🔙 Quay lại trang tải sách pdf ebook Ôn Luyện Bồi Dưỡng Học Sinh Giỏi Hình Học Không Gian Ebooks Nhóm Zalo TRUdNG TRUNG HQC PH(f THONG CHUYEN CHU VAN AN - HA NOI 0454 L PHANHUYKHAIICHUBIEN) CHlf XUAN DONG - HOANG VAN PHU - CU PHUQNG ANH On • BOI DUONG HOC SINH GINHA XUAT BAN TONG HOP THANH PHO HO CHf MINH ' oCdrl not ctdu ' Nham giup ca c em hoc sinh trung hoc pho thong noi chung, ca c ban hoc sinh gidi Toan noi rieng c6 them tai lieu de hoc t^p tot mon Toan trong nha trUdng, cung nhi/chuan bj day du kien thufc phuc vu cho cac ki thi tuyen sinh vao Dai hoc, Cao d^ng va ca c ki thi Olympic ve Toan cac cap, nhom giao vien Toan trudng PTTHChu Van An - Ha Noi chung toi bien soan hai bo sach sau: - Bo sach chung gom 6 cuon: /. O n \uyen boi dudng hoc sinh gidi hinh hoc khong gian. 2. On luy$n boi difdng hoc sinh gidi hcim so. 3. On luySn boi difdng hoc sinh gidi phuang trlnh bat phuang trlnh. 4. On luySn boi dUdng hoc sinh gidi hinh hoc giai tich. 5. On luyen boi difdng hoc sinh gidi tich phan, to hap va so phifc. 6. On luy$n boi difdng hoc sinh gidi lucfng giac, bat ding thifc, gia tri Idn nhat, gia tri nhd nliat.. - Bo sach luyen thi ve mon Toan bao gom cuon: /. Bdi difdng hoc sinh gidi lifdng giac. 2. Boi difdng hoc sinh gidi cac bai toan ve day so'. 3. Boi dudng hoc sinh gidi ham so va da thifc. 4. Boi difdng hoc sinh gidi so hoc. 5. Boi dudng hoc sinh gidi hinh hoc to lidp. 6. Boi dudng hoc sinh gidi bat ding thUc va cac bai toan cUc tri. Chung toi cho rang hai bo sach nay se dap ilng duoc mot so li/cfng I6n ban doc. C a c ban hoc sinh phd thong trung hoc noi chung, ci c ban hoc sinh gi6i Toan noi rieng, cung nhi/ ca c thay c6 giao day Toan deu c6 the tim dUdc cho minh nhufng dieu CO ich trong ca c bp sach nay. Mcic du tap the tac gi5 da ra't nghiem tuc trong qua trinh bien soan, nhiing do dung liiong cua bO s^ch qu^ I6n nen chi c chi n trong Ian dau ra mit ban dpc khong the tranh kh6i nhQng khiem khuyet. Mong nhan duac gop y cua ban dgc xa gan de bp sach tot hon trong ca c Ian tai b3n tiep theo. Xin chan thanh c3m on! Cac tac gi5 Nha sach Kliang Viet xin Iran trong gi&i thieu t&i Quy doc gia va xin lang nghe mgi y kick dong gop, decuon sack ngay cang hay hem, bo ich hoTi. Thu xin gi'd ve: Cty TNH H Mgt Thanh Vien - Djch V u Van Hoa Khang Vi?t. 71, Din h Tien Hoang, P. Dakao. Quan 1, TP. HC M Tel:(08)39115694-39111969 -3'< i i968 -39105797-Fax : (08)39110880 Hoa c Email: [email protected] . , Cty TNTIII MTV DWII Khimg Viel DU6N G THAN G V A WA T FHAN G TRON G KHON G GIA N QUA N H E SON G SON G I. TO M TA T L i TH U YE T 1, Ca c tien de ciia hinh hoc khong gian: _ Qua hai diem phan bic t trong khong gian c6 mot va chi mot du'cfng thang ma thoi. _ Qua ba diem khong thang hang c6 mot va chi mot mat phang ma thoi. _ Mo t du'cfng thang co hai diem chung vdi mot mat phang thi nam tron trong mat phang ay.;,'., .•„;-)~ \ • ,-MA ifiOi b lid !M(,) 4. Di]f(Vng thang song song V(?i mat phang: iif^hTa: Di/dng thin g a goi la song song vdi mat phang (P) va k i hieu a // (P) neu nhiT a va (P) khong c6 diem chung. ! Dinh l i 1: (Tieu chuan song song) 11)^4 Difdng thang a (khong nam trong mat phang (P)) song , song vdi (P) khi a song song vdi mot diTdng thang b bat k i cua (P) (hinh 4) Dinh l i 2: Gia stir du-dng thang a song song vdi mat phang (P). Khi do mpi mat phang (Q ) di qua a ma cat (P), thi giao tuyen cua hai mat phin g (P) va (Q) se song song vdi a (hinh 5) 5. Hai mat phang song song 1 EUl^, > Hinh 4 (P) n (Q) = A // •V inn Hinh 5 Dinh ni^lua: Hai mat phang (P) va (Q) goi la song vdi nhau, ne'u nhiT (P) ( Q ) khong C O diem chung. ^ Dinh li I: (Tieu chuan song song) ' ' '* " —•^^'^''^^^bT Ne'u a va b la c$p diTdng thing giao nhau cua (P); con a' va b' la cap dtfcfng thang giao nhau cua ^<.>\ (Q) , sao cho a / / a' ; b / / b' ; thi khi Hinh 6 4 Ctij TNIIII MTV DWH Khang Vm g do (P) va (Q ) se la hai mat phang song song (hinh 6) Dfnh li 2: Neu (P) va DViduQ ^' ' ( Q ) la hai mat phang 3^,!%, ^ song song, va (R) la mot mat phang sao cho (R) cat (P). Khi do (R) cung cat (Q ) va giao tuyc'n A cua (R) vdi (P) se song song vdi giao tuyen A' cua (R) vdi (Q) (hinh 7) , Binh l i 3: Cac mat phang song song djnh ra tren hai cat tuyc'n nhffng doan thang ti le (hinh 8) Dpcm. B a i 5. Cho ba diem A, B, C cung phia doi vdi mat phang (P). DiTc^ng thang BC cii (?) tai mot diem D. ChiJng minh rang it nha't mot trong hai diTdng thang A B , AC , cat (P). ^ ; Gia i Neu A, B, C thang hang va do BC cat (P) tai D nen hien nhien ca A B va AC deu cat (P) tai D (hinh 14). Neu A, B, C khong thang hang. Khi do goi (Q) la mat phang xac dinh bdi A, B, C. Do A e (Q) ma A € '= (P) nen (Q) ^ (P). Ma t khac (Q) va (P) c6 chung nhau diem D, nen (Q) n (P) = A va D e A. R6 rang I, trong (Q) thi AC va AB khong the ciing song song v di A (do qua mot diem A CO duy nha't mot dufdng thang song song vdi A). Hinh 14 ,-( .(|...-:,.f,, \f M l , von V ay mot trong hai du-cfng thang AB , AC phai cat A (tiJc la cat (P)) ==> dpcm. B a i 6. Cho hai du-dng thang a, b cat nhau tai diem M . Hai du-dng thang c, d khong CO diem chung va tu-cJng iJng song song \6i a, b. Chrfng minh c va d cheo nhau. . ; \i , -v H*II»UJ ^iuj' y i « . i ' ' I ' I , t i.rvi .'ill,. ' ! .' I , I, Cty TNIIII MTV DVVII Khang Viet Gia i Goi P va Q la hai mat phang tu-dng iJng xac djnh bdi a, c va bdi b, d. < m'hih tOni KIH» th 'v Do c, d khong c6 diem chung (P) va (Q) la hai mat phang phan biet. Ma t khac (P) va (Q) c6 diem chung la M , nen (P) n (Q) = A, d day M e A.Trong (P), ta co a // c ma a n A = M , nen c n A = I . TiTdng tif trong (Q), ta c6 d // b ma b n A = M , nen d n A = J. V i c va d khong CO diem chung nen hien nhien >,V, q : Ta CO c e (P), d n (P) = J va J ?t c, nen c va d la hai d^dng thang cheo nhau => dpcm. j/[ DfU r/.vi j Qn;:;fk-) isucj ujit uU:j.;<')ri • mhi:i r Bai 7. Cho bon diem A, B, C, D khong dong phang. Goi I la diem tren nufa diTcJng thang BD nhu^ng khong thuoc doan BD. Trong (ABD) ve mot du-cJug thang qua I va cat hai doan AB , A D Ian lu-dt tai K va L. Trong (BCD) diTdng thang qua I va cat hai docin CB, CD tiTdng iJng tai M va N . Gia su" B N n D M = O,; BL n D K = 02 ; L M n K N = J. Chu-ng minh ba diem A, J, O, thang hang. Gia i ,iHi bb ah J = Ul\n i^lA [\'' Theo gici thiet ta c6: —-^'•"•*r""/* BNnD M = 0| =>0 | G B N ' -yi'i / ; =^0 | e (ABN) fLM n K N = J =^ J e K N 'MA 9 1 -"'^ I, ivy(C (1) n> J € (ABN ) (2) I DTnhien A e (ABN ) (3) Tir(l) , (2), (3) suy raO,,J, A ^ cung thuoc (ABN ) (4) TiTdng tir, 0 | G D M O, e (ADM)(5 ) J G L M =^ J e (ADM ) (6) O,, J, Aci:mgthuoc(ADM ) (7) Hai mat phang (ABN ) va (ADM ) c6 chung I nhau diem A, nen chac chan (ABN ) n (ADM ) = A fg^j n >f <- m Tir (4), (7) suy ra A, 0|, J thang hang. B a i 8. Trong msil phSng (P) cho hinh thang ABCD (BC // AD ) va mot diem S € (P). Ma t phcing (Q) di dong chiifa diTcJng thang SB va gia su" cat SC, SD tifdng Boi diCdng HSG Ilinh hoc khoruj cjian - Phan IIiuj Khdi dng tai M, N. Mat phang (R) di dpng chita diTdng thang CD vii gia su" ciit SA, SB tiTdng iJng tai P vii Q. 1. Chi?ng minh MN, PQ luon di qua mot dicm co' djnh khi (Q) vii (R) di dong nhu" tren. 2. Goi I = AN n BM, J = CQ n DP. ChiJng minh duTdng thang noi I , J luon di qua mot diem CO djnh. ':].*••!> 3. Goi K = A M n BN, L = CP n DQ. ChiVng minh r^ng cac diTcJng thang noi K, L cung luon di qua mot diem co dinh. • ivfi :;i,<(0;ii v.fc;,!ffeiU i,•.{.e, siiSui . • Giai ' ' n f.' - i - 1. Gi3 sur AB n CD = E, vay E co djnh. Nhu" vay M, N, E cung nam Iren hai m;ll phang (ABMN) vii (SDC), do do M, N, E nam tren giao tuye'n ciia hai mat phiing ii'y, vi the M, N, E thang hang. Vay ciic du'dng thiing MN luon di qua dicm co dinh E => dpcm Hoim toan tu'dng tu" ta co P, Q, E cung nam tren hai mat phang (DCPQ) va (SBA), do do P,Q, E nam tren giao tuye'n ci'ia hai miit phiing ify, vi the' P, Q, E thang hiing.'