dung tien de 5 de dirge phep sijf dung cac dinh ly so cap. Tir do ta xay diTng phep dUng hinh tiTcfng quan giffa cac yeu to dugc chuyen doi vdi cac yeu to da dugc chuTng minh sSn (Dinh ly va He qua) trong mat phang (a) de dan den eac ycbt. Trong chuyen de phep chijfng minh phan chufng cung thong thirdng dage sut dung. Hien nhien viec siif dung cac tien de khac trong bai toan van ngam hieu la luon luon duoc dat ra. 45 N BA O C N AO TI BA c cA . n 5 8 i B a tg Don . 60° c go t mo u nha i viS o ta t mo i do z O, Oy , Ox a ti a b o Ch : g rSn h min g Chufn . do a ti a b n tre f ti T thu C, B C B + B A / b . dien m ta c g6 t mo h than o ta z O, Oy , Ox a ti a B / a i Gia n&ng a kh 2 6 c a t , (a) g tron g phSn g don z O, Oy , Ox a ti a b f sO a Gi / a h° 120 = z x9 i th ° 60 = ydz c g6 t mo u nha i vd o ta a d z O, Oy a ti i ha ) thiet a gi c go t mo h than o ta g chun y ha g phan g don g khon z O, Oy , Ox y Va z , p , O od g dUn a v B AOA t xe h eac g ban , phang n toa i ba e v n toa i ba a Du / b : g phan c ho h hin g tron 6 c a d a t , do ) mp(OAB n tre ' AOBB ) (1 B O + A O > B 2 A : tir g Tuan ) (2 C O + A 2AC^O ) (3 C O + B 2BC^O + B O + A 2(O ^ ) +BC C A + B 2(A > = ) (3 + ) (2 + ) (1 e v o the g Con (dC O + B O + A O O B + C A + B A >= 6 8 i B a mig Chufn . nhau g ban u de t ma n bo a cii h tic n di^ 6 c n die f tij o Ch . nhau g ban g chun a cu i Gia o the ) (a g xuon A ADC . ADCB , ADAB c ca i tra , (ABC) = ) (a n Cho : o ch o sa ) ve h hin f (nhi A C , BC s AD: = i AD - ) Al : 2 BD = i BD = D 2AB = N E => 2AB ^ NC + EC (vi N , C, E chua thSng hang) 2AB < 2CD A B ^ CD (dpcm) Bai 88 Cho tur dien ABCD c6 AB = CD = a; AC = BD v^ A D = BC. Xi c dinh vj tr i diem M tren canh AB d6 cho AMCD c6 chu v i nho nhat, xac dinh gia tr i nho nha t cua chu v i do. Giai Trai ADAB xuong mp((x) = (ABC) vdi true tra i A B th i : BC = AC = BD AAB C = AABD M C = M D AC' = BC = AD Luc do chu v i 'f ciia z\MCD la : r = M D + MC + CD = (M C + MC) + CD r » C'C + a Do do : min'f = (C'C + a) xay ra <=> C , M , C thSng hang Vay M HE Mo = C'C AB , th i min ^ = (C'C + a) (ycbt) Bai 89 Cho tiir dien ABCD c6 AC = A D = BC = BD = 1; AB = a; CD = b; con M , N Ian lifort la trung diem AB va CD. Ti m tren canh A D diem P sao cho tong (PM + PN) dat gia tr i nho nhat. Giai Cac tam giac ACMD (can ta i M) va AANB (can ta i N) thuT tif ciing trung tuyen M N va cung la dUcfng cao fMNlAB MN^CD lYai AACD tren mp((i) s (ABD) vdi true trai la AD difoc AADC AC - 1 DC = b Goi N ' la trun g diem canh D C => P N = PN ' Luc do tong : I = PM + P N = P M + PN ' > MN ' De y thay : < ASTD - A5ct) = 90° AXT ) AJCD ^ 90° AMD = A5?t) - 90° => Tijf giac AMDN ' noi tie'p trong dUcfng tron dudng kin h A D = 1, nam trong mp(ABD). => min S = MN ' xay ra kh i va chi kh i M , N' , P thSng hang. Hay P ^ Po = MN ' A D (ycbt) Bai 90 Cho hin h chop S.ABC, cac goc phang dinh S dou bang a (0 < a < —) con canh ben SA = 1. 3 Churng minh rang : AB + BC + CA ^ ^2(\ cos 3u) . 47 i Gia utf thij ) (SBC =) (a g phan t ma n le C ASA , ASAB : c gia n tar c ca i Tra B S i tra e tru c ca o the f ti r thu C ASA = i ASCC a v B ASA = , ASBB c dug . SC av , BjC » j CC + C B + , BB = A C + C B + B A = I : o d c Lu i v(J i ABiSC g tron n cosi m ha y l h din g dun p a i kh y e D a 3 = ^ BjS? a 2SBi.SCi.cos3 - f -SC ? SB = ? BjC >=i ) 3a s co - 1 2( = a 3 s 2co - 1 + 1 = ) 3a s co - 1( 72 = , BjC >= , B ) (dpcm ) 30 5 (1-00 72 > A C + C B + B A = S >= 1 9 i B a m die a b a v c go g vuon t mo i do z O; Oy ; Ox t bie n pha a ti a b o Ch ) (* 1 = C O + B O = A O : o ch o sa z O; Oy ; Ox kh i do g khon C OAB n die f tij a cu n pha n toa h tic n die h min g ChiJn / 1 ) (* n kie u die a tho 6ch. + C AB * A <3B = ; C BA * AC 6 - i (5Af =j I ; h Tin / 2 i Gia g phuon g ban o n g xuon C AOB n die J ti i tra ) (OBC =) mp(a g Tron / 1 f tij a cu B O h can a chuf M O a v 1 = M O h can 6 c N OMD g vuon h hin B O : 6 c a t , x = C Ot dS i kh y na g dUn h eac o The . dien T tu a cii C O h can C AND = B AOA B AMD = C :\OA C ADB = C AAB ^ ^ 1 = C O + B O = A ; O C SAAB + C S^OB + C SAOA + B S/^OA = p St C SADB + C S^OB + ! SAMDI + C SAND = p St >= ) (dpcm t cons = 1 = O SMDN = p S, >= : 6 c a t n tre o The / 2 G BA + C (5A - AB 6 = i I ) (ycbt " 90 = M MD = B CD . B 5lD + C ?fl) =i I >= ^= 1/ Ta CO : A C = B D = b A D = B C = a TueJng tif: ADIC can Giai AACD = ABDC AJ = BJ AAJB can JUA B IJID C ^ IJ la doan vu6ng goc chung cua hai canh AB \k DC. Mpt each tuang tu => (dpcm). 2/ Suf dung phUcfng phap trai nhU sau, dung APQR c6 cac canh di qua cdc dinh cua ABCD va song song v AAPQ vuong tai A Tuang tu : AAQR; AARP cQng vuong tai A. Ta CO : AP2 + AQ2 = PQ2 = 4a2 A Q2 + AR2 = RQ2 = 4b2 AP2 =2 (a2+b2-c2 ) A Q 2 =2(8^4 - c^-b^) A R2 =2 (b2+c2-a2 ) "^ABCD = -v APQR 24 AP.AQ.A R V ABCD -V2 12 VCa^ + b^ - c^Xa^ + - b^Xb^ + - a^) (dvtt). Chuyen de 6 : X A C DIN H V A TIN H CA C LOA I GO C TRON G KH6N G GIA N Loai 1 : GOC CUA HAI DUdSG THAN G THONG KHONG GIAN L PHirONG PHAP Co SOT cua phuang phap tim goc cua hai dUcfng thing (a), (b) can thuc hien 2 budc ca ban : • Bi : Xac dinh : " = cp la goc dUng tCr diem O tuy y c6 the d tren (a) hoSc (b) (dat tinh Uu viet cho bai toan khi dUng va tim do Idn g6c) hai tia Oa', Ob' thU tU song song vdi hai ducfng thing (a), (b) c6 yeu cau tim goc". • B2 : Tinh do Icrn goc cp bkng cac dinh ly va tinh chat cua hinh hoc phing hay dinh ly ham O Ghi