^-^''»^^ * r=, ,vr i Nhu" viiy cac du'dng thang PQ ciing luon di qua diem co' dinh E => dpcm. * 2. Vi AN n BM = I , do do noi rieng I e (SAD) (vi I 6 AN, ma ANe (SAD)), I e (SBC) (vi I e BM). TiT do I thuoc giao tuye'n cua (SBC) va (SAD). Vi CQ n DP = J => J e CQ =^ J e (SBC), J e DP ^ J e (SAD), vay J thuoc giao tuye'n ciia (SBC) vii (SAD) suy ra I , J, S thang hiing. NhU" the dirdng thang noi I , J luon di qua diem co' dinh S => dpcm. 3. Giii su" AC n BD = O => O co dinh. Vi AM n BN = K ^ K e AM => K e (SAC) K e BN => K e (SBD), vay K nam tren giao tuye'n cija hai mat phang (SAC) va (SBD). Tu-dng tir, do CP n DQ = L cung nam tren giao tuye'n cua (SAC) vii (SBD), con O cung thuoc giao tuye'n cua (SAC) vii (SBD) (do AC n BD = O) => K, L, O thang hang. Noi each khac cac du^dng thang noi K, L luon di qua diem co dinh O ==> dpcm. ' ' • W''^^ • • - , CUj TNIin MTV nVVII Khancj Viet Bai 9. Cho tiJ dien ABCD. Goi A,, B,, C,, D, tu-dng tfug ha trong tiim cua cac tam giac BCD, ACD, ABD va ABC. ChiJng minh rang AA,, BB,, CC,, DD, dong qui tai diem G vii ta co: fb'ruA. > ; / / , ^ AG BG CG DG 3 ' AA, BB, CC, DD, 4 • • * v., Giai • ( Goi A|, B| IMng rfng lii cac trong lam cac tam giac BCD, ACD vii M la trung dicm cua CD. Thco tinh chii't trong tam tam giac MA, _ MB, _ 1 ta co: MB MA Do do Iheo dinh li Talet dao, ta co A.B, // AB. A,B, MA, 1 Tiifdo:—!-^ = (1) n\ AB MB 3 Trong (AMB) ro rang BB, n AA, = G. Vi A,B| // AB, nen lai iheo dinh li Talet, la co (diTa viio 1) A^^A^^_1 ^ AG ^3 GA AB 3 AA, 4 AG' 3 TiTtJng ur, la co CC, n AA, = G' va = - AA, 4 AG" 3 DD (4) i n AA, = G" vii AA, (3) I Tif (2) (3) (4) suy ra G, G', G" trung nhau.luTc la AA,, BB,, CC,, DD, dong AG BG CG DG 3 , qui tai G va AA, BB, CC, DD Chii y: = — => dpm. 1. Diim G noi tren goi lii "trong tam ciia Itf dien ABCD". No lii sir md rong cua khai niem trong lam ciia tam giac. 2. Ta CO each khac xac dinh trong tiim ciia lufdien ABCD nhiTsau: Cho {({ dien ABCD. Goi I , J, E, F, K, H liin lircn lii trung diem ciia AB, CD, AC, BD, AD, BC. Khi do IJ, EF, KH dong qui lai G vii G ' ' chinh lii trong tam cua liJdien ABCD. Boi (hcdng IISG Hhih hoc khong gian - Phan Iluy Khdi ofc:De thiiy IKJH la hinh binh hanh va '{ lEJF la hinh binh hanh. .V Tir do IJ va EE cung nhiT IJ va K H att nhau tai trung diem cua moi du"5ng. i NhU" vay IJ, HK, EE dong quy tai mot ,j diem G => dpcm. Bay gicf ta chtfng minh G la trpng tarn , cua turdien ABCD . -"mXi in Trong (ABJ), gia sur A G n BJ = A). A p dung djnh l i Menelauyt trong tarn g.acABA,.tac6:^.-?i.M. i ' ; . IB JA, GA ^-f^^^.^' f AM : ,BU BJ A, G JA, GA = 1 A AI , do — = 1 IB (1) L a i ap dung dinh l i Menelauyt trong A BIJ, ta c6: B A, JG l A = 1 A .J GI A B JG , . l A 1 . BA, ^ Do — = 1 va — = -,ne n ^- = 2 £ 1 '^^^ 0 , A GI AB 2 A,J (2) (1) chu'ng to A | la trong tam cua tarn giac BCD. R I A G I Tir (2) suy ra — = 3, v i the iCf (1) c6 = - : JA, GA 3 A G A A, a c i Va y theo bai 9, G la trong tam cua tiJ dien ABC D => dpcm. 3. Nhiic hii dinh l i Menelauyt (xem hinh hoc Idp 10). , . ^: Cho tam giac ABC. M , N , P Ian lifdt la ba diem tren cac du'dng thang AB, BC, CA sao cho M , N , P thang hang. Kh i do ta c6: A M B N CP , " * M B NC PA n 0 maiO .1 Bai 10. Cho ttf dicn ABCD. Goi I va J Ian lu-dt la trung diem cua AC va BC. Tren B D lay diem K sao cho B K = 2KD. 1. Xac dinh giao diem E ciia dUc^ng thang CD vcti (UK) va chu'ng minh DE = DC. 2. Xac djnh giao diem F ciia du'dng thang A D vdi (UK) va chu'ng minh FA = 2FD. 5. Chu-ngminhEK/ZU 4. Goi M , N la diem bat k i tiTcJng uTng tren AB, CD. Tim giao diem cua M N vdi (UK). .-.l^ ; •.> i • Gia i 1. Trong (BCD) gia sCrCDnK J = E , •r,^, ^^l^ „,^n, n, ^CDn(UK ) = E , ^^^^ ^lsmiLuui\(> Theo dinh l i Menelauyt trong N , tam giac BCD, ta c6: D K BJ CE • = 1. (1) K B • JC • ED ^ D K 1 , BJ , , , , . . V \ oho (•.V'' : \ i D \ Do = -v a — = I(g/t ) nen tu'(I) suy ra K B 2 JC ^ C F — = 2 CE = 2ED ED DE = DC •dpcm. 2. V i E e DC => E e (ADC). C Trong (ADC), gia s^ EI n A D = F F e EI => F e (UK ) ^ A D n (UK) = F. Trong tam giac ADC la i theo dinh 11 Menelauyt, ta c6: * CE DF A I EDFAI C = 1 'hh ltd J (2); AI •iiirMiMrAiA'^i;'^ Tircaul,tac 6 • ^ = 2,c6n ^ = l (g/t), nen tif (2) c6: ^^'^ 7 "f''^ ED IC — - i F A = 2FD => dpcm. F A 2 „. , l_ 3. Theocau2th i — = - 2 , FA 2 D K „(in j^>:iaff'Bril Theo gia thie't ta c6 - ^ ' 1 ^ KB 2 FA DB dao), ma IJ // A B => F K // IJ => dpcm. _ 4. La'y M e AB , N e CD. '''^ j Trong (DAC) gia suf A N n IF = A' . Trong (DBC) gia suf B N n KJ = B' . Trong (NAB) gia stir A'B'nN M = P. ' f. Do P G A'B ' P e (UK), ma P e MN , dieu do c6 nghla la M N n (UK) = P.B a i 11. Cho ti? dien A1A2A3A4. Goi G,, G2, G3, G4 Ian liTcft la trong tam cua camat A2A3A4, A1A3A4, A1A2A4, A1A2A3. M la mot diem bat k i trong khongian. Goi M| , M2, M3 , M4 liTdng i?ng la cac diem doi xiifng ciia M qua Gi, G 1 ituj in oc oiuj gian - an uy i Gj, G4 . ChuTng minh A|M| , A2M2, A3M3, A4M4 la bo'n diTdng thang dong qui lai mot diem. yil! ; Giai Goi N la trung diem cua A3A4, the thi ta CO (do G|, G2 lU'dng iJng la cac trong lam ciia cac tam giac A2A3A4, A1A3A4) ]^.^.i^G,G2//A,A2 , N A 2 NA , 3 va theo dinh li Ta-lct cung c6: ^ = 1 . (1) A,A 2 3 Mat khac trong AMM2M1 , ta c6 G1G2 la diTcfng trung binh, nen: M,M2 = 2G|G2. (2) Tur(l), (2)suyra:M|M2= jA,A 2 (3) Vay A1A2M1M2 la hinh thang vdti hai day la A1A2, M1M2 c6: L\, ,.:0 !•)! AH. (B M1M2 = -AjA j ,3 . • Vi the hai di/cJng cheo A,M|, A2M2 c^t nhau tai mot diem S, trong do: 2 SM, _SM2 ^ ^ 2 ^ SA| SA 2 3 TiTdng tir cac doan th^ng A,M, , A,M,; A|M, , A4M 4 c^t nhau tiTc^ng tfng tai S', S"trongd6: ^.^ . S;:M , _ S;;M^ _ 2 '^^^ ••• S'A , S'A , S"A , S"A , S M | S'M , S"M | 2 TCr (5) noi rieng suy ra: Dicu do CO nghla la AiMi, A2M2, A3M3, A4M4 dong quy tai mot diem S, va t - ' ^ u- w .u SM, SM 2 SM^ SM4 2 ^ diem nay chia chung theo ti so: = = = = —. Do la dpcm. SA, SA 2 SA3 SA4 3 Bai 12, Cho hai doan th^ng ch^o nhau AB, GD. Goi I va J Ian liTdt \h cic trung diem cua AB va CD. r* H nh A + BD 2I «» * i* Ctfj TNim MTV DWH Khang Viet Giai Trong (ACD) di/ng hinh binh hanh ACED. Vi J la trung diem CD ncn A, J, E thang hang va CO AJ = JE. Trong (ABE) de thay BE = 2IJ . Do AB va CD cheo nhau, nen B, D, E khong thing hang. Tu" do ta co: BE < BD + DE =>2IJ< BD + AC. ; Bai 13. Cho hai mat phang (P) va (Q) cat nhau theo giao tuyen A. Lay M (P), N 6 (Q) sao cho M va N deu khong thuoc A. Tim tren A diem I sao cho MI + IN la be nhat. Giai Trong (Q) ha NA 1 A (A e A). Tren (P) difng tren niJa mat phang bcf khong chiJa doan NA| 1 A vii AN| - AN Noi MN, cat A tai diem I can di/ng. ; > That vay do hai tam giac vuong NiAI va NAI bing nhau, nen IN] = IN Lay diem K tuy y tren A (K ?t I). • ti/ A Trong tam giac KMN, ta co: d j j , .^j ; MK + KN| >MN, =MI + IN| , hayMK + KN, >MI + IN. (1) DoAN,AK = ANAKnenN|K = NK(2). TCr (1), (2) suy ra: MK + KN > MI + IN dpcm. , i ..Big .jKfudi gnfi'i Loai 2. CAC BAITOANVE THIET DIEN A, Phrfcfng phap xac djnh giao tuyen bSng hai diem chung, Nhu" ta da biet de xac dinh giao tuyen cua hai mat phang, ta chi can xac djnh hai diem chung A, B cua chiing. DU"dng thing di qua A, B chinh la giao tuyen can tim. Xac djnh thiet dien vdi mot khoi da dien thiTc chat la viec tim giao tuye'n cua thiet dien can tim vdi cac msit cua khoi da cho. Thi dy 1. Cho hinh ch6p tam giac S.ABC. Goi M, P Ian lifdt la cac trung diem AN 1 cua SA, SB con N la diem tren AB sao cho: = - AB 4 Ve thiet dien tao bdi (MNP). : * : ^ Boi (hcclng IISG IRnh hoc khoiig gian - Pluin IIuij Khni Giai Trong (ABC): NP n AC = E. Trong (SAC): EM n SC = Q Khi do MQPN la thiet dien phai dyng. Bay gid ta xac djnh vi tri cua Q tren SC. Trong tam giac ABC, theo djnh li Menelauyt, ta c6: AN BP CE _ j N NB ' PC • EA AN__1_ ' Do NB . AN V I AB BP CE _1_^ ~3 4 . = 1, nen thay vao (1), ta c6: = 3 (2) PC EA Trong tam giac SAC, lai theo dinh li Menelauyt, thi: CE A M SQ EA• MS •QC = 1 (3) ' U:^^> A)Mm «.rf (Q) 3no-!T Tir (2) vh ^ = l nen thay vao (3) c6 = 1, hay ^ = - MS , QC 3 ^ SC 4 'ilo ..^4 1. Viec diTng thie't dicn vdi mot khoi da dien da cho di/cJc tie'n hanh theo 2 biTdc: - Birdc 1: Ve thie't dien. - - < i v'- - ' ^"-^ - Birdc 2: Xac djnh chinh xdc vi tri c^c dinh ciia thiet dien. De lam dieu n^y ngirdi ta thirdng sijT dung hai dinh li cd ban la djnh li Ta-let va djnh li Menelauyt. 2. Cc1n lull y cac dieu sau day khi giai mot bai todn ve thie't dien: ^' ' - Phai luon coi mSt phang la v6 han, thi du (ABC) chiJ khong phai la tam giac ABC. - Trong khong gian de tim giao diem cua hai du'dng thang trufdc het phai lim xem chung c6 ciing d trong mot mat phang hay khong? Thi du trong bai loan , tren, ta phai trinh bay: ^. Trong (ABC), ta c6 NP n AC = E ^. ( Do Ih dieu can thie't, neu khong la rat de bj ngp nhan. 3. Ketqua A N SQ trong bai tren van dung, neu N la diem bat ki tren A B A B S C (mien la N khong phai la trung diem cua AB) . That vay theo dinh li Menelauyt, trong cac tam giac AB C va SAC , ta c6: A N B P CE N B P C C A = 1; C E A M S Q CA'M S Q C = 1 i Cty TNHH MTVDVVII Khang Viet A N B P C E ^ C E A M S Q NBPCC A CAMSQ C A N ^ SQ QC do B P _ A M N B A N P C " M S = 1 AB SC ..I,.,,.,.,,., , 4. Thifc ra neu N la trung diem AB, thi ta van thu lai ke't qua tren. Tuy nhicn each di/ng thie't dicn la khac vdi each tim giao diem ciia cac di/dng nhu" trong bai tren! Thi du 2. Cho hinh chop ti? giac S.ABCD, day ABCD la hinh binh hanh. Gpi M, fi"'5^\ hhH M dnlh ac, > f.T . . . B F C P 1 SQ 2 , Vi — = 3; = = _ (suy iir 2) F C PS 2 QB 3 Vay SO 2 SB 5 Vi tri cac dinh cua ngii giac ihict dien hoan loan xac djnh. Chiiy: Co the thay MN, AB va RQ dong qui lai mot diem. (cac ban tuT gisii ihich vi sao?) , 5 V .... r.- J V T h i du 4. Cho lang tru tarn giac ABC.A'B' C day la lam giac deu. Goi O va O' Ian hMl la cac lam cua day ABC.A'B'C' . Gia su- M vii N Ian Imi la trung diem cua A'B ' va BC , con P la diem nhm ircn O' O sao cho Difng thiet dien tao bdi (MNP). 'f T^t^ityT O 'P 1 O ' O 6 Cty TNHH MTV DWH Khang Viet Gia i Goi M' = C O n AB. Trong (C'MM'C): MP nC' C = Q. ^, Trong (BB'C'C) ; QN o B'B = E. Trong (ABB'A'): E M n AB = R. Trong (BCC'B') : E Q n B' C = F. Trong (A'B'C) : MFoA' C =S. Khi do MSQNR la ngu giac thiet dien phai diTng. Bay gicJ xac dinh vi tri cac dinh cua thiS'tdien. Do = - ma theo dinh li Ta-let, ta c6: O ' O 6 O'P MO" 1 _ ..^ , = = - => Q la trung diem cua C C . C ' Q M C 3 ..'ti.v ; De thay B E = QC=-CC'=-BB' . ^ VH- M M .i " 'If' i Hit , „ ,1, 2 2 Theo dinh liTalet, thi BR _ E B _ 1 BR _ 1 1 I o p \M r 1'] Do F C = N C = -B C = -B' C =^ = 3. m s^^^ Ap dung djnh li Menelauyt trong tam giac A'B'C , ta c6: ^ .^j.,,.,] |.{ g B F C S A' M F C SA ' MB ( 1 ) i.', >!lijfj"iy iy'Jp l-U B F 'A' M , C S 1 C S 1 F C MB SA ' 3 CA ' 4 Vay vi tri cdc dinh cua ngu giac thiet dien hoan to^n xdc dinh. Nlian xet: 1. Qua vi du nay, ta thay viec xac dinh giao diem "dau tien" la rat quan trong (d day do la giao diem Q). TiT giao diem Q nay, cac giao diem con lai di/dc xac dinh mot each khong lay gi lam kho khan. 2. Nhtf da noi den 5 phan tren viec lim giao diem ciia hai diTcfng thc^ng trong khong gian triTdc het phai xem chiing c6 d trong ciing mot mat phing nao hay khong? Trong cac bai loan trifdtc dieu nay de nhan thay. d bai tap nay de tim giao diem cua MP va C C ta phai nhin ra chiing d trong cung mot mat phang (do la (MCCM')) . Dieu nay khong phai de dang nhin thay ngay. 1 Boi duc khong gum - Phaii Iluy Khdi T hi du 5. Cho hinh chop tu" giac S.ABCD, day la hinh binh hanh va O lii lam cua day. Goi M , N, P lUdng iJng la trung diem cua AB, A D va SO. Ve thiet dien tao bcfi (MNP). \ -C.. ; Trong (ABCD): M N n AC = E. ', Trong (SAC): EP n SC = Q. / Trong (ABCD : M N n CD = F, / MNnB C = H. Trong (SDC): FQ n SD = R. Trong (SBC): HQ n SB = K. Giai Vay MNRQ K la thiet dien phai difng. ^ Ta c6: EO = -A0 = -OC . 2 2 F "' O r, ^ oQ OE n Q'O CO SP Trong tam giac SOC theo dinh Ii Menclauyt, thi: .— . = 1 (1) EC QS PO D « ? | = T ; = 1. ncn tir (1) suy ra ^ = 3 ^ = -i . H H ^MiMj EC 3 PO ^ QS SC 4 iff 1 'Y De thay FD = A M = - A B = -DC . 2 2 CO SR D F Trong tam giac SDC, theo dinh l i Meneiauyt, thi . = 1 (2) QS RD FC Vay R la trung diem cua SD. TiTdng tiT K la trung diem cua SB. Nhqnxet: (I ) - . |.,..2 ™ 1. Mo t Ian nffa qua v i du tren, ta thay ro vai tro quan trpng cua viec xac dinh giao diem "dau tien " (c( day la giao diem Q). - - _ - i, ^ 2. Vdi bai toan nay, ta c6 each khac de xac djnh thiet dien (xem phan sau). B. SiJ dung tinh song song de xac dinh giao tuyfin cua hai mat phang. PhiTcfng phap xac dinh giao tuyen giffa hai mat phang bang each suT dung tinh song song diCa tren menh de cd ban sau: Neu a // (P), thi moi mat phdn}^ (Q) chiia a ma cat (P) thi neu goi A la giao tuyen cua (P) va (Q), ta CO A //a. Cty TNHH MTV IJ\ KhntigVift T hi du 1. Cho tuT dien S.ABC, M la mot diem tren SB. 1. Dyng thiet dien qua M , song song vdi SA va song song vdi BC 2. Xac dinh vi tr i cua M de thiet dien la hinh thoi. '!•' 3. Xac dinh v i tr i cua M de thiet dien co dien tich Idn nha't. .: l a W 'iVi .M V- YitA V Giai i v ,A oum •• 1. V i thiet dien qua M v^ // SA, // BC nen trong (SAB) ke M N // SA (N G AB), va trong (SBC) ke M Q // BC (Q e SC). Khi ay (MNQ ) qua M va song song vdi SA, song song vdi BC. Bay gid ta se md rong (MNQ ) thanh thiet dien. V i MQ//BC ^ MQ//(ABC) ' => (MNQ ) n (ABC) = NP trong do NP // BC (P e AC) Vay MNP Q la thiet dien phai diTng. V i M N // SA => M N // (SAC) =^ (MNPQ) n (SAC) = MQ , trong do M Q // NP. V i the MNP Q la hinh binh hanh. 2. Tir cau 1. suy ra thiet dien MNP Q la hinh thoi khi va chi khi: M Q = M N (1) The o dinh l i Talet, ta cd: M Q SM M Q = BC SB SM.BC SB (2) , M N B M • M N = SA.BM SA(SB-SM ) (3) SA SB SB SB TiS (2) (3) suy ra (1) o SM.BC = SA(SB - SM) SA.SB o SM = . BC + SA Tir (4) suy ra M hoan loiin xac dinh. 3. Tacd : SMNPQ = MN.NP.sinMNP >ijfii i) lib iv iff'- . (4) . ,. 1 day . Do MN P = a, d da y a la goc giffa S A va BC la hang so. Tir (5 ) suy ra: SMNpgniax <=> MN.NP ma x M N NP •max r;,jit*; SA BC ( 5 ) "AM; • (6) ^ M N NP Do + SA BC M N NP ^ + ^ = 1, nen tir (6) suy ra: SB SB . max » — = — = - o M la trung diem cua SB. SA BC SA BC 2 uu.' j fta-Hj rJUSi iiyi ! .('! '''.V ghbij - tri-' i f'i'i s ns>!^nBfi nsriq 6'> v • 21 Boi diKJiig IISG Hinh hoc khdng ginii - Phan liny Khdi Nhdn xet: ' 1. Khi M \h. trung diem cua AB, ta nhac lai Irong v i du cua muc A. Xe t thiet dien tao bcfi (MNP), khi M , N , P tiTdng uTng la trung diem cua SB, AB, AC. Luc nay ta khong the tim giao tuyen bang phu'dng phap xac dinh cac giao diem cua hai during th^ng nhu- trong muc A, v i ly do d day M N // SA, NP // BC. Ta phai suf dung phiTdng S f phap diing tinh song song £v ,'JIA M) A3 'A ; // \ nhu- da trinh bay nhiT tren. • I X <, ^ Ke t hdp vd i v i du cl muc A, A I', itv/yj i ta c6 ket qua sau: A) c~ (NA^NB ) • •/ Cho hinh chop lam giac SABC. Khi do vdi moi v i tri cua N tren A B va gia suf SC n (MNP) = Q, thi ta iuon .... . A N SQ CO he thu^c: (NA = NB) mi 2. Vd i V I du 5, (5 muc A ta c6 the suT dung phU'cfng phap trong muc B nay dc giai lai no nhiT sau: - \ V i M N // B D => M N // (SBD) =^ (MNP) n (SBD) = A, trong do A qua P va A // BD. V i the trong SBD qua P ke XR // BD. Do P la trung diem cua SO, nen X , R tUdng i^ng la trung diem cua ^ SB,SD. Trong (ABCD) ta c6 M N n DC = E . -J - Trong (SCD) ta c6 ER n SC = Q. . SQ 1 MNRQ X la ngu giac thiet dien phai difiiii de thay U SC 4 j Lcfi giai nay c6 phan nao ddn gian nhu" IcJi giai da diTng trong v i du 5 muc A. CUj TNIIIl MTVDVVII Klutng ViH 3. Qua v i du nay ta thay trong mot bai toan xac dinh thiet dien, ngu'di ta thuTcJng ket hdp mot each nhuan nhuyen ca hai phU'cfng phap da neu. -/••. , T h i du 2. Cho hinh hop ABCD.A'B'C'D' . Goi O va O' Ian liTOt la tam ciia hai day ABCD, A'B'C'D' . P la diem tren 00 ' sao cho — = i . Di/ng thiet dien qua P song song vdi AC va song song vdi B'D. ' Gia i V I thiet diOn qua P va // AC nen trong (AA'C'C) qua P ve M N //A C (M e AA' , N e CC ) Ti/Ong tif V I thie't dien con i,, //B'D , nen trong mat cheo BDD'B'quaPkcEF//B' D (E e B'D'; F e DD'). Mat phang xac dinh bdi MN va EF la mat phang qua P va song song vdi AC vii B'D . Bay gid ta md rong no lhanh thiet dien. jCffilifl/ifl'?;'.1-' yi';* V i MN// A C =^MN// A ' C =>MN//(A'B'C'D') Vv;.?