chii : 0 < (p <90°, neu (p > 90° thi ycbt se chi ro tim goc tu. 49 N BA O C N TOA I BA C CA LI 3 9 i B a C A, BC m die g trun a l t lug n Ia K, J, I i Gg . ABCD n die T tu t mo o Ch . CD a v B A a cu c go o r h din y Ha / 1 : K U c gia m ta e d D C a v B A a giuT n kie u die a r y Su / 2 / c I i ta n Ca / b I i ta g Vuon / a i Gia ) cSSTclT = k Jl = p ( > = D C/ / K I ; AB / / J I : y e D / 1 D C L _ B A ^ < " 90 = p c > <= ) 90" - d K 2a/AU D C = B A « ) IK = U ( K 2b/AJI > <= K I = J I a v ) 90° = p ( =i ( K AJI / 2c 4 9 i B a D C 1 B jA c go g vuon i th u de n die t tii t mo a cii i do h can c ca r&ng h min g ChiJn i Gia , BC m dig g trun a l u t T thu P, N, M 6 c a h can D ABC u de n die r tu t Xe B :MN//A , - , - , , = : h bin g trun g duan t cha h tin o The ^ D IMP//C den die i ti i v, cp a l g cun i la n co u nha i do h can p cS c ca a cu c go c (ca = D N = B Ni v Ni ta n ca D ABN y e D a t o ch e Pythagr y l h Din . Ni ta n ca D ABN o ca g dUcfn a l P N ^ 2a • '• = ^ BP - ^ BN = ^ NP (a) f ^ ^ V y L 2 4 2 A/rD , 2p 2MN.MPcosc - ^ MP + * MN- = ^ NP : a t o ch n cosi m ha y l h Din ^ 2a = p( C0S^ MN^_+_MP^_NP P 2MN.M 2 2 0 = . (ycbt) ) nhau c g6 g vuon u de n die r tii i do h can c (ca " 90 = p( = ^ MP + ^ MN • tinh the c6 : 1 u ch i Gh O 2 \ /_ 2N j l 2 2a' hi Xac djnh va tinh goc cua SD va BC. d Gpi I va J Ian Itfat la trung diem cua SA va SC. Hay dinh ro goc cua IJ BD. Giai a/ Da CO : SA 1 B C ADZ/BC ma SA = AD = a => ip = (SSTBO = SAD = 90° (ycbt). > ASAD vuong can tai A (ycbt). hi Tuang tif: AD // BC => P = (SBTB O = SD X = 45" (ycbt). fIJ// AC d Tuang tu : IJIB D [BD 1 AC => Y = CETTBIT) = fioC = 90 ° (tinh chat diTcrng cheo hinh thoi). Bai 96 Cho tiJ di6n ABCD. Goi M, N, I Ian lutrt Ik trung diem BC, AD vk AC. Cho AB = 2a, CD = 2a, ^ vk MN = aVs . Tinh : (p = CSSTctJ). Giai Theo tinh chat dtfdng trung binh trong tam gidc 'lN//-CD = aV2 2 IM // = - AB = a 2 Xac dinh dtfoc : (p = = (S^"SlJ) Ap dung dinh ly ham cosin : IM^+IN^-MN ^ 2a2+a2-5a2 => cosip = = -= <=> cosrn = 2IM.IN 2.aV2.a Goc nhon tao bdi AB CD 1^ 45° (ycbt). Bai 97 <=>

= . B) i ta n ca g vuon B ASA i (v ) (ycbt ° 45 = p ( >= t ma i ha y tha y e d a v B A m die g trun a l I i Go / b . AB n tuye o gia o the ) (SAB 1) (ABC : g phan ) deu C AAB i (v B A 1 I C ^ ) (SAB 1 I C n xie g ducjn a l : CS u chie h hin a l : S I ] (SAB) ="Tscr i § d = p = 3 tan( : O C a T 2 iS 3 aV 3 aV I C I S 9 9 i B a m die t Xe . C a v B t bie n pha m die i ha o ch ) (a g phin t ma g Tron . (a)] , [AC = ] (a) , [AB : g ran h min g Chiln . ;CCB = C X B i Gia ) (1 Ai ta n ca C AAB => B iCc = C ^CB : y e D . (a) g xuon AT ti a h c go g vuon g difcrn a l H A g Difn n xie diiting c ca a l : C A ; B J A . do u t j thi o the u chie h hin c ca a l : C H ; B [ H ) (1 C A = B A ^ = f Ti bi da o d 6 c n xie g dudn c ca a cu g ufn g tuan u chie h (hin C H = B H >= vuoc go h can - n huye h (can ) 90° = i (f C AAH = ) 90° = i (f B AAH >= (dc) (ABC))T(ACr(ABC) 5lABr ) Ufng (tuan i Xcf = I ABI • Phrfdng phap 1 Xac dinh (p bSng cdch diftig theo din h nghia. • Phtfcfng phap 2 Xac dinh gian tiep cp bSng din h ly di§n tich va hin h chieu. n PhUctag pha p 3 Trong thUc te, ngiTcri ta diftig va tin h goc cua 2 mat phSng bang dinh ly ba dUcfng vuong g6c : Goc cua dUcfng xien va hin h chieu la goc cua mfit phin g chieu (a) va mat phang hi chieu (P) tao bdi dudng th^ng ® va dudng thin g ® : cp = I^Atl = [tSlilpn 0 Ghi ch u : 0 < cp ^ 18(p= (h = K^JfT^ la goc nhi dien (d). 0 < (p <9(f => (pla goc cua hai mat phang (a) va. (fi). n. CAC BAI TOAN C O BAN Bai 100 Cho tijT dien deu ABC D canh a. Tinh goc phing cua nhi dien (A; BC ; D). Gia i 1/ Goi M la trung diem C B va G la trong tam ABCD, ta c6 : A G 1 (BCD) • A M : l a difdng xien [ D M : la hinh chieu Ma C B 1 DM => B C 1 AM (dinh ly ba dudng vuong goc) => (p = (MA =7X^85115) Taco : A M 2 3 2 6 coscp = B ai 101 aVa aV 3 1 (p = arccos 3. (ycbt). Cho hinh chop S.ABCD c6 day la hinh vuong ABC D canh a; SA 1 (ABCD) va SA = a. Tinh cac goc phang cua nhi dien sau : a/ (SBC; ABCD) hi (SBC; SDC). HUdng d&n a/ Ta CO : SA 1 (ABCD) S B : l a dtfdng xien A B : l a hinh chieu => (p = SB A = (SBC ; ABCD ) la goc nh i dien trong ycbt. ASA B vuong can ta i A => (p = — (ycbt). 4 hi Hai tam giac vuong SB C va SC D bang nhau trong khong gian nen c6 chung chan dUcfng cao M cua hai difcfng cao B M va D M theo thiJ tu do ; => P = BSit) = (SBC ; SDC ) la goc nh i dien trong ycbt. 53 : 6 Tac' 2 a Wf ^ BC ^ SB ^ B M : 6 c a t , ABMD g tron n cosi hkm y l h Din ' 2 a = P cos' BP - ' DM + ' BM M 2BM.D _ _3 _ . 2f •i ' 4 a t 27. (ycbt) — = p : a l n die i nh a cu u t c G6 3 2 10 i B a z yO ; 120" = y xO o ch o sa n gia g khon g tron z O ; Oy ; Ox a ti a b o Ch . a = C O = B O = A O o ch o sa C; B ; A m die c ca y la t lucf n Ia y a a t i c g6 g vuon g ducfn n cha a cu i tr i v a v C AAB a cu g dan h hin h din c Xa /1 . (ABC) vdi o ta ) (OCA a v ) (OBC a m o d c go c ca h Tin /2 i Gia a = A O = C A > = u de C ,\AO / 1 a-j2 = C B O d n ca g vuon C ABO ° 120 = e XOf 6 c O d n ca B AAO 3 0AVa aV = - 2 = BA ^ CB + * AC= = * AB^ ^ C. d g vuon C AAB >= H c go g vuon g dUcm n cha n ne C O = B O = A O : y Be i ngoa n tro g duto m ta a l h chin ) (ABC g xuon O tis a h . (ycbt) B A a cu m die g trun a l H y ha C AAB p tie . CA a v C B a cu m die g trun a l t lug n Ia J; I i Go /2 C A 1 J H a v C B L_ HI : O C aT A L _ J O a v C B 1 1 0 : 6 c a t c go g vuon g thSn g dadn a b y l h din o D a =l CJif : a l ) (ABC vdi ) (OBC a cii c fg6 < >= p = l Ojt : a l ) (ABC i vd ) (OAC a cu c g6 a I H' 45 = a 1 = ^ =0 ^ = — = a tan : o ch H d g vuon I AOH Giai 1/ Ta CO : fBOlO A (O la tarn hinh vuong) B O X AM (gt) Cho nen : B O _L (AMO) => B O 1 0M_^ TiTong tif ta cung c6 : B O 1 ON (M, BD,N) = MoN 2/ (MBD) 1 (NBD) » Mo!