*;),,- A?, ; giao tuyen cua thiet dien vdi (A'B'C'D') sc qua E va // MN (tiJc la // A'C ) V i the trong (A'B'C'D' ) qua E kc RQ // A' C (R e A'B' ; Q e B'C).,, , , ^ MRQN F la thiet dien phai diTng. 1^^; Bay gid la xac djnh vi tri cac dinh cua ihie't dien. A ' M _C N _ Q'P _ 1 Ta c6: A ' A C C O'O ~ 4 ' • Gia O'O n B' D = I =:> O'P = PI . D F = Pl=ioO'=>-5ta , 1 4 DD ' 4 Ro rang do E la trung diem ciia BO' R, Q Wdng u'ng la cac Irung diem ciia A'B'v a B'C . " ^ 'iH':'-- • Cac dinh ciia ngu giac thiet dien MRQNF hoan loan xac dinh. n ; ! 'fhi du 3. Cho hinh hop ABCD.A'B'C'D' . Goi O la giao diem cua hai difdng cheo A' C va B' D con M , N Ian liTdt la trung diem ciia A D va B'C . Difiig thiet dien tao bdi (OMN). Gia i " Trong (A'B'C'D') : M O n B' C = P. Khid6(MON)n(A;B'C;p' ) = NP. ^ ^ .ujTiil - • Boi dUt'mg IISG IRnh hoc khdng gian - Phan Iliiy Khdi Vi (ABCD) // (A'B'C'D') .^.5- Nen (MON) n (ABCD) = MQ, trong do MQ // NP. A' D6 tha'y P va Q tifdng iJng la trung diem cua B' C va AB. Trong (ABCD): QM n CD = E. Trong (DCC'D'): NE n D'D = R. Trong (ABCD): QM n BC = F. Trong (BCC'B'): EP n BB' = X. Khi do MRNPXQ la luc giac lliici dien pfiiii dtJn^, De thay ED = AQ = AB CD = D'N R la irung diem cua DD'. F 2 2 Tu-clng tiT X la trung diem BB'. Nhqn xet: ;'>tKji(i.j .wiin- aiu>i;ii'Hull .U'fcl \ Ta Ihay Irong vi du nay da dong thdi stJ dung ca hai phu'cfng phap difng Ihicdien: phifdng phap tim giao diem chung cung nhif phU'dng phap suf dung tinsong song (trong bai nay sijf dung cac ke't qua ve hai mat phang song song). Thi du 4. Cho hinh chop S.ABCD day la hinh bmh hanh. M la diem tren AC(khac A va khac C). Difng thie't dien qua M song song vdi BD va song songvdi SA. Gia sur AC n BD = O. Xc l hai triTdng hdp sau: ('C!"^'a'A) '^jvn\] iV 1. Neu M e (OA) {}A^O\U^ k) iRl;!-! Vi thiet dicn qua M vii // BD, nen ' trong (ABCD) qua M vc EF // BD (E e AD, F G AB). Vi thie't dien qua M va // SA, nen trong (SAC) ke MQ // SA (Q e SC). Vay (QEF_ la mat phang qua M song song vc'Ji BD va SA. , ,^ c w&h Md rong no thanh thiet dicn nhif sau: .jj Vi MQ // SA => MQ // (SAD) ^ (QEF) n (SAD) = E¥,,,,^ anui ori'3 .€ nh id ddayEP//MQ(liircEP//SA,Pe SA) /^-fi -y,^>j^^ Vi MQ // SA ^ MQ // (SAB) => (QEF) n (SAD) = FR„, , j^,^ , d day FR // MQ (Itfc FR // SA, R e SB) PQRFE la ngu giac thiet dien phai difng. , " '"^ H'A ) anoiT SQ _ AM _ AE _ SP ^ AF _ SR , DiTa vao dinh li Talet ta c6: SC AC AD SD AB SB Cti) TNHH MTV DWH KItang Viet Vay 5 dinh cua thiet dicn xac dinh theo vj tri cua M nhif sau: ^ A E SP SQSRAF T AM ^ AD SD SC SB AB A C 2. Neu M e (OC) (M ^ O; M ;^ C) V! r-. Lap luan nhu" tren, ta difng thie't dien nhU' sau: Trong (ABCD) ke qua M: EF//BD(E G CD, F e BC). Trong (SAC) qua M ke MQ // SA (Q G SC) Khi do QEF la tarn giac thie't dien phai difng . u -r^ , ' DE AM CF CM SQ AC 3. Theo dinh li Talet, ta co: = ; = ; = - DC AC CB AC SC NeuM^O . - ..v.^ nc, J dncm. 1 = : ^ + M^ + M^>3 .JMA' MB ' M C SA SB SC SA SB SC , SA SB • SC 2. Theo ba't dang thiJc Cosi va can I , ta c6: M A ' MB ' M C M A ' MB ' M C 1 =o 1 >2 7 SA • SB • SC SA SB ' SC 27 < . „ „ , MA ' MB ' M C 1 N M P M Q M 1 Dau "= " trong (1) xay ra <=> = = = - » = = —Tr = r ^ ^ SA SB SC 3 NA PB QC 3 o M = G, vdi G la trong tarn cua lam giac ABC. M A ' MB ' M C ., , . V 1 ,. • V u'lu - Vay dai liTdng . . nhan gia In Idn nhal bang — km va cm km M la trong tarn cua tarn gidc ABC. • i, . SA SB SC ' 27 T hi du 2. Cho hinh hop ABCD.A'B'CD' . Hai diem M . N Ian liTcJt nam tren hai canh A D va C C sao cho j^M = .£!!L Chtfng minh r^ng diTdng th^ng M N M D N C song song vdi mat phang (ACB') ^ , A . Gia i Ve MP // AC (P e CD) Thco dinh l i Talet, ta c6 M D PD V I the tu" gia thie't suy ra: CP C N -^-.^ — = — :^PN//D C (2) •««•> PD N C Theo tinh chat hinh hop, ta c6 D C // AB nen tir,(2) ta co PN // D C ^ PN // AB ' Tir (3) va M P // A C => (MNP) // (ACB') Do M N e (MNP), nen tiT (4) suy ra M N // (ACB') => dpcm. Ctij TNHH MTV DVVII Khang Viet f h i dy 3. TCf cac dinh ciia tarn giac ABC, ta ke cac doan thiing AA', BB', C C .song song Cling chieu, bang nhau va khong nam U-ong mSt phang ciia tam giac ABC. Goi I , G, K Ian IiTdt ia trong tam cua cac tam giac ABC, AC C va A'B'C . 1. Chu-ng minh (IGK) // (BB'CC) . A' 2. Churngminh(A'GK)//(AIB'). Giai 1. Goi M va M ' tifdng iJng la trung diem cua AB , A'B' . Theo tinh chat trong tam tam giac, ta c6: C'K _ C I 2 C M C M 3 ~ 3 V i (ABC) // (A'B'C) , nen (CMM'C ) c^t hai mat phang (ABC) va (A'B'C ) theo hai giao tuyen song song =:>CM ' //CM . ' Tir(l)suyraKI//C C (2) Goi E va F ttfdng vlng la cac trung diem cua BC va CC . V i I , G tuTdng iJng la trong tam cua cac tam giac AB C va ACC , nen ta c6: *^ A I A G 2 . • AE AF 3 Vay theo dinh l i Talet dao, ta c6: IG // EF (3) Tir (2), (3) suy ra (IKG) // (BCCB' ) => dpcm. 2. Do A I n BC = E, nen (AIB') chinh la (AEB'). Goi N la trung diem cua AC, thi trong hinh binh hanh AA'C C de thay A', G, C thang hang. „j^../..,f; Do vay (A'KG ) chinh la (A'CJ) (J la trung diem ciia B'C) . ji; Ro rang A' J // AE; JC // B'E, do do (A'JC) // (AB'E), nen ta co (A'KG ) // (AIB') => dpcm. Thi du 4. Cho hai nu^a dUcfng thang chco nhau Ax va By. M va N la hai diem di dong tren Ax va By sao cho A M = BN. DiTng mat phc^ng (P) qua By va song song vdi Ax. DiTcfng thang qua M va song song vdi A B cat (P) tai M' . Goi I la trung diem cua M'N . Chiang minh rling I nam tren du^dng thang co djnh. 27 Bdi dudng HSG Hinh hoc khdng gian - Phnn Hug Khdi Giai > K e Bx ' // Ax. Khi do Bx ' la dtfdng thang c6' djnh. Ma t phing xac djnh A bdi By, Bx ' chinh la mSt phang (P), vay (P) = (By, Bx') , Ke MM'//A B (M ' e Bx'). ihu* B Ta C O A M = BM ' Tir A M = B N => BM ' = BN. Trong tam giac can M'B N (xet trong (P), do IM ' = I N => I nam tren Bt, d day Bt la tia phan giac cua x'B y . R6 rang Bt co dinh => dpcm. T h i du 5. Cho ttf dicn ABCD . Goi G la trong tam tam giac BC D va M la diem nam ben trong tam giac BCD. Du'dng thang qua M vsi song song vdi GA Ian liTdt cat cac mat phang (ABC), (ACD), (ADB) tai P, Q, R. 1. ChiJng minh rang khi M di dong trong tam giac BCD, dai liTc^ng: MP+MQ+M R GA la hang so'. "^n W FA ir. 2. Xac djnh vi tr i cua M , de tich MP.MQ.MR dat gia trj Idn nha't va hay tinh gia trj ay. Gia i 1. Lay M ben trong tam giac BCD. Gia .sij^ M G n BC = I ; M G n C D = J; M G n DB = K. Qua M ke Mx // GA Trong (AIJ): M x n A I = P - ('.Q (do chinh la diem M x cat (ABCD). ,f;,yfi. Tifdng tir trong (AIJ): M x n A K = R, M x n AJ = Q. \ Tren mat phang (BCD), ta c6: -IM.ICsinJI C e IG 1 -IG.ICsinJI C ''Gic Ti/dng liT, ta c6 I M S MIB IG 'GIB Tir (1), (2) va theo tinh cha'l cua day ti so bang nhau, ta c6: Cty TNHII MTV DWH Khang V,v, I M IG ^GB C Do G la trong tam tam giac BC D ncn ta co: 3GB C BCD Thay (4) vao (3), ta c6: :/'''A I M _ 3S[y|[jC IG 'BCD (4) (5) M!rfnit/-'"ni . MP Do G A // Mx , nfen theo dinh l i Talet, ta co: Tir(5)va(6)suyra:^^3-^^'^ c IG GA GA 'BCD g ci:! 1 t .«Da A) \i ^ :>! 5 r V . M Q ^ Hoan toan tiTdng liT, ta co: = 3 GA ^BCD \m-i i^v A mp aftvA ^'nob ib {''1; M R = 3 GA SBCD Cong tCfng ve (7), (8), (9) vdi Im y SMBC + SMCD + SMBD = SBCD, ta c6: MP+MQ+M R - = 3 = const => dpcm. i' •, G A 2. Theo bat dang thiJc Co-si, ta co: MP + M Q + M R > 3. ^MP.MQ.MR , | v j Ap diing cau 1, ta co: ' ' GA > ^MP.MQ.MR MP.MQ.MR < GA ' ..'•c. ^ (10) Do dd tir (10) ta C O max (MP.MQ.MR) = GAI ' ^'^^ ^ * "^'"^ Dieu nay xay ra khi va chi khi MP = M Q = M R o M la trong tam ABCD O M = u . T hi dy 6. Cho tiJ dicn S.ABC vdi cac diem M , N, P di dong tren SA, SB, SC tiTdng ufng sao cho SM 1 SN 1 SP 1 v di k = 2, 3... SA k'S B k + l'S C k + 2 ChiJng minh ci c giao tuyen cua (MNP) vdi (ABC) khi k thay doi luon luon song song vdi mot du'dng thang co dinh. Giai Dirng hinh binh hanh SABI va SBCK. Gia sijf tren (SAB) thi SB n M I = N ' '^^.^ .^^^^ ^5„,f^ ,,,,.|, • . Theo dinh l i Talet, ta c6: B6i (iKcifng IISG Ilinh hoc kitong gian - Pluin Hug Khni SN' SM N ' B B I ^ 4 (doB I = SA) B I k TO do: SN' I SN' 1 N ' B + SN' SN' = SN. k +1 SB k +1 V i N ' nam giffa SB nen N' s N. NhiTvay (MNP) luon di qua I co'dinh TOdng tir (MNP) luon di qua K co dinh. Vay (MNP) luon di qua diTdng thang co' dinh IK. Do SK // BC; SI // AB ^ (KSI) // (ABC). L a i C O (MNP) n (SKI) = KI , (MNP) n (ABC) = A. V i (KSI)//(ABC) =^ A//K I => dpcm. 'If/' T h i du 7. Cho hinh chop S.ABCD, day la hinh binh hanh tSm O. Mo t mat phang (P) di dong luon qua A va song song vdi BD . (P) cat SB, SC, SD Ian liTcJt tai E, F, G. Ma t phang (Q) qua EG va song song vdi B D c^t SA tai H. 1. Chtfng minh EG//BD. 2. ChiJng minh H F luon song song vdi mot diTdng thang cddjnh. - Giai fi.;;':.:,; . • 1. D o B D // (P) nen (SBD) chiJa B D va c^t (P) theo giao tuyen EG, do d6: EG//BD. 2. Ro rang E e (Q) n (SBD), va B D // (Q), nen giao tuyen cua (Q) v<3i (SBD) qua E va song song \di BD, hay do la dUcfng thang EG. Gia surl = EGnSO=> I e SGc(SAC)=> I e (SAC). M a t khac I e EG c (P) => I e (P). TO do suy ra I e AF = (SAC) n (P). TOdng tir cung co: I e C H = (SAC) n (Q). Vay I la giao diem cua SO, AF, C H tren (SAC). Ap dung dinh l i Xeva trong tarn giac SAC ta co: SF CO A N , . CO . = l,ma -^—-\ FC O A HS O A SF SH FC HA ' TO do theo dinh l i Talet dao suy ra F H // CA, tiJc la H F luon song song vdi du'dng thang co djnh dpcm. Cty TNIIH MTV DVVII Khang VuH Xhi du 8. Cho hinh chop tiJ giac S.ABCD va diem M S nhm cung phia S doi vdi mat phang (ABCD). Goi I , J, K, L Ian lUOt la trung diem ci'ia AB, BC, CS, DA. Goi (P), (Q), (R), (1) Ian lUdt la cac mat phang qua SI va song song vdi MK , qua SI va song song vdi ML , qua SK va song song vcti MI , qua SL va song song vdi MJ. ChiJng minh rang cac mat phang (P),^(Q), (R), (T) cung di qua mot diTdng thang. . , Giai Do IJ la dU^ng trung binh trong tam giac ABC, nen IJ // AC va IJ = AC TOdng tirLK//A C va L K = AC ml Ji s V i the IJK L la hinh binh hanh. Gia sur I K n JL = O, thi O la trung diem cua IK vii JL. Trong (MIK ) ve hinh binh hanh MINK , , ,i ! thi O cung la U-ung diem cua MN , do vay i i LJM N cung la hinh binh hanh nen IN // KM ; I M // KN ; JN // LM ; JM // LN . Mat khac M K // (P) va I N // MK , hdn the do I e (P) (do (P) qua SI) nen suy raI N e (P)=> N e (P). • .^Miii"/ \f<3ikiis,m , Lap luan tifdng tuf co N e (Q); N e (R), N e (T) Do M N n (ABCD ) = O => M va N nam khac phia doi vdi (ABCD). Do S, M nam cung phia v6i (ABCD) => N va S nam khac phia doi vdi (ABCD), tir do suy ra N ; t S. Vay di/dng thang qua S, N chinh la diTcJng thang thuoc ca bon mat phang (P), (Q), (R), (T). Do la dpcm. ; , ; ' ) \ til , I' ' ' [1, 1 U f,' . . 1 I' I / ' I!' \\ f ' ^f' I 31 B()i dicoiuj IISG Ilhih hoc khong gian - Phan IIuij Khdi Cnirc?N€ 2 . QUAN HE VUONG GOC -J J I. TOM TAT LY THUYET 1. Goc giS-d hai duTcfng thang trong khong gian Cho a va b la hai diTdng thang trong khong gian. Lay mot diem M trong khong gian. Qua M ve hai du'cfng thang a' // a va b' // b Khi do neu gia tri a (a < 90") la goc tao bSi hai dUdng thang a', b' thi ta cung noi a va b tao vdi nhau mot goc a. Khi a = 90", ta noi rkng a va b vuong goc vdi nhau. 2. Dufcifng thang vuong goc vdi mat phang - Dudng thang a gpi la vuong goc vdi mat phfing (P), ne'u a vuong goc vdi mpi dirdng thang cua (P) (hinh 1). Hinh 1 Hinh 2 - Neu dudng thang d vuong goc vdi hai dU'dng thang cat nhau a va b eijng nam trong mat phang (P), thi d vuong goc vdi (P) (hinh 2). > - Qua mot diem O cho trUdc c6 duy nha't mot mat phang (P) chu'a O vuong goc vdi mot dU'dng thang d cho trUdc. (hinh 3) d •"J . 0 Hinh 3 O Hinh 4 - Qua mot diem O cho trUdc cd duy nhat mot dudng thang d di qua O vuong goc vdi mot mat phang (P) cho tru'dc (hinh 4). :> \j - Dinh li ba dU'dng vuong goc: Cho dU'dng thang a c6 hinh chie'u a' tren mSt phang (P). Khi ay dU'dng th^ng b nkm trong (P) vuong goc vdi a khi va chi khi no vuong goc vdi a' (hinh 5). - Goc gii?a dU'dng thang va mat phang: Hinh 5 Cty TNHH MTV DWH Khnng ViH * Neu dU'dng thang a vuong goc vdi mat phang (P) thi la noi goc giffa a va (P) bang 90". * Neu dudng thang a khong vuong goc vdi mat phang (P) thi ta goi goc giiJa • ^ a va hinh chieu a' cua no tren (P) la goc giffa a va (P) (hinh 6). i;.*'--''' Mat phang vu6ng goc vdi mat phang Goc giiJa hai mat phing la goc giffa hai dirdng thang Ian lU'dt nam trong hai mat phang va vuong goc vdi giao tuye'n cua hai mat phang ay (hinh 7) Hai mat phang gpi la vuong goc neu goc Hinh 6 giffa chung bang 90". mAA -jpil lab iuw, nc Hai mat phang vuong goc vdi nhau khi va chi khi mot trong chung chu'a du"dng thang vuong goc vdi mat phang con lai. Neu (P) 1 (Q), thi bat cur di/dng thang a nao thupc (P) ma vuong goc vdi giao tuyen cua (P) va (Q) se vuong goc vdi (Q) (hinh 8). .^'^'^ vi: , •.;:::v-.,:v Hinh 8 Hai mat phang (P), (Q) c^t nhau Cling vuong goc vdi ji 11/' (R), thi giao tuyen cua (P) va (Q) se vuong goc vdi ^ (R) (hinh 9) . Khoang each 4'' Hinh 9 Hinh 10 Cho hai duTdng thang cheo nhau a va b. Khi do neu M e a, N G b va MN 1 a, MN 1 b, thi MN gpi la dU'dng vuong goc chung cua a va b. Luc do MN chinh la khoang each giffa a va b (hinh 10). ^ Neu b nam trong (P, ^ n X b va a // (P), thi d(a; b) = d(a; (P)). Neu a // (P) va M la diem luy y nam M tren a, thi d(a; (P)) = d(M; (P)). CJ day d(a; (P)), d(M; (P)), Ian liTdt la khoang each giffa a va (P), giffa M va (P) (hinh 11). Hinh 11 Boi dicSng ITSG Htnh hoc klionfj niitu - Plum Hiiij Khdi 11. CA C BA I TO A N V E KHOAN G CAC H A . Khoang each tuf my t diem tdi mo t dUofng thang, hoac tiif my t diem t(Ji m a t phang. , Cho diem M va diTcfng lhang A ( M g A). Goi H la hinh chie'u ciia M tren A. Khi do M H chinh la khoang each i\i M tdi diTdng thang A. M H = d(M,(A)) „,„„^^,.„^,^„_^„, , . , ;,.i...>.„^,;,.u.. Cho diem M va mat phang (P). Goi H la hinh chieu cua M tren (P). Khi do M H ehlnh la khoang each tiJf M tdi mat phang (P). „• M H = d(M,(P)) * • '•>6'9 uhn o?'.? Til l du 1 . (De thi tuycn sinh dai hoe khoi D - 2012) ifti I Cho hinh hop du'ng ABCDA'B'C'D ' c6 day la hinh vuong, tam giac A'A C vuong can, A' C = a. Tim khoang each tiif A de'n (BCD') theo a. i . u/. i Gia i Tam giac A'A C vuong can lai A va A' C = a => AA ' = AC = A B = iV2 72 2 2 2 ^•••r.vrM?-, yvi Ke AH l A' B (H ' e AB) >'-r.j4;':>i'L; V i A H 1B C (do BC l(ABB'A' ) => A H 1 (BCD'A'), tu-c la A H 1 (BCD') Trong tam giac vuong ABA' , ta co: 1 1 1 4 2 6 T- + A H ^ AB ' A'A ^ a^^ ^ a^ A H = 01 V ay d(A,(BCD')) = 1^6 Nhdn xet: Xe t each gitii khac sau day (bhng phu'dng phap the tich) . w 1 r^.^c 1 asjl 1 a a -d^^ I urn Taco : Vp,. .R C = -D'D.SAO P = . = (1) D.AB C 3 ^'^'^ 3 2 22 2 48 Ta c6: D'B C h\m giac vuong tai B, nen VA.D'B C =^h.SD.Bc , ^ day h - d(A, (BCD')). Ta CO D'B C la tam giac vuong lai C, ne n Cty TNHH MTV DWH Khang Vm SpBC = iBC.D' C = -.