^ = 90" icfoA-?rot; = 90° (1) f ,,f^ AM 2x tan MOA Nhitag : A O ~ Lnl ^ = ^ = ^ ^ cotNOC = ^ O C 2y Do do (1) tirang ducfng vdri : tan SfoA = cotKoC <=> — ^ = <=> xy = — (ycbt). a42 2y 2 m. GlAl TOAN THI Bai 104 (DA I HOC KHO I B - 1975) Cho hin h chop S.ABCD, day ABCD la mot hin h cha nhat, mat ben SCD vuong goc vdi mSt phSng day va la mot ta m giac vuong ta i S, SCt) = a. Ma t ben SAB tao vdi mat phftng day mot goc p = 90^* ~ tt. Goi SH, SE theo thiif ti i la difdng cao cua cac tam giac SCD, SAB. Biet SH + SE = m, tin h : a/ The tich ciia hin h chop S.ABCD b/ Tong dien tich ciia hai mat ben SAD, SBC. Giai a/ Ha SH 1 CD. Vi mat ben (SCD) 1 (ABCD) => SH 1 (ABCD). Ha HE 1 AB; kh i do theo dinh ly ba diTdng vuong goc ta c6 SE chinh la difcrng cao cua ta m giac SAB. Trong tam giac vuong HSE => SH = SEsinP = SEsin(90'' - a) = SEcosa Do : SH + SE = m SH 1 + cos a ; SH = mcosa 1 + cos a (a day a * 90°, v i ASCD vuong ta i S) Lai CO : AD = H E = SH.cotp = SH.tanu Trong tam giac vuong HSC ta c6 : HC = SHcota Trong ta m giac vuong HSD, ta c6 : H D = SHcot(90*' ~ a) H D = SHtanu (vi ASDC vuong ta i S). DC = H D + HC = SH(tana + cota) Luc do dien tich hin h chO nhat day la : 7J = DC.DA = SH^tana (tanu + cota) = SH^d + tan^a) = SH^ cos^a 55 : a l D S.ABC p cho h hin h tic e th o d o D ^ ^ -SH = « =isH. V 3 3a cos m^ cosa) + 1 3( a cos^ ) g6c g vuon g dudn a b y l h din o (the D A ± D S O C a T / b H S HS . (ycbt) vuA ASD >= a v = DSa cos i sinSDf) Hi ta g vuon H ASD i ( v H S: — = C S a v Hi ta g vuon C ASH 6 c g cun a t u t g Tifan a n s i : 6 c a t y va r Nhi . -SHtana = D AD.S - = ) dt(ASAD H S a cos isH^- = ^ cos 2 . -SHtana = C -BC.S = dt(ASBC) H S . 2 ±SH = 1 ^ -SH = ) dt(ASBC + ) dt(ASAD 2 < 1 ^ -SH = ) dt(ASBC + ) dt(ASAD 2 a cos 2 a n s i 1 a sin + — y cosa a s co a cos + a s m J a s co = ) dt(ASBC + ) dt(ASAD) cosa + a (sin m . (ycbt) ^ cosa) + 1 2( ) 1977 - A NH - C DLfCJ - Y C HO I (DA 5 10 i B a h can u de c gia n tar t mo a l C AB y da 6 c C V.AB p cho h hin o Ch ) (VAB , (VAC) n be t ma c ca n co , (ABC) y da t ma i vdt c go g vuon ) (VBC a l Di Go n/3. g ban o d o s 6 c u nha g ban n die i nh c go i ha h than ) (ABC . AC vdi c go g vuon E De k , (ABC) g phan t ma . VD i vdr c go g vuon D A g ran h min g ChuTn / 1 . 2DE = E V g ran h min g Chijfn / 2 C V.AB p cho h hin a cii n pha n toa h tic n die h Tin / 3 i Gia : 6 c a t , nen C B n tuye o gia o the ) (ABC ±) (VBC : O C a T / 1 . (dpcm) D V 1 D A > = ) (VBC 1 D A > = C B 1 D A E V_ J C A > = c go g vuon g dudn 3 y l h din o The / 2 Ta CO : V D = VEVS 2 DEV 3 3 a V i V E = 2D E = 3 a aVs Stp = -a . 2 4 2 + a. •tp - — V -J T ^;_-V X -r (ycbt). S,p = ^ ( 3 + 6V3) = ^( 1 + 2V3 ) 8 8 B a i 106 (DA I HO C SLf PHA M TP.HC M - KHO I B - 1977) Cho tar n giac can OA B (O A = O B = a), dftt AO S = G. Ti J O t a difti g difdn g thin g vuon g goc vd i ma t phan g (OAB) , v a tre n do la y doa n O C bSn g a. a/ ChtTng min h rkn g ACA B l a tar n giac can. Tin h cac goc v a die n tic h S ciia ACA B the o a vk 0. b/ Tif C t a difng mo t ducfng thin g thin g goc vdri ma t phan g CA B v a tre n do t a la y doan C D = CA . Tinh cac goc v a die n tic h S' ciia ADA B the o a v a 0. Tin h sinG de S l a trun g bin h con g ciia S' va dien tic h cua AOAB . d Gpi I l a trun g die m AB . Chiln g min h rkn g bo n die m I , O, C, D ciing nk m tron g mo t ma t phlng . a/ CB X = ClAfe = arcco s 1 . 0 —;=sin — V2 2 ACA B ca n ABt! = — - arcco s 2 —;^sin — V2 2 G => S = a^sin - J l + cos^ - (ycbt) 2 b/ I5BX = r5Afe = arcco s 1 . e —sin — 12 2, A D S = — - arcco s 2 1 . —sm — 2 2 S' = a si n — 9 0 V? (ycbt). 2 f d Doc gi a ti f giai . 3 +co s — ; sinG = 2 4 B a i 107 (DA I HO C KIE N TRU C TP.HC M - 1994) Cho hinh vuong ABC D canh a trong mat phlng (P). Ha i diem M ; N Ia n lua t d i dong tren hai canh C B v a CD . Da t : C M = x v a C N = y . Tre n dUorng thing A t vuong goc vdi (P), la y diem S. Ti m lien he giuTa x v a y de : 1/ Cac mat phlng (SAM ) v a (SAN ) tao vd i nhau goc 45° . 2/ Cac mat phlng (SAM ) v a (SMN ) vuong goc vdi nhau. Giai 1/ Ta CO : S A ± (ABCD ) S A l AM S A l A N 57 - = N (SAN)]T1!A ; [(SAM) >= 4 - = p + a: ;tac6 a =j rJAJ ; P = M BA : t Da 4 = 1 =) p + a tan(p tan + a tanp tana.tan - 1) (1 = a tan y - a : a M = p tan X - a a ) (1 o va y tha ; X - a + o y a - 1 =) x - a y)( - a( a a a ) y - a x)( - a ( - ^ a =) y - X - a a(2 ><= . (ycbt) a 2 = x > «= N M. 1 M A> = ) (SMN 1) (SAM : O C a T / 2 ' MN + ' AM = ^ AN « ^ y + ^ x + ^ x) - a ( + ^ a = ^ y) - a ( +' a ><= . (ycbt) ) y - x a( = ^ x ><= ) 1998 - I TA N VA G THCDN O GIA C HO I (DA 8 10 i Ba a v y da a cu h din t mo a qu n die t Thie . deu c gia f tO p cho h hin o Ch caa giO c go h Tin . day h tic n die a nijf g ban h tic n die 6 c o d h din i vd n die i Gia 2 aV = C A> = a = B A : Dat i do n be h can i vd c go g vuon a v Aa qu p cho h hin a cii n die t Thie . N; Mi ta : 6 c a t , (SBD) g ph^n t ma g Tron D MN//B ) (SAC 1 D B: Ma K A 1 N M > = K A 1 D B ^ C MNIS C 1BD1(SAC)=>BD1S Mat khac : AO ^ = iCCS = a 1 ^ (2) J^JvJ => 00' = —aV2 cota = 1 - cot^o 2 B D => MN = (l-cot^a).aV2 1 r n Ta CO : S^MKN = g ^ ABC D ; ^ " < 2 o AK. M N = a^; 0 4.sin^a - sina - 2 = 0; I + V33 s m a = 8 t h i thoa : 0 BD 1 SO. Khi do : S^snu = - SO.BD 2 BD = aV^ Vdi : SO = VSA^ + AO^ = 1 2 (ycbt). 