-N/D'D2+DC 2 = - 2 2 Tir do suy ra h = d(A, (BCD')) = a2 — + — 2 4 ',1, a^73 Cac ban hay so sanh tinh hieu qua cua hai phu'dng phap. !•< > T hi du 2 . (De thi tuyen sinh Da i hoc khoi B - 2011) >' • ' • Cho hinh lang tru ABCDAiBiCD , co day ABCD la hinh chff nhat vdi AB = a; A D = aVs. Hinh chieu vuong goc cua A| tren (ABCD) triing vdi giao diem O cija hai du'cJng cheo AC, BD cua day. Bie't rang hai mat phang (ADD|Ai) va (ABCD) tao vcti nhau goc 60". Tim khoang each tCf B, den mat phang (A,BD). Gia i Do tinh chat cua hinh hop, nen ta co: ]J2 ^ ^ AiBiCDlahlnhbinhhan h ^''^ Sik^ ^^^'M:.; i A, => B,C//A,D B,C//(A,BD). , Vith e "w)i./fv * d(B,,(A,BD)) = d(C,(A,BD)). (1) Taco : A|0 1 (ABCD) , 'W^ u => (BA,D) 1 (ABCD). V i (BA|D) n (ABCD) = BD, cho nen neu ke CH 1 BD ( H e BD) =>CH1(A,BD) . h Tir do di den d(C, A,BD)) = C H (2) Trong tam giac vuong BCD, ta co: ill ^ 1 1 1 I CH^ BC^ CD^ 3a^ 1 3a^ •CH- ^ C H = aV3 2 (3) Tir(l)(2 ) (3) suy ra d(B,, (A,BD)) = ^hqn xet: Trong viee tinh khoang each tif B, den (A|BD), ta khong can sit dung den gia thiet: ((ADD,), (ABCD)) = 60". (*) Gia thiet nay diing de tinh the tich cua lang tru ABCD.A.BiCiD, . Viec tinh the tich nay la phan dau trong de thi noi tren. vil Thid u 3. (De thi tuyen sinh Da i hoc khoi D - 2011) ' ^ Cho hinh chop tam gide S.ABC day la tam giac vuong AB C tai B va AB = 3a, BC = 4a. Bie't rhng m5t phin g (SBC) vuong g6e vdi (SAC). . , , Gia sur SB = 2aVs va SBC = 30". Ti m khoang each tiir B den (SAC). Bdi ditdiuj HSG ITinh hoc khdng girui - Phan Huy Khdi Giai - V U.'J H Ta C O (SBC ) _L (ABC ) v i (SBC ) n (ABC ) = BC , do do nci i k c S H 1 B C ( H e BC) , th i S H 1 (ABC) . K c H E 1 A C ( E e AC) . . • ^: a --..A f The o din h l i ba diTdng vuon g goc, la c6: SE 1 AC . Tu" do suy ra A C 1 (SHE ) ^ (SAC ) 1 (SHE). B d(H , (SAC)) = HE . (1) Ta C O S H = SB . sinSB H = SB.sin30" = aVs , B H = SB.cosSO" = 2a^/3.— = 3a 2 K e B K 1 A C ^ B K // HE . C H = a. The o din h l i Talcl , la c6: H E _C H 1 1 B K ~ 1 B C 4a 1 4 1 Matkha c - + - B K ' AB ' BC ' 9a ' 16a^ 2 5 Tird o HE^ = ^^ 9a^ V , yij .((a*ft0^3ib'~'((da,A)''^,€ilj 16 25 Tron g la m giac SHE , la c6: B 1 H P^ SH^ H E ^ 3a^ 9a ' H P = 3a W7 » B T K e BB ' 1 (SAC) . D o — = — = 4 H P H C =:>BB ' =4H P = 6a_ V 7 -( 3 ) I B 1 lit ' 0 3 ^DS ^H T i r (1), (3) suy ra d (B ; (SAC)) = BB ' = 1 TWd u 4. (Ba i loane d ban) 'V gncn' ' C h o li? dic n OABC , trong do OA , OB , O C do i mo t vuon g goc vd i nhau. K cO H 1 (ABC) . ' rf!Or :.h'»f|- 1. ChiJng min h H la triTc ta m ta m giac ABC . rfi&t qMvn 2. Chtfng min h he thiJc: = —L. + _i _ + _ L ^ 3^^'' i'^'^^^ = 3a O H ' OA ' OB ' OC ' • Ctij rmiH MTV DWII Khamj VH't Giai J KeOHl(ABC ) . A H n B C = M •j/ii>o B C 1 AM . Ti/dn g t y B H 1 A C => H la trifc ta m ta m giac AB C 2. Xe t trong la m giac vuon g AOM , la co: B ^ ^ ' (1) r2 • O H ' OA ' OM ' Tron g ta m giac vuon g BOC , do O M 1 B C ne n c6: i 6IJ '6^ :t8 4 1 ': 1 O M ' 1 1 - + O B ' O C .2 • (2) 1 Tir(l)(2)suyr a ^ O H Nhdn xet: 1 O A^ - + -1 . - + • O B 1 1 1. H e IhiJc 1 1 - + • diTcfc sur dun g nhie u kh i gia i cac ba i O H ' OA ' OB ' OC ' loan vc li m khoiin g each. 2. Xe t th i d u sau (D c th i Da i ho c kho i D ) C ho ti? dic n ABC D c6 can h A D vuon g goc vd i ma t phan g (ABC) . G i a sur A C = A D = 4cm , A B = 3cm, B C = 5cm . T i m khoan g each tCf A de n (BCD). u visig tf) qorirj rlnW od D J »ih •sd lie Kii J = u A ; « - V i A B = 3cm , A C = 4cm , B C = 5c m => AB C la la m giac vuon g la i A . Nhi r va y do A D 1 (ABC ) =>AD1AB ; ADIAC . T t r do ap dun g ke l qua tre n la c6: 1 1 •,(a:>?i)»-.Ji'ignJsorl)! fnf"" A H ' A B ' AC + ' AD'"9^1 6 16 = > A H = 6x/34 17 Bdi dUf'Jntj IISG IFmh hoc khong gian - Phan Ilutj Khdi Thi du 5. Cho hinh king tru diJng ABCA'B'C'day la tam giac ABC vuong taj B. Gia siir AB = a, AA' = 2a, A'C = 3a. Goi M la trung diem cua A' C va I l;i giao diem cua AM va A'C. Tim khoang each tiT A den mat phang IBC. Giai I • • Trong tam vuong A'AC, ta c6: ' ' f^"'^^'k-^-My AC = VA'C ^ -A'A ^ = V9a^-4a^ = aVs . W<:i--.\X:'-. Tii do irong lam giac vuong ABC, ta c6: ,j ^ B C = VAC'-AB ^ = V - V - a ^ = 2a. it,i^:uOk Nhanlhay (IBC) = (A'BC) Theo dinh li ba du'dng vuong goc, la c6 BC ± A'B (do BC 1 AB). Tir do kel htJp vdi BC 1 AA' )Oi suyra BC1(A'AB)^(A'BC)1(A'AB) . J ( B Vi (A'BC ) n (A'AB) = A'B, nen neu ke AK 1 A'B (K e A'B) => AKl(A'BC ) j,,^,,^ , I .g i . u =>d(A;(A'BC) = AK. (1) .t » Trong tam giac vuong A'AB, ta c6: 1 1 1 1 1 5 1 • 1 • • •! A K ' A'A ' A B ' 4a' a 2aV5 4a' 'V - + ti; ' H O A O AK = -5 Tir(l).suy rad(A, (ICB)) = 2a75 5 111,1 I I I Thi du 6. Cho hinh chop It? giac S.ABCD c6 day ABCD la hinh thang vuong, trong do ABC = BAD = 90"; BA = BC = a; AD = 2a. Gia sijT SA = aV2 va vuong goc vdi day (ABCD). Goi H lii hinh chieu cua A trcn SB. ' ' * Tim khoang each tiT H den (SCD). s - • ;; *r Giai De tha'y trong hinh thang vuong ABCD, ta c6: AC = a>/2. Do AB = 2a • => ABC la tam giac vuong can tsii C 4 =>BC1AC . Theo djnh li ba du'cJng vuong goc ta c6: AI^VTV - " \ S C I CD. • ,1/!:,,;^.; '^^^^i, , , Gia sijf DC n AB = E. . v^^'-' Ta CO B la trung diem cua AE. :) hit ti'Y Cty TNHII MTV DVVII Khnng Viet Vi BC 1 (SAC)=> (SDC) ± (SAC). -r. ; ; Do (SDC) n (SAC) = SC, nen neu ke P^! i a/ , \ >;?/, ] / , , AK 1 SC => AK ± (SCD) •..'« H A => d(A, (SCD)) = AK. j tm^m^'i Ta C O AB n (SCD) = E Q; => d(A, (SCD)) = 2d(B, (SCD)) (2) (do AB = BE) Ke AHISB . Ta C O trong lam giac vuong SAB: SA ' = SH.SB p. . I ^ V -• ...., VI SB = VsA^ + AB^ = ^(aV2) + a^ = a73, j,.^„,, \^ ^ SA^ 2a^ 2a 2V3a nen SH = — r = = ~i = = SB aV3 ^/3 3 ^ Theo dmh li Talet, la co: d(B,(SCD)) _ BS _ a>/3 _ 3 d(H, (SCD)) ~ HS ~ l^S " 2 Tir(l)(2 ) (3) suy ra d(H,(SCD))-|d(B,(SCD)). (3) 2 1 d (H, (SCD)) = (A. (SCD)) = ^ d (A, (SCD)) = - AK . . (4) r- r ^ Do SA = AC = aV2 => AK = 'W2. — = a . '• • I (5) Tir(4)(5)suyra d(H,(SCD)) = |. . Trong ihi dii n^y de tinh d(H,(SCD)) la da hai Ian thong qua tinh khoang each tCr diem khac den (SCD). ,^ a„ , Trirdc het d (H, (SCD)) = | d (B, (SCD)) Sau do .siir dung d(B, (SCD)) = ^cl(A, (SCD)) 4- ' ' Thi du 7. Cho lang tru di^ng ABC.A'B'C c6 day la tam giac vuong lai B co AB = a; AC = nS. Mat phang A'BC tao vdi day ABC goc 60". Goi M, N tiTdng tfug la trung diem cua BB' vii BC. Tim khoang each tCf B' den mat phang AMN . ^ . N^, • , .^^ "SJ^ , O.--" - • 391 Boi dudiig HSG mtih hoc kh6ng gian - Phnn Iluy Khdi Giai Ta c6 A' A ± (ABC), AB ± BC nen theo dinh li ba diTcJng vuong goc c6 A'B 1 BC. Tijf do suy ra A'B A chinh la goc tao bdi hai mat phang (A'BC) va (ABC). Theo gia thiet ta co A^BA = 60" (-^> => AA ' = AB.tan60" = aVs. Theo dinh li Pitago, ta c6: BC = VAC^ - AB^ = V3a^ -a^ = aV2. ' ''•'4' Vi B'B n (AMN) = M ma MB' = MB, nen d(B',(AMN)) = d(B,(AMN)). (1) Ta CO BA, BM , B N doi mot vuong goc vdi nhau, nen theo ket qua cua —7 boort! vi du 4 (vi du ccf ban), ta c6 ne'u goi r h = d(B, (AMN)) thi:, j;,- ' M^. /B H 1 1 1 • + -1 1 • + - 2 doBM = —;B N = aV2 h^ BA^ B M " BN ' . MA ' a" 3aj 13 ^. ,.^^(a^^),A)b-^. ^ = ((a:)^).H)b 3a 3a^ -^h = aV39 (2) 'Xsls «• >JA,<= SVJ! «3 A = A2-OQ 13 , 13 Tir(l)(2)suy ra d(B',(AMN))- a N/39,U4,032).Hill fn vii^ CcXi; ..i. . 13 Thi du 8. Cho hinh vuong ABCD va tam giac deu SAB canh a d trong hai mSt phang vuong goc vdi nhau. Goi I, J, K Ian lufdt la trung diem cua cac canh AB, CD, BC. Tim khoang each tiT I den mat phang (SDK). • ^•<' , i" ; v '\u> Giai ."• Do SI 1 AB => SI l (ABCD). Trong hinh vuong ABCD de thay KD 1IC vh gia stf KD n IC = H. Ta CO KD 1 SI (do SI 1 (ABCD), tiT do J suy ra KD 1 (SIC) (SKD) l(SIC). J - Vi (SKD) n (SIC) = SH, nen neu ke IE 1 SH (E e SH) thi IE 1 (SDK) A y r ang e => d(I, (SDK) = IE. (1) Trong tam giac vuong SIH, ta c6: 1 1 1 — T,T 2 • (2) : I' I. lE^ SI IH^ DS thay SI = aV3 Trong hinh vuong ABCD, ta c6: IC = a +- — =— — • , ,, ~ , v, s 4 2 vijiot ii:CH = aV5 IH = IC - CH = aVs a>/5 ^ 3aV5 nt^vK'o.;ij5 uifT - 2 5 ~ 10 • ngfroff • Ttr do thay vao (2) va c6 1 4 20 _ 3aV32 • !w^i ! . = 1 ^=4>IE = lE^ 3a^ 9a' 32 • A)*,, 'in 3aV32 "'"^ '^'^ ''^ ^^''^^ yi)b,.rifiJi.•lyiv.uSidn.iL.. Vay d(l,(SDK)) = ."^> t •r^.ll.l/ub i^ ' Thi du 9. Trong miit phang (P) cho du-cfng tron tam O, di/dng kinh AB = 2R. Tren diTdng thang d vuong goc vdi (P) Vdi A lay diem S va SA = R>/3. M la mot diem tren du-dng tam O, sao cho goc giffa SM va (P) bang 6O". Goi D, E Ian imi la hinh chieu vuong goc cua A tren SB, SM. Tim khoang each tu" S den (ADE) va tiT A den (SBM). '\, Giai ' •i-jin'iiib Theo gia thiet ta c6 SMA = 60" . ^ ' ((^1} Ta CO AMB = 90", nen theo dinh li ba diTcJng vuong goc suy ra SM 1 MB. Tur do BM 1 (SAM) => (SBM) 1 (SAM). Vi (SBM) n (SAM) = SM, ma AE1 SM => AE1 (SMB) => d(A, (SBM)) = AE. (1) Ta CO AM = SA.cot60" = R73. — = R. • pi/,'.. , ^ TCrdo , / AE = AM.sinAME = R.sin60" = RV3 2 Tir (1) (2) suy ra d(A, (SMB)) = RV3 Ta CO AE ± (SMB) AE 1 SB. Bdi dttdng IISG mnh hoc khdng gian - Pluin TIiuj Khni Vi A D ± SB ^ SB ± (ADE) d(S, (ADE)) = SD. (3) Ta CO trong tarn giac viiong SAB 1 1 1 1 A D ^ SA ' AB" 3R' • + -1 7 A D = RN/12^_ 2Rs ^ 4R' 2RV2T Tir(3 ) (4) CO d(S,(ADE)) = - 12R' V7 -~r- ( Qua cac th i du tre n ta ni t r a du^ofc ke't luan sau dSy: De giai bai toan tim khoang each tiir mot diem M den mot mill phc^ng (Pta thirdng tien hanh theo cac biTdc sau: - Timmotmatphang(Q)churaMsaocho(Q)l(P ) - Tim giao tuyen A cua (P) va (Q) - Trong (Q), ke M H 1 A. Khi do: M H = d(M; (P)) C^n lull y them cac dicu sau day: 1 ; r —'>« - Do i khi viec tinh d(M, (P)) thifcJng thay bang viec tinh d(N, (P)), trong d, d l nhien viec tinh d(N, (P)) lii de hdn so v6i viec tinh d(M, (P)), ngoai r N H biet diTdc ti so: k = , d day M N n (P) = H. M H ki U Kh i do d(N,(P)) d(M,(P)) = k . - No i rieng ta suf dung ke't qua sau: Neu M N n (P) = I va I la trung diem cua MN , thi ^ ;*l :. d(M, (P)) = d(N, (P)) - Vie c dijng ket qua cua bai toan ccf ban (thi du 4) cung hay su" dung de tinh khoang each lit mot diem de'n mot mat phang. B. Khoang each giffa hai dufofng thang cheo nhau - Cho hai du'cfng thiing cheo nhau a, b. Doan " thang M N ( M G a, N G b) goi la diTdng vuong ' ' goc Chung ciia a, b neu nhxi M N 1 a, M N 1 b. K h i do ta noi M N la khoang each giiJa hai * du'cfng thang cheo nhau a, b va ki hieu: !: d(a;b) = MN . ^ ^ ' De giai bai toan tim khoang each giffa hai diTdng th^ng cheo nhau a, b ta cd cac each giai thong dung sau day: ' Cty rmiH MTV DVVII Khnng Viet C^ch 1: Tri/c tiep dung dinh nghTa. Cach nay suT dung khi ta c6 the xac dinh diTdc du-dng vuong goc chung M N ciia a va b. Khi do d(a;b) = MN . > .4{'Vi.'- Thi du 1. (De thi tuyen sinh Da i hoc khoi A - 2012) , , Cho hinh chop S.ABCD c6 day la lam giac deu canh a. Hlnh chieu vuong goc cua S tren (ABC) la diem H thuoc canh A B sao cho H A = 2HB. Goc giffa dirdng lhang SC va (ABC) bang 60". Tinh khoang each giffa SA vii BC theo a. Gia i Ta c6: SCH la goc giffa dffcJng thin g SC va (ABC), nen SCH = 6O". Ke Ax // BC. Ke H N 1 Ax va HK 1 SN. r Taco : BC//(SAN) , nen d(BC, SA) = d(BC; (SAN)) = d(B; (SAN)). (1) . 'tfni':, : .['.. • • ; • • f,i. Bdi ditdng HSG Hlnh hoc khdng giaii - Phnn IIuij Khdi Tu- HA = 2HB B A= -H A 2 => d(B, (SAN)) = -d(H; (SAN)). (2) Ta CO HN 1 Ax va Ax 1 SH (do SH 1 (ABC)) ^ Ax 1 (SNH) ^ Ax 1 HK. Ket hdp vdi HK 1 SN suy ra HK 1 (SAN) d(H, (SAN)) = HK . - (3) ' Ta CO AH = - AB = — ; HN = AH..sin60" = — 3 3 3 2 Trong tarn giac vuong SNH, ta c6 HK.SN = SH.NH SH.NH SH.NH HK = SN VSH^+HN^ (4) m cua Ta CO SH = HC.tan60", ma HC = VcD^ + HD^, d day D la trung die AB ^ CH = + aV? Thay lai vao (4) CO HK = _ 3 V42 ;jnT)rif aV42 Tir (1) (2) (4) (5) di den d(BC, SA) = !J fJfifrf tliUX .^fjf'> n^ig rasfi •' Thi du 2. (De thi tuyen sinh Dai hoc khoi A - 2011) ; ^,,ir . Cho hinh chop tam giac S.ABC, day la tarn giac vuong can tai B, trong d6 - AB = BC = 2a. Gia suT hai mat phang (SAB) va (SAC) ciing vuong goc vdi day (ABC). Gpi M la trung diem cua AB. Mat phang qua SM va song song vdi BC cit AC tai N. Biet rang hai mat phang (SBC) va (ABC) tao vdi nhau" goc 60". Tim khoang each giffa hai du'dng thang AB va SN thco a. Giai Ta CO (SAB) n (SAC) = SA, nen tiT giii thiet suy ra SA 1 (ABC). Mat phang qua SM va song song vdi BC se c it (ABC) theo giao tuyen MN // BC N la trung diem cQa AC. i ( Cty TNHH MTV DWII Khang Vm Qua A kc dirdng song song vdi BC, qua N ke "Vn , ^ du'dng song .song vdi AB, chung cat nhau d • H.vacatBCdE . . ; • ' ^'V E ^„ , ,. ,.„,., . , Ta CO AB // HE va do SN e (SHE), nen d(AB, SN) = d(AB, (SHE)) = d(A,(SHE)). (1) Ta CO SA 1 HE (do SA 1 (ABC) ma HE e (ABC)), Lai CO HE 1 AH (theo each difng), (1 GI K 'I: Tir do suy ra HE 1 (SAH) => (SAH) 1 (SHE). ' ' - Vi (SAH) n (SHE) = SH, nen neu ke AK 1 SH (K e SH), thi AK 1 (SHE). Do do d(A, (SHE)) = AK . ,»r»J> v (2) Vi BC ± AB, nen theo dinh li ba du-dng vuong goc ta c6 SB 1 BC, tuf do SBA la goc giuTa hai mat phang (SBC) vii (ABC), nen theo gia thiet: SBA =60". •xXffuu- ' ai:-tr.'cuii Ta CO SA = AB.tan SBA = 2a.tan60" = 2a 73. Di thay AH = BE = BC = a. Trong tam giac vuong SAH, ta c6: 1 1 1 1 AK^ SA^^AH ^ 12a' 2aV39 Vay d(AB, SN) = 13 12a^ 13 AK = 2aV39 13 13 Thi du 3. Cho lang tru diJng ABC.A'B'C day la tam giac vuong c6 BA = BC = a, ccinh ben AA' = aV2. Goi M la trung diem ciia BC. Tinh khoang each giffa hai difdng thang AM va B'C. ' ' A ,rHAy. Giai Gpi E la trung diem cua BB'. Khi do ta co B'C // EM B'C // (AME) =>d(AM,B'C) = d(B'C, (AEM)) (1) Do B'B n (AEM) = E ma E la trung diem cua B'B, nen d(B', (AEM)) = d(B, (AEM)) (2) A' TCr B'C // (AEM) va tir (1) (2) suy ra \ d(AM, B'C) = d(B, (AEM)) (3) ' j^i^J^ Do BA, BE, BM doi mot vuong goc vdi nhau nen neu gpi h la khoang each tff B de'n (AEM), thitacd: h^ BA^ BM^ BE^ a^ a^ a^ rii.nl ^'5l.^^ Boi dicCfiuj IlSa IPmh hoc khong gian - Phnn Hug Khdi h = »V7 (4) TCr (3) (4) di den d(AM , B'C) = »V7 I 11 7 hi ' Id.' 1 > i T hi du 4. (De ihi tuyen sinh Dai hoc khoi B) Cho hinh chop tuT giac deu S.ABCD canh day bang a. Goi E la diem doi xiJng cua D qua trung diem cua SA. Goi M , N tiTcfng iJng la trung diem cua A E va BC. Ti m khoang each theo a giffa hai du'dng thang MN , AC. Gia i ^1/ • M / If ; bh'ia Goi P la trung diem cua AB. Ta c6:'»'' ' i ilmb r«-riJ (I' m si _ E MP//EB. (1) \K>sf-Jn S V i DASE la hinh binh hanh nen SE // D A va SE = D A => ^^^^^^ SE // BC va SE = BC ^ SEBC la hinh binh hanh =>EB//SC. (2) /,'^.-A > Tiif(l )(2 )suyraMN//SC . (3) /^l^^^-'-'"^ M N // (A'BC). Tird o d(AT , MN ) = d(MN, (A'BC)) = d(M,(A'BC)) . (1) Gia sur AB ' n A' B = I => Al l A'B . MatkhdcviBCl(BAA'B' ) => BC 1AI . 4 6 Ctg TNIIH MTVDVVn Kluuig Viet TCr do Al l (A'BC) =>d(A; (A'BC)) = AI . (2) Xa C O A M n (A'BC) = B va M la trung diem cua AB , nen ,-; s . j(A , A'BC)) = 2d(M, (A'BC)). (3) • :: , Tir (2) (3) suy ra d(M,(A'BC) ) = id(A,(A'BC) ) = ^AI . (4) ^: 1 I Fy yifi,l= -A'B = -A'Byl2=^, nentLr(l)(4)tac6 : 2 d(A'C,MN ) = 4i Thi du 6. Cho hinh chop tu" giac S.ABCD day lii hinh thoi canh A B = Vs, diTdng chco AC = 4; SO = 2>y2 va vuong goc vdi day ABCD , d day O la giao diem cua AC va BD . Goi M la trung diem cua Ccin h SC. Tim khoang each giCTa hai du'dng thang SA va BM . Ta C O M O // SA => SA // (MOB) ^ d(SA, BM ) = d(SA, (MOB)) = d(S,(MOB)). V i SC n (MOB) = M , ma M la trung diem cua SC nen .d \\'d gar' Giai (1) d(S,(MOB)) = d(C, (MOB)). (2) A Ta C O BO 1 AC (do ABC D la hinh thoi, BO 1 SO (do SO 1 (ABCD)) ' BO 1 (SOC) lu-c BO 1 (MOC) => (MOB) 1 (MOC). !^>' uyg J V V i (MBO ) n (MOC) = OM , do do neu ke C H 1 O M ( H e OM) thi CH 1 (BOM) => d(C, (MOB)) = CH(3) „v : i, acj Ta c6:0U=— = - & (2V2) +2^ ->/3 1 M C = -S C = -S A = N/3 => OMC m tarn gi^c can OM C dinh M . 