2 2 = — .av2 . a = 2/ Ta c6 : BD _L (SAC) BD 1 SC.(dpcm) 3/ Difng : C H 1 SO Vi : BD 1 (SAC) => BD 1 CH ; nen : C H 1 (SBD). Hay goc tao bdri SC va mat phin g (SBD) la CSb . Ta CO : ASAO to ACHO aVe C H C O aV2 = S =^ C H = SA SO 59 . (ycbt) - n arcsi = > = i = — = ^ sinC : do i Kh 3 3 CS ) 1999 - I NQ A H A KHO Y C HO I (BA 0 11 i B a C AB c gia m ta a l y da a v o ca g dudn a l A S 6 c C S.AB p cho h Hin . 60° g b^n ) (SC n die i nh c go e d a m ti a = 5 XS i Go . 45" i Gia C S 1 J B g Dito C A 1 I B ) ( 1 I SAIB : SAl(ABC) C BUA ) (2 C SL _ I B > = ) (SAC _ J I B >= . IJISC > = ) SC±(BIJ ^ Tir(l)&(2) : lik C S h can n die i nh g ph^n c g6 , do i Kh = = p( . I i ta g vuon U AB o D . deu c gid m ta a nuT a l U AB » ° 60 = l 6j : n Ne 1 4 1 3 V = I B o • o « J B2 BJ 3 ^ BI 2 ) (3 B i ta g vuon C ASB : c kha t Ma C B = B S > = Bi ta n ca g vuon C ASB B A BA = a sinC B BS : 6 c Bi ta g vuon C AAB g Tron 1 1 1 a BC.sin = BA + a ^ sin ^ BC ^ BC ^ AB) (4 1 + — = e JSf : 6 c Ji ta g vuon B ASJ g Tron 4 J .B 2 V = B S > = Ji ta n ca g vuon B ASJ >=^ BC ^ BJ D E Tl/dNG Tlf Bai 111 (DAI HOC TONG HdP TP.HCM - KHOI C, D - 1994) Tiif dien OPQR c6 cac mat d dinh O bang Iv. Goi A, B, C theo thuf tU la trung diem cac canh PQ, QR, RP. ChuTng minh : a/ Cac mat tiif dien OABC la nhCtng tam giac b^ng nhau. b/ AABC nhon. d tanA.tanB = 2 biet nhi dien canh OA cua tuf dien OABC la nhi di^n vuong. HitGfn g dan a/ OA = |PQ va BC = |pQ ^ OA = BC OB = AC; OC = AB => dpcm. b/ AABC w APQR => dpcm. CH^ d Tif tanBtanC = HA.HO (1) Ta CO : HC^ = 2HA.H0 thay vao (1) => dpcm. Chuyen de 7: CAC LOAI THIET DIEN TAO THANH V d l VAT THE HINH HOC cAc DEV H L Y GIA O TUYfeN SONG SON G • DLi : Mot mat ph&ng tiiy y (y) song song vdi giao tuyen (d) cua hai mat phing phan biet (a) va (P) thi (y) cat trd lai (a) va (P) Ian lUcft theo cac giao tuyen a, b va : a // b // d. • DL2 : Mot mat phang tuy y (y) cat Ian luot hai mat phing song song (a) // (P) theo hai giao tuyen song song a // b. L PEnrONG PHA P Ca sd cua phifcfng phap la sCt dung dinh ly giao tuyen song song va so diem chung cua hai mat phing trong vat the bang each thuc hien hai budc cO ban : O Bi : Tim tat ca cac giao tuyen ma mot mat phing cat tuy y (a) trong gia thie't c6 the cat duac tat ca cdc mat (mat ben, mat day) cua mot vat the hinh hoc. • B2 : Noi lien tie'p va khep kin cac doan giao tuyen do tren vat the hinh hoc ta difac hinh da giac phing goi la thiet dien. 61 dan sdu lam ra dien thiet chia phdn the cd Igi tien de : u ch i Gh O o ta e th t va h can t mo h quan y qua e s t c^ g phSn t Ma : 1 g Dan • hc hoa h can t mo i vd g son g son e s t c^ g phan t Ma : 2 g Dan • . dien t thie a cu g ph^n t ma t mo i vd g son g son e s t ca g phan t Ma : 3 g Dan • . the t va a cu g phin t ma i ha n ch^ e s t cd g phan t Ma : 4 g Dan • cig thin g dudn t mo vdi c go g thin e s t ca g phin t Ma : 5 g Dan • to c a cu g phin t ma i vd c go g vuon e s t ca g phan t Ma : 6 g Dan • vd song song sU cd biet dd md dien thiet sat khdo Khi : i chi i Gh O sd viec cho gidi ly do Diiu du. day logi phdn diigc se dien thiet day tinh dgt Igi nhitng muon Id do dieu Du sdch. trong nay tri vi N BA C O N TOA I BA C CA . n H CAN T MO H QUAN Y QUA E S T CA G PHAN T MA : 1 g D^n N DIE T THIE A R O T A 2 11 i B a tg tron D Ca v C B h can i ha a cu m die g trun a l t lug n Ia J, I i Go . N a v Mi ta \\sgt n Ia B A a v D A h can c ca t ca J I a qu ? N IJM c gia l ti h Hin / a ? h han h bin h hin a l N IJM c gia l ti e d o na i tr i v di pha Na v M hi i Gia J I / / D Bi v ; BD / / ) (a n ne J I h quan y qua ) (a y e D / a D B/ / N M = ) (ABD n) (a >= () MN a v J I y dd i (ha N IJM g than h hin a l c dug n nha n die t Thie : 6 c i pha ckn h han h bin h hin a l N IJM g than h hin i Kh / b D AAB h bin g trun g difdn a l N M > <= N M = JI 6 df ti T thu o the B A a v D A m die g trun a l f ti l thi N, M ^< 3 11 i B a ABCD(AB/g than h hin a chil g phin t ma i ngoa d S m die t mo o Ch ' Uvldng di n Thiet dien tijy y thong thUcfng la hin h thang ADM N (AD // MN) . Thiet dien do la hin h bin h hanh kh i va chi khi : fM = MQ : trung diem S A [N = Ng : trung diem SD (ycbt). Bai 115 Cho mot hin h vuong ABCD, S la mot diem d ngoai mSt ABCD. Goi A', B' 1^ trun g diem SA va SB. Mo t mdt phang (y) lifu dpng qua A'B' cat SC va SD ta i C va D". a/ Hinh tin h thie t dien ma (y) cat hin h chop S.ABCD tao than h ? hi Goi I la giao diem cua A'D' va B'C. Ti m quy tich I kh i C vach doan SC. d Goi J la giao diem cua A' C va B'D'. Ti m quy tich J kh i C vach doan SC. HU'drng dfi n aJ Thiet dien la hin h thang A'BC'D' ma hai day A'B' CD' (ycbt). hi Quy tich I la hai ti a nam tre n dudng thin g xSy bo di nhOfng diem tre n do na m trong doan SSQ , vdi So = CB' o xSy va Sx c xSy = (SAD) n (SBC) (ycbt). c/ Quy tich J la doan : ^ SOi c SO = (SAC) o (SBD), ' trong do Oi = DB' ^ SO (ycbt). D' D Dang 2 : MAT PHANG CAT SE SONG SONG Vdl MOT HOAC HAI CANH CUA VAT THE TAO RA THIET DIEN Bai 116 Cho tuT dien ABCD va mot diem M bat k y tre n canh AC. Mo t mat phin g (P) qua M cMt cdc canh BC, BD va A D ta i N , R va S. Hay noi ro hin h tin h thie t dien trong mSi trtfcfng hcfp sau : a/ (P) song song vd i CD. b/ (P) song song vdi A B va CD. Gia i a/ Kh i (P) // CD = (ACD) n (BCD) (P)n(ACD ) = MS//C D (P)o(BCD ) = NR//C D Difng hin h bang each : keo dai MN : MN n A B = E noi E R : E R n A D = S Thiet dien nhan difoc la hin h thang MNR S (MS // NR) (xem h.l) . (h.l) ~-<§> ^ (h.2) 63 : i Kh / b : (BCD) o ) (ACD > = D C/ / ) (P : (DAB) o ) (CAB > = B A / / ) (P D C / / S M = ) (ACD n ) (P D C / / R N = ) (BCD n ) (P " B A / / N M = ) (CAB n) (P B A / / R S = ) (DAB n ) (P . (h.2) S MNR h h^n h bin h hin a l c ducf n nha n die t Thie 7 11 i B a B A t CD.Bie vdi ) g6c g (vuon o gia c trU B A 6 c D ABC n die f tu o Ch sg son ) (P g phin m&i e k M a Qu . x = M A i bd h din M m die y la CA . T a v , R, Ni ta D A a v D B , BC h can ? D ABC n die f tu t cS vk n die t thie h tin h hin m Ti / a . x a v a o the o d n die t thie a cu S h tic n die h Tin / b ? y na t nha n liJ o s i tr h Tin . nhat Idn i tr a gi t da S e d x h Din d . luan n Bi^ . sSn) o ch g duan o s : ^ (m ^ m = S e d x h Tin / d n dS g Htfdrn : 6 c a t , M a qu ) (a i kh y e D / a = ) (DAB o ) (CAB = B A / / ) (P : ) (BCD ) (ACD = D C / (P)/ B A / / N M = ) (CAB ) (P B A / / T R = ) (DAB n) (P D C / / T M = ) (ACD ) (P D C / / R M = ) (BCD o ) (P : a nOT n Ha D ABIC D MN//AB;NR//C R MNIN B . (ycbt) T MNR t nha a ch h hin ^ 1 c dua n nha n die t thie o d o D : 6 c a t s Thale y l h din o The / b = D C/ / T M : B A / / N M M A T M C A DC M C N M . CD = T M « . AB = N M ><= MA CA C A C B A C• v a o the h tin T MNR t nha a ch n die t thie a cu S h tic n di$ : 6 d c Lu (ycbt) (1 ) x - a x( = N MT.M = ) S(x = S : 6 c ) (1 g tron , x = M Ai kh y tha n Nha d a < X < 0 a < X - a < 0 . x - a a v x m a g khon ' so i ha o ch y Cauch T BD g dun p A ) (1 < ) x - a x( = ) S(x) x - a ( + X f) (2 — < ) S(x ><= 4 ] a ; [0 e — = x > <= x - a = x > A B ± CD (ycbt). S M A M b C D A C M N C M A B C A ^ SM = - x c ^>MN = -(c-x ) c De y AB ; CD co dinh khong gian v i tuT dien la c6' the. r:> (p = SIN U = rZOTcTr) = (const) Goi S la dien tich thie t dien bin h hanh MNRS, ta c6 : S = S(x) = SM.MN.sincp = ^^^^.x.(c - x) BDT Cauchy c j, ab.sin(f> f x * c - x ^ b(x) < 2 — ' 5 i S(x)< ab si n (? (1) Dau dang thiJc trong (1) xay ra<=>x = c - x<=>x = — 2 => max S = xay ra « • x = — due do M trun g diem AC) Osxs c 4 2 Vay max S = tiTong uTng (a) qua 4 trung diem 4 canh AC, BC, BD , A D theo thOt tU n«xsc 4 do; tiJotng ijfng MNRS la hin h tho i (ycbt). Bai11 9 Cho ttif dien deu SABC canh a va mot mSt phin g lOU dong (TI ) qua S song song vdi BC, Cat AB a M va AC 0 N . a/ Hin h tin h thie t dien ciia (T I ) vcJi tiif dien? b/ ChiJng to (TT) chufa mot difdng thSng co dinh. c/ Dat A M = X . Tin h tong y cac bin h phifcfng cac canh tarn giac SMN , ve do th i y = y(x). Gia i a/ Thiet dien nhan diioc la ta m giac SM N can ta i S (ycbt). b/ De y tif S diTng (d) // BC (d) CO dinh ^ \(d) - (SMN) o (SBC) Do do (TT) chiifa (d) co dinh (dpcm). c/ Ta C O tir dinh ly Thales, kh i M N // BC. MN AM r,. ^ AM X => = <=> M N = BC. = a. - = X BC AB AB a Xet ASAM = ASAN (vdi chu y SX ^ = SA?? = 60" ) SN = SM = VSA^ ^ AM^ - 2SA.AMcos60° 4 -2ax. -2 SN = Va ^ - x^ - ax Do do : y = MN ^ + SM^ + SN^ = x^ + 2a^ + 2x^ - 2ax => y = 3x^ - 2ax + 2a^ co do th i la cung Parabola ASB (ycbt). y 2a^ B a^ A / 2 2 r o a X 6 5 0 ^aiJ2 D m die t mo a v B O a cu m die g trun a l C, Oi ta g vuon O AAB o Ch g don u lu ) (a g phan t ma t Mo AC. vdi o gia e tru D O o ch o sa c gia n tar . S , R, N, Mi ta t lua n Ia B O, BD , AD , OA t ca ? S MNR c gia f tu h tin h hin o r i no y Ha / a h tic n die h Tin . x = M Ot da a v a = D O = C O = A O m the t bie o Ch / b ? y a cii t nha n Id i tr d gi a v X av din UUdng . (ycbt) S a v Mi ta g vuon g than h hin a l S MNR / a . MNRS h tic n die a l yi Go / b X = M O : t da i Kh 2 xV = S JM X - a S R + NM S .M = y > = X - a = N M i => S ;R t X - a2 f ,.^fil::^^ ^(4a-3x„3x ; 2 V 2 1 21 a 2 2 _ 2 V J . u_. 2 _ 2 V(y — = x > <= x 3 = x 3 - a 4 > <= ^ a — = y x ma y ha ^ a — < y 3 3 a (K Os 3 1 12 Bai i ngoa d S m die t mo y la ) Idn y da a l B (A D ABC g than h hin o Ch duo ta n die t thie m Ti . Q, P i ta C S , SB t ca D A h quan y qua i ( g phan . DP o Q A = J n da g HUcTn . (ycbt) D APQ i lo c gia f tij a l n die t thie i th g dUn h eac o The ) (SBD n ) (SAC = O S > = D B o C A = Oi Go ) (SAC c Q fA : y D e) (SBD : c PDO S e J = P D n QA : i Kh S ^ J ^ S iQ^Ps a l J h tic y qu c dug a t , dao n pha m la f ti a gi c Do . (ycbt) O S n doa 2 12 i Ba i Go . hanh h bin h hin a l D ABC y da c6 D S.ABC p ch6 h hin o Ch Tiicfng ti f nh u va y t a cun g xac din h diiac gia o diem : F = S D o mp(MNP) . Vay thie t die n cua hin h cho p S.ABC D vd i mp(MNP ) \k ngfl gii c MENP F (ycbt). D^ng 3 : MA T PHAN G CA T S E SON G SON G MQ\T MA T PHAN G CU A VA T TH E Bai 123 Cho hin h than g ABC D c6 ha i dd y A D v a BC . Go i S l a die m ba t k y nS m ngoa i ma t phin g (ABCD). Tre n doa n A B la y die m M . Qu a M diTng ma t ph^n g a son g son g vd i ma t phIn g (SBC). Xac din h tin h cha t thie t die n MNP Q m a a ca t hin h cho p S.ABC D ta o thanh . Gia i De y kh i (a) qua M v a don g thcfi (a) // (SBC ) !(a)//SB (a) n (SAB) = MN//SB , x,,o ^ f(a) n (ABCD) = MQ//B C => <(a)// BC• ! 1(a) n (SAD) = NP//BC (a) // SC => (a) r^ (SCD) = PQ// SC Noi lie n tie p cac can h giao tuye n ta o than h di/dng khep ki n tre n hin h cho p S.ABCD . B Ta duoc thie t die n l a hin h than g MNP Q (2 da y M Q // NP ) (ycbt) Bai 124 Cho hin h vuon g ABC D v a ta m giac deu SA B na m tron g ha i ma t phan g kha c nhau . Go i M la mot die m lifti don g tre n doa n AB . Qu a M k e ma t phln g a song son g v6i ma t (SBC) , a/ Churng min h thie t die n vd i hin h ch6 p S.ABC D nha n daoc lei hin h thang . b/ Ti m quy tic h giao die m I ciia ha i can h be n hin h than g no i d tren . Hvldng di n I S a/ Doc gia ta gia i v a chufng min h duac thiet dien l a hin h than g MNP Q (da y Id n la MN ) (dpcm). b/ Quy tic h I l a doan SI Q son g son g \6i AB va DC , tron g do DNII Q l a hin h bin h hanh (ycbt). Bai 125 \ p /.-.• \ ' / ^ \ \ / A / f Cho ti i die n SAB C m a dd y AB C l a mo t ta m giac vuon g goc ta i B . Ch o S A = 3a , A B = 4a , BC = 3a. Ti r mo t die m A' tre n can h S A sao cho SA' = x , ve mo t ma t phin g (P) son g son g vd i mat phIn g AABC . Ma t phin g na y ca t S B v a S C Ia n lifa t ta i B ' v a C . Tin h ch u v i v a die n tic h thiet dien c6 daoc the o x. Thie t die n l a AA'B' C (6 = 90° ) Theo din h l y Thale s t a c6 : SA' lA-B' = AB.- SA - 4a. — = — X 3a 3 {B' C = BC — - 3a. — = X SA 3a S A I = 5a.-3a 5 A ' C = AC — = V(3a2) + (4a2).- = — X 3 67 ) (ycbt x 4 = x — + x +x — =' ^^Tic : i v u Ch . (ycbt) ^ -x = x .-.x. - = C -A'B'.B' = ' S^T,C : h tic n Die 3 3 2 2 6 12 i B a a l M. AB m die g trun a l I i Go . a h can C SAB u de n die r tii o Ch . (SIC) / / ) (P g phin t ma t mo e v Ma Qu . AI n doa thie a cu yi v u ch h Tin hi ? i g h hin o the n die T tu t ci ) (P t Ma aJ . (ycbt) Mi ta n ca N APM a l n die t Thie /a : u sa u nh s Thale y l h din o the 6 c a T hi . SI = P MM AI Ax . V3 = i ^.J a 2 . x V3 = P M = N M x 2 = — a. = — SC. = N P : u t g Tifon a I A • ^ : a l n die t thie a cu yi v u ch o d o D . (ycbt) a V ; l)x + a V( 2 = x 2 +) .x 3 1/ ( 2 = y G SON G PHAN T MA I HA N CHA E S T CA G PHAN T MA : 4 g Dan i TH T VA N DE 7 12 i B a , Ax i Go . CD a v B Aa l y da i ha 6 c D ABC g than h hin t mo o Ch g phin t ma g tron m na g khon a v u chie g cun g son g son g thSn g dudn , C, B' , A' i ta t li/g n Ia n tre g thin g dudn a nOf 4 t ca y k t ba ) (P g phin . dien t thie a cu h tin h hin o r h din y Ha /a g than h hin o che g difdn c ca a cu m die o gia a l t lifo n Ia ' Oa v Oi Go hi . Dt / / z C / / y B / / x A / / ' OO ot i Gia ) (1 ) Dt ; By)//(Cz ; (Ax y 6 D /a ' A-B = ) By ; (Ax n ) (Pj ' CD = ) Dt (Cz; n) |(P ^ Bai 128 Cho hinh binh hanh ABCD. Tif nhOfng dinh cua hin h bin h hanh nay ta ke nhiJng niia dudng thang Ax, By, Cz, D t song song cung chieu khong nk m trong ma t ph^ng ABCD. Mp t mat phang (P) cat 4 di/cfng th^n g tre n ta i A , B', C , D'. a/ Chutng minh (ABA'B ) // (CDC'D'); (BCC'B') // (ADD'A') b/ Tut giac A'B'C'D' la hin h gi ? Giai a/ De y : fAB// CD A'A'// C'C (ABA'B) // (CDD'C) (dpcm). TircJng tif : (BCC'B') // (ADD'A ) (dpcm). ,, ^ , !(P)n(ABA'B') = A'B' ..r.. „ r^.j^. l(P)n (CDC'D'). CD ' ^^BZ/C D va { ABCD ' la hin h bin h hanh (ycbt). Bai129 Cho hai hin h binh hanh ABCD va A'B'C'D' khong nkm cung trong mat ph4ng. a/ ChiJng min h (AA'B) // (DD'C). hi Mot mat ph^ng bat ky (P) c^t AB , B'A , CD ' va CD ta i M , N , R, S. Ha y n6i ro hin h tin h thiet dien nhan dUcfc ? Hvldng dltn DQC gia tif giai. Bai 130 Cho hin h chop S.ABCD c6 day la hin h bin h hanh. Goi I la trun g diem SA va M la diem lifu dpng tren canh AD . Xe t N la diem thupc SM thoa I N // (ABCD). a/ Tim quy tich cAc diem N . b/ Trong cau nay gia suf M chay tren ca chu v i hin h binh hanh ABCD. Tim quy tich cua N . Giai a/ Ta CO : I N // (ABCD) I N e (SAD) I N //A D N e I J : la dudng trun g bin h cua ASAD. K hi : fM s A N s I 1M = D=> N = J Dao la i lay diem tuy y N Q e IJ . => INo//(ABCD ) Vay quy tich N la doan I J (ycbt). b/ Goi E, F thuf tU la trun g diem SC va SB. TifOng tif ta c6 quy tich cua N la chu v i cua thie t di^n hin h bin h hanh IJE F (ycbt). 69 THG OJCIN T MO \lQl C G6 G THAN E S T CA G PHAN T MA : 5 g Dan 1 13 i Ba n die t thife' m Ti . AB 1 C B; AB L _ A S 6 c D S.ABC p ch6 h hin o Ch ? h than o ta p cho h hin t ca i Gia ) (AB 1 ) (a , Ma qu ) (a y e D C (a)//B A "(a)//S C (a)o(ABCD)=^MN//B : o d o D C PQ//B = ) (a)o(SBC A PM//S =) ^CSAB Icajr ) bupc g rin i gh n ca g (khon N Q =) (SCD o ) (a - D^: y tha M P a v P Q, NQ , MN n tuye o gia c ca p tig' n lie i No . kin p khe g chiin i (vd P MNQ g than h hin a l c dUd n nha n die t thie y Va . ben h hin g tron a nh m da o t c difa , QP) > N M : y da i ha 2 Bai13 g vuon n ca C AB c gia n tar ; AB 1 A S 6 c D S.ABC p ch6 h hin o Ch do Mi ta B A 1 a t ma a m n die t thie m Ti . AB n doa g tron g don u Itr ? h than o ta p cho i Gia : a v Ma qu ) (a : O C aT ) (SAB e A S / (a)/ > = B A 1 ) f(a ) (ABC c C A / (a)/ BC 1 ) (a 1 A MQ//S = ) (a)n(SAB C MP//A - ) («)n(ABC . APMQ a l c dira n nha n die t Thie 3 13 i Ba C n ItJ y da g than h hin a l D ABC y da 6 c D S.ABC p cho h hin o Ch a qu a t ma a m n die t thie m AB.Ti n tre g don u lo m die t mo a l Mi Go ? h than o ta D S.ABC p cho h hin i Gia < a v Ma qu ) (a y e D D B / (a)/ > := D C 1 ) (a , BD X DC A S / (a)/ > = D C i ) (a , SA 1 D :C M Gia i Mat phin g ((5) 1 A B ta i I do do : :(P)//Ax = (BAM)o(AMN) : :((!)// A y = (ABN ) o (BMN ) r::> '(P )^(BAM ) = IL//A x '((3) o (AMN ) = J K // A x '(P) ^ (ABN ) = I J // B y 1 (P) o (BMi\ L K // B y i K Thiet dien nhan diTcrc kh i ((5) X (AB) tao thanh vcri tijf N dien ABMN la hin h binh hanh IJK L (ycbt). Dang 6 : MAT PHANG CAT S E VUONG GOC Vdl MAT PHANG LIEN Odl CUA VAT TH^ L PHirONG PHA P Ca sa cua phaang phap la quy doi bai toan dimg thiet di^n vuong gdc v&i mat phang (a) thanh bai toan dying thiet di^n song song v&i mqt dU&ng thang, ma dU&ng thang do vuong gdc san v&i mdt phdng (a) da cho trong gid thiet tim thiet di^n : sau do ap dung dinh ly giao tuyen song song va phuong phap difng thie t dien (ycbt). n. CA C BA I TOA N C O BA M Bai 135 Cho hin h chop S.ABCD c6 SA 1 (ABCD); SA = a va ABCD la hin h vuong canh a, ta m O. DiTng va tin h dien tich thie t dien qua SO va vuong goc vdi (SDC). Gia i Qua O ditog EF // A B i . (SAD) vdi E e AD ; F e BC ^ EF 1 (SAD) Do do thie t dien nhan duac la ASEF {t = 90") ycbt. Goi S la dien tich thie t dien do : => S = -EFSE -= - a 2 2 l l 2 Bai 136 1 (ycbt). Cho hin h chop S.ABCD c6 ABCD la hin h vuong ta m O, canh a. Dudng cao hin h chop la SO. Goi I la trun g diem CD va (p = [ SI ; (BCD)] ((p > 45°). Xet mat phSng a qua A B va vuong goc (SCD). Xdc dinh thie t dien ciia a va hin h chop va tin h dien tich thie t dien theo a v^ (p. Gia i Goi J la trung diem A B =?. CD 1 (SIJ) Ma CD c (SCD) => (SIJ) J_ (SCD) Ha J H 1 (SCD); H e SI Diing EF qua H va EF // A B => EF 1 (SIJ) vdi E G SD; F e SC Thiet dien nhan difoc la hin h thang can ABE F (day \dn AB; dUcfng cao JH ) Trong AIH J (fi = 90") 7 1 ^ SI--—p ( 2cos o asin = p Usinc --- J •iH p acosc = ) IJcos <= ) tu c g6 p (2( p acos2i - = F E > <= : a l n die t' thie a cu S h tic n Die ^ a 1 1 cos2= ) (1 ) BC//(P ^ P MQ//N ) MQc(BAD • : tif g Tuon ) (CAD c P N ) (P / / D A > = P M, MQ / / D A = ) (CAD n ) (BAD )(2 P N = N M > = i tho h hin a l Q MNP )(3 C B / / N M : 6 c a t C AAB g Tron N A N M Nhif vay mSt phSng (P) c6 tinh chat tren la ton tai va ta c6 each difng mat phang (P) nhu sau: AN AD Chia AC thanh hai doan theo ty so : (7) CN BC Prong maet phaung (ABC) kcu MN // BC Trong macl phaung (CAD) kcu NP // AD Difng mat phang chuTa MN va NP cat BD tai Q, do chinh la mat phing (P) phai difng v^ MNPQ la hinh thoi. That vay : Vi : MN // BC NP//AD : MN AN BC AC NP CN AD AC <=> AC.MN = AN.BC <=> AC.NP = AD.CN (8) (9) Tif (7) va (9) = Tif(7) va (10) AC.NP = AN.BC >MN = NP. (10) Dong thcri : ,P)n (DBC) = PQ / / MN ^^^^ (P)o(BAD)=MO//NP Bai 138 (DAI HOC SLf PHAM - 1976) Day cua hinh chop S.ABCD la hinh chuT nhat ABCD c6 AD = BC = 2a va AB = DC = 5a. Goi H la hinh chieu vuong goc cua dinh S tren day ABCD. Bie't rang diem H nam tren doan thing IJ, trong d6 I la trung diem cua AD va J la trung diem cua BC. Cho SH = 2a; I H = a. 1/ Chijfng minh r^ng tam giac ISJ vuong. Tinh gia tr i g6c nhi di?n tao bdi hai mat ph^ng (SAD) va (SBC). 21 Tinh dien tich xung quanh va the tich hinh chop. 3/ cat hinh ch6p bdi mot mat phing vuong goc vdi I J tai diem K. H6i thiet dien la hinh gi ? 4/ Gia sijf IK = x. Goi y la di§n tich cua thiet dien MNPQ (M, N, P, Q la bon dil m cua mat phing thiet di$n 1 (ABCD) vuong goc cat bon canh cua hinh ch6p). Tinh y theo a va x (x6t trudng hgp K nkm giaa I va H hoac K nam giffa H va J). Giai 1/ Xet AISJ ta c6 SH la difcrng cao : JHI.H J = HI.(AB - HI ) = a(5a - a) JSH^ = 4a2 SH^ = HI.H J => AISJ vuong goc tai S (dpcm). Ta c6 :(SBC) n (SAD) = x'Sx S J 1 B C ^ S J 1 x'Sx ^ |SI 1 AD => S I 1 x'Sx => = a = [(SBC); (SAD)] = xS ^ => = a = 90" (dpcm). 2/ Dien tich xung quanh S^, cua hinh ch6p S.ABCD : S,, = 2dt(ASAB) + dt(ASBC) + dt(ASAD) 73 R S X D C X i = ) dt(ASCD = ) dt(ASAB2 : 6 c a T . CD n tre S a cu u chie h hin a l R o d g Tron ^ 5a = 2 4a + ^ a = ^ SII + ^ HR = ^ SR aS = SR >= ^ = s aV X a 5 X - = ) dt(ASCD = ) dt(ASAB 2 2 ^j5 2a = 2ayfE X a 2 X - = J S X C B - = ) dt(ASBC 2 2 s V ^ a = s aV X a 2 X ^ = I S X D A X - = ) dt(ASAD 2 2 ) (ycbt s V ^ 8a = s V * a'" + s V ^ 2a + S V ^ 5a = >= : D S.ABC p ch6 h hin a cii V h tic e Th . (ycbt) = ^x2ax5ax2a = ^dt(ABCI))xSll = V t thie p ha g trUcfn i ha 6 c a t D C a v B A n tre N, Ma cii i tr i v y tij y c D /3 . AD / / C B/ / ) (MNPQ > = J I ± ) (MNPQ n die t Thie , SC c (hoa A S , SD , CD , AB t cS ) (MNPQ g phan t Ma i ta ) SJ c (hoS I S t ca Q P . Q , P , N , Mf ti f thti o the ) SB : O C a t , K' D A / BC/ / / Q P / / N M a v N M a l y dd i ha g than h hin t mo a l Q MNP y Va m la ) (SIJ g phan t ma n nha p cho h hin y tha y e D . PQ . NP = Q M > = g xufn i do g phan t ma . (ycbt) n ca g than h hin t mo a l Q MNP n die t Thie a 2 = N M 6 c a T. HI n tre d K : i TH • / 4 x 2ax K SHxI , „„ ' KK . IK I II I SI I II x a- K U ' SK P Q x 2 = ' KK « - ) 2ax^^2(a-x D A a a 1 11 I S : a l Q MNP g than h hin a cu y h tic n die y na p hg g Tri/dn „ a 2 + x 2a-2 N QP+M x 2 X = K K X ^ -^ = y 2 2 ^ Trirdng hop nay dien tich y ciia hin h thang MNP Q la : x - a y =^^.KK - = -l -+ 2a 5 a - X 2 y = rx + 3a^ 4 5a ' = -(-x' ^ + 2ax + 15a^). 8 Bai 139 (BA I HOC Si/ PHA M - KIN H TE - TA I CHIN H - 1977) Trong mat phftng (P) cho hinh chCf nhat ABCD c6 cac canh AB = 2a; A D = a. TiT A ke di/dng vuong goc vdi mat phang (P) va tren do lay doan AS = 3a. 1/ Tinh cac doan SB, SC, SD theo a. 2/ Chijfng min h rkn g cac tarn giac SDC va SBC la cac tam giac vuong. 3/ Mat phang chijfa DC cat SA ta i M va SB tai N . I lay xac dinh tin h chat cua tOf giac CDMN. Dat A M = x. Tin h dien tich cua tuf giac CDM N theo a va x. Gia i 1/ Ta CO : . SB' = SA' + AB ' = 9a' + 4a' = 13a' ^ SB = asf\3 (ycbt) . SC' = AC ' + SA' = AB ' + BC ' + SA' => SC' = 4a' + a' + 9a'= 14a' => SC = as/u (ycbt) . SD' = AD ' + SA' = a' + 9a' = 10a' => SD = aVio (ycbt). 2/ Theo dinh ly 3 difdng vuong goc, ta c6 : SA1(1')1 3a - X AD ID C I SAl(P ) . SD 1 DC => ASDC vuong ta i D (dpcm) AB 1 BC => SB 1 BC => ASBC vuong ta i B (dpcm). 3/ Tinh chat cua tiJ giac DCNM . Goi (a) la mat phang qua DC; (a) cSt SA ta i M va cat SB ta i N . DC // AB ^ M N // DC // AB DC 1 (SAD) =i. M N 1 (SAD) =:> M N 1 M D Vay DCN M la mot hin h thang vuong (hai day DC // M N chieu cao la DM ) (ycbt). Dien tich cua tuf gidc DCN M la : dt(DCNM) = '^"^""^ ^ X D M D M = N/AD'+AM " = sfa" + X' Ta CO : M N S M AB SA M N = 2a(3a-x) 2(3a-x) 3a 2a + (3a -x ) dt(DCNM) = dt(DCNM) = 6a - x Va" + X" (ycbt). 7 5 A V H CAC G KHOAN G DAN C C A 8: de Chuyen G CHUN C GO G VUON G DirdN KG TRON H CAC G KHOAN G DA« C CA l Dpjf C XA A t Tina : 1 l to^ T MO N DE M DIE T MO Q T H CAC G KHOAN H TIN : 1 g Dan P PHfl G PHirOM L : u sa f nhi n ba r cc c bud 2 n hie c thif n ca p pha g phifan a cii sa r Cc pt mS t mo g difti h eac g ban , (a) i vor H Mc go g vuon n doa h din c Xa : i B • ] (a) ; d[M = H M> = ) (d 1 H Ma h , (d) n tuy§' o gia o the sa c ho h hin y l h din c ca p phe g bSn h tin c difa H M = ] (a) ; d[M : 2 B • doan ddi do la goi con (a)] d[M; each Khodng : 1 i chi i Gh O goc. vuong diiimg 3 ly dinh trong goc vuong the hay thiic cong b&ng MH t'lm con ta nay Sau : 2 i chi i Gh O tich di$n phap phUctng (xem the vat tich) dien (hay tich tich). the phap phUctng va M ILBAITAPCOBA 0 14 i B a 2B = B A a v Bi ta g vuon c gia m ta a l y da C S.AB p cho h hin o Ch . (SAC)] ; d[B h Tin . (ABC) i Gia ) (SAC : c A S; (ABC) ± A S o D . BC n tuye o gia o the ) (SAC 1 ) (ABC H i ta C A 1 H B a H . (SAC)] ; d[B = H B >= : 90°) =i (f C AAB g tron g luan e thur e H 5 1111 1 • + •^ 4a ^ 4a ^ BC ^ AB ^ H B = ^ HB" 4 3 = B H « = ) (SAC ; d[B : y Vaa 2. (ycbt) 1 Bai14 Theo dinh ly Thales M N B M 2 S A ' B S ~ 3 M N = -SA = - h (2) Thay (2) vac (1) => I H = -.- h = - h 2 3 3 Vay d[I; (ABC)] = ~ (ycbt). O Bai 142 Cho tuf dien ABC D c6 AB = CD ; B C = BD con A C = AD. Dang A H ± (BCD), a/ Chiifng minh H nam tren trung tuyen BI ciia ABCD. b/ Xac dinh d[A; (BCD)]. Giai a/ Theo tinh chat dudng trung tuyen cung la dudng cao ha til dinh cua tarn giac can, ta c6 : j™ ^ ^ => C D 1 (ABI); ma C D c (BCD) [CD X BI => (ABI) 1 (BCD) Ha dudng cao A H cua tuf di^n ABC D =^ A H 1 C D => A H c (ABI) hay H e BI (ycbt) b/ Theo cau tren => A H = dlA; (BCD)] (ycbt). Bai 143 Cho hinh chop S.ABC c6 SA 1 (ABC) va AABC deu canh a. a/ Tinh dlB; (SAC)]. b/ Gi a sOf SA = h. Tinh dlA; (SBC)]. Giai a/ Nhan xet thay (ABC) 1 (SAC) theo giao tuyen AC . Ha B H 1 A C tai H. d[B; (SAC)] = B H = (dtfdng cao A deu) b/ Goi M la trung diem BC , ta c6 : Mk BC c (SBC) => (SAM) 1 (SBC) theo giao tuyen SM. Dirng A K 1 SM => A K 1 (SBC) d[A; (SBC)] = A K 1111 1 Trong do : A K ^ SA' ^ AM ^ h^ J _ 4 h^^Sa ^ A K ^ 3aV 4h' Vay d[A; (SBC)] = , (ycbt). 4h^ 77 4 14 i B a 90 ^ C XB C XB a v a 3 = D S = B S = A S 6 c D S.ABC p cho h hin o Ch . 2a = D B t bie i kh ] (ABCD) ; d[S h Tin i Gia h kin g dudn n tro g ducfn n tre a D, B > = " 90 = KDC. - C XB : y e D . AC m die g trun a l Hi Gp D H = B H = A H >= 3 . SD = B S = A S6 ci la t thie a Gi . .\ABD p tie i ngoa n tro g di/dn e tru a l n So d o D /) (ABCD E E ) (ABD 1 HS D^^^^^^.^H S = l (ABCD) ; dIS >= ) 90" i (f A ASH g Tron 2^[2.a , ^ -/to - ^ a' ^ ^|9a'^ ' HA'' ^ VsA'- = H S >= C . (ycbt) a . 2 V 2 = l (ABCD) ; dIS y Va (HA G THAN G Dl/CJN T MO T TL H CAC G KHOAN I TIK : 2 g Dan G PHAN T MA T MO l Vd G SON G SON P PHA G PHirON L (« / / ) (A g thSn g ducrn f t£ h eac g khoan m ti p pha g phiftrn a cii o s o C : c biTd i ha m la a t) (u n de a cu t vifi U U h tin a tho i pha g (nhifti y y tij M m die t mo y La : i B • . («) / /) ((5 n tre y ha ) (a / /) (A n tre ) nay u sa n toa h tin h trin a qu ycb g tron m ti n ca h eac g khoan a l H M => (a) _ J H Ma H : 2 B • . biet) =* d[By; (CAx)] = BA = 2a (ycbt) Tuang tu ha AH 1 BC tai H d[Ax; (CBy)] = AH = 2a (ycbt). Bai 146 Cho Ax, By la hai tia vuong goc vdi mat phang hinh thoi ABCD d cung mot phia. Tinh d[(DAx); (CBy)l biet hinh thoi canh a c6 dien tich bSng a^Vs Giai D6 y thay hai mat phftng (DAx) // (CBy) va chung ciing vuong g6c v(Ji mp(ABCD) theo cac giao tuyen AD va BC theo thil tu do. Chon A e AD va ha AH 1 BC; tai H AH = d[(DAx); (CBy)] Lai CO vdi S la dien tich hinh thoi ABCD canh a. S=i.AH.BC « ^ = l.AH.a <^ AH = ^ a 2 4 2 2 Vay : dl(DAx); (CBy)] = (ycbt). y D Dang 3 : TINH KHOANG CACH TLT MOT DIEM DEN MOT DUCING THANG LPHU'ONGPHA P * PPi : Co so cua phaong phap thuf nhat de tim khoang each tiS diem M den mot dudng th^ng (A) trong khong gian ta can thiTc liien hai bUdrc : • Bi : TiJ di6m M can tim khoang each trong khong gian ha dadng vuong goc M H vdi dudng thang (A). • B2 : Do dki M H = dlM; (A)] la khoang each can tim. (Xem h.l) M (\ \ (h.l) 0 Ghi chii : Thuc te ngitai ta con thuc hien tinh khoang each til diim den mot diiang thang nhu sau, theo 2 phuang phdp nUa : • PP2 : Tim mat phang,(a) qua M va vuong goc vdri (A) tai H => JMH = d[ M ; (A)]| (xem h.2) • PP3 : Tim each ghep (A) la di/orng thuf (3 ) nfim trong (a) va |d[ M; (A)] = MH | la do dai difcfng xien (T)' (xem h.3) •k PP4 : Thinh thoang ngUcfi ta con tim d[ M ; (A)] bSng cong thuTc tinh dien tich hinh ph^ng (phuang phap dien tich). 79