2 2 Ke M K 1 OC => K la trung diem cua OC nen M K = ^S O = V2. Trong tam giac MOC, ta c6 MK.O C = MO.C H a. C H = MK.O C V2.2 iS M O (4) Tir (1) (2) (3) (4) di den d(SA, BM ) = 2 ^6 47 di ditdiig HSG Hinh hoc khdng gian - Phan Iltuj Khdi Uidii xet: Trong cac thi du tren, de tim khoiing each giiTa hai duTcfng thang cheo nhau a, b, ta deu siSr dung each 2 hoac each 3. Durdi day se trinh bay cac thi du ap dung cdch 1 de tim khoang each giffa hai dU'cfng thang eheo nhau. Caeh nay diTa vao viee xac dinh truTc tiep dU'dng vuong goc chung cua hai du'dng thang cheo nhau. Nguyen tac chung de giai bai toan xac djnh du'dng vuong goc chung cija hai du'dng ihing cheo nhau a, b nhU' sau: , _^ Xae dinh diem M G a, N e b sao cho M N 1 a, M N 1 b. Khi do M N la du'dng vuong goc chung eua a va b. Va'n de la d cho lam the nao de xac dinh du'dc hai diem M , N? , , ' ^ , ., . , A ^- 0 ti l qM'i rfnirt «d O .h y b ($• . Phiicfng phap long quat ta giai nhiisau: „ , Difng mat phang (P) chtfa a va song song vdi b. ^^"^ "' '^''^ ^' '''' b B Lay mot diem B tren b ke BB ' 1 (P) (B' e (P)). Trong (P) qua B ' diTng b' // b. Gia siJa nb ' = M . TurMk e MN//BB'( N G b). b' K hi do M N la du'dng vuong goc chung cija a va b. V K hi a va b c6 ca'u true dac biet (thi du nh\i a 1 b,...) thi ta lai c6 cdch xuT ly rieng tiTdng iJng va ddn gian hdn phep giai tong quat neu tren. , rhi du 7. (Trirdng hdp dac biet khi alb ) ^' ((Bfy^) "m ^ f( iUM) ^Mr Trinh bay each duTng difdng vuong goc chung vdi hai du'dng cheo nhau va vuong goc vdi nhau. >\f < - j JutA) -± On SUi v .^uc,. j . c^ti <^ • im uvlo ^ .11) UO±ru> -jiMh oh ,M5'^^ (DOM) n Cho a va b ch6o nhau v^ vuong gdc vdi nhau. DiTng mat phin g (P) qua b va vuong goc vdi a. Gia suT a n (P) = M . Trong (P) dirng M N 1 b Khi do M N la diTdng vuong gdc ^ chung cua a va b. rhiduS . S ! Cho hinh chop S.ABCD day la hinh vuong ABC D canh a. Goi M va N Ian imt la trung diem ciia c^c canh A B va AD . Gia sijT H la giao diem cua CN va DM . Biet SH vuong gdc vdi mat phang (ABCD) va SH = aVJ. Ti m khoang each giiJa hai du'dng thing D M va SC theo a. . , j - ,| , -^' i Ctg TNHH MTVDVVn Khaufj ViH Giai Trong hinh vuong ABCD , ta c6: ^AMD = DNC I : => NC D = AD M => D M 1 C N ^ , > Ma t khac D M X SH (do SH 1 (ABCD)) =^ D M 1 (SNC) D M 1 SC (nhiT vay D M va SC la hai difdng thang cheo nhau va vuong goc vdi nhau). D M n (SNC) = H, vay tCf H ke H K 1 SC (trong (SNC)) Theo thi du cd ban 7 thi H K chinh la .oi ! riui difdng vuong goc chung cua SM va SC. tfA '|i;iyuf!( Nhir vay d(DM , SC) = HK. i;iiK.;frti nil t' XhA M') Ta cd 1 H K ' SH ' S H = aV3(g/t); D H C ' III'} 61 i l i . • con HC = DC. cosDCN = DC — = D C ' a ' ^"2aV^ C N /DN^+DC ^ Thay vao (1 ) va cd: 1 1 9 H K ' 3a' 4a' 12a' H K = 2aV57 1 9 d(DM , SC) = 2a^/57 Nhqn xet: Thi du 8 la mot minh hoa sinh dong cho bai toan tdng quat nam trong thi du 7. ' ' - ,/tt i o ! ij b iri ! T h i dy 9. • • ^"'^ ' ' Cho hinh lap phu'dng ABCDAiBiCD , canh a. Tim khoang each giffa hai dirdng thang Ai B va BjD . Gia i Tac d AB,1A,B , A , B 1 A D (do A D 1 (ABB,A,) => A, B 1 (B|AD) => A,B 1 B, D (nhu'the A|B va Bi D la hai diTdng thing cheo nhau va vuong gdc vdi nhau) Gia sijTBA, n AB , = H , ., , =>A,Bn(B|AD ) = H. Boi diC(iiig IISG Hinh hoc khdng yinn - Pluin Iluy Khdi Theolhid u 7,ke HK±B,D , thi H K la di/dng vuong goc chung cua A ,B va B,D => d(A,B, B,D) = HK . Ke AE1B,D( E e B,D),thiHK = -A E Trong tam giac vuong BjAD , ta co: 1 1 1 AE^ AB? ' AD^ 2a^ .AE=^=.H K = i 2a^ Vayd(A|B, B|D) = aV6 Nhdn xet: Day cung la mot minh hoa sinh dong cho thi du 7. ir?! (;:>ti T T hi du 10. Cho hinh lap phi/dng ABCDA,B|C,D, canh a. Goi M , N , P Ian liTcft la trung diem cua BB, , CD, A,D|. Ti m khoang each giffa hai di/cJng thang MP va C,N . ^ Giai G oi E la trung diem cua CC| =>ME//BC^ME//A|D , • Trong hinh vuong CDDiC, :^ tathay DiElCi N (xem chiJng minh tiTdng tif trong thi du 8). M a t khac C. N 1 M E (vi M E // (CDD|C,)). Tif do la C O C|N 1 (MED.P) => C, N 1 MP. V ay MP va C|N la hai di/dng lhang cheo nhau va vuong goc vdi nhau. Gia su-C N n ED, = H => C, N n (MED,P) = H . "'' Kc H K J. MP. ; ,.K,t . i• " - ••' " Theo thi du cd ban 7, ta c6 H K lii du'dng vuong goc chung cija MPvaCiN , nen d(MP,C,N ) = H K (1) Ta C O EH = C,E. sin EC,H .a 2C, N a 2 : 'oti) Cl''. y a 2 ' . F H = EF-E H = 2ED,-E H = a^/5-a>/5 9aV5 10 10 Cty TNIIII MTV DVVII Khany Vici Theo dinh l i Talet, ta c6: H H ' FH 10 V ay HH ' = —E M = 10 E M 9a . 10 • EF \S 10' ''i^S Ai-1 :itix ! (fivij j ,(f,;;., ., ;>s Ta C O H ' H K = HPQ (goc c6 canh tuTdng iJng vuong goc) Tu" do trong tam giac vuong H'HK , thi: '•ium.t i}><>m*ipi' HSiil fl'iif If' a%/5 H K = HH' . cos H ' H K = HH' . cos HPQ = 9a QP _ 9a lOM Q ~ 10' 5 ^ + ^ (2) Tir(l)(2 ) suy ra d(MP, C,N) = 3a>/30 20 9a >/5 _3aV30 10\/6~ 20 . T hi dy 11. Cho hinh lu: dien deu ABC D canh a. Hay xac dinh khoang each giiJa A B va CD . ^ Giai , Goi M la trung diem cua CD. 1 Do ABC D la tuT dien deu, nen trong cac tam giac deu ACD , BC D ta co: B< IAMICD , BMIC D 1 CD 1 (AMB ) CD 1 AB . ' ' 'Va y A B va CD la hai diTcfng thing cheo nhau va vuong goc vdi nhau. ^Ta C O CD n (AMB ) = M , vi ihe ncu ke M K 1 A B ( K e AB) , thi theo ihi du I; C d ban 7, M K la di/dng vuong goc chung ciia A B va CD, nen ?d(AB,CD ) = M K (1) Ta cd A M = B M = a^/3 Do do trong tam giac can MA B dinh M , ta cd: M K = VMA^-AK ^ 3a^ a' aV2 4 4 (2) Tilf(I)(2 ) suy ra d(AB, CD) = &y/2 z han xet: Trong cac Ihi du 8 - 11, ta da linh khoang each giffa hai diTdng thang cheo nhau va vuong gdc vdi nhau a, b bilng each van dung thi du cd ban 7. 3di dudng HSG IRnh hoc khdng gian - Phan Iluy Khai Trong cac Ihi du dU'ofi day, ta xac dinh dUling vuong goc chung cua hai du-clng thang cheo nhau a, b U-ong cac tnfdng hdp Ichac (a va b khong vuong goc v6i nhau. rhi du 12. Cho hinh chop S.ABC D c6 day AB C la tam giac vuong can tai B ( BA = B C = 2a), canh ben SA = 2a va vuong goc vdi day (ABC). Tinh khoang each giiJa hai dtfdng thang AB va SC. wi ,;.«.: Gia i Goi M, N Ian liTdt la trung diem cua SC, AB. ^ Ta C O AB 1 B C =^ SB 1 B C (dinh li ba dUdng vuong goc) TO do trong hai tam giac vuong SA C va S C S B C suy ra MA = MB (vi cung = — ) ^MNIA B (1) ^-V,:^, i Ro rang fc^ SA N = ^ NB C (SA = B C = 2a; NA = NB = a) =^ NS = N C =>NM1S C (2) Tur (1) (2) suy ra MN la diTcfng vuong goc chung cua AB va SC , nen d(AB,SC ) = MN (3) Trong tam giac vuong MA N ta c6 MN = ^MA^ - AN^ = SC^ AB ' (4) Do SC ' = SA ' + AC ' = 4a' + (2a^/2) = 12a', ' ^'-'^^ ' ' ' CM J. K « ,«;>.!. MA nen tii (3) (4) suy ra d(AB, SC) = a^. I ^C j , , j Whan xet: Trong thi du tren ro rang AB va S C ch^o nhau nhuftig khong vuong goc vdi nhau. Do bai toan c6 cau triic dac biet nen viec xac dinh trifc tiep dtfdng vuong goc chung cua AB va SC trong thi du nay la ddn gian! r hi du 13. Cho hinh chop S.ABC D co day ABC D la hinh vuong canh a, SA = h va SA vuong goc vcti day (ABCD). DiTng duTcJng vuong goc chung cua S C va A B, tir do tinh d(SC, AB) - • ^ ^ Trong (SAD) ke AK1S D (K € SD). K Trong (SCD) kc K E // C D (E e SC). Khi do E K // AB Trong mat phang (BAKE ) (do E K // AB) ke EF//A K (K e AB). Do AB 1 (SAD) !::> AB 1 A K . B Cty TNHH MTV DWII Khang ViH nenAK//E F => AB ± E F (1) Ta C O D C J. (SAD) (SDC) _L (SAD) ">"^'' 'AA.iuihh Vi (SDC) n (SAD) = SD, nen A K 1 SD =^ A K ± (SCD) => A K 1 S C ma A K // E F => EF 1 SC. (2) ^ m-'^^ " Tif (1) (2) suy ra E F la duTdng vuong goc chung cua SC , AB. ' • Cung tijf do ta c6: d(SC, AB) = EF . (3) '. - • O^A ' nr De thay EKA F la hinh binh hanh nen E F = AK . (4) ^"'3 • Trong tam giac vuong (SAD) ta c6: 1 1 + 1 1 1 A K ^ SA ' AD ' A K = ah 7^ a^+h^ Tir (3) (4) (5) suy ra d(SC, AB) = j i 3M -Ma ill =: Jii <= dvR ••' • ah V a ' + h^ Nhdn xet: Neu bai toan chi doi hoi tinh d(SC, AB) ma khong yeu cau difng dtrdng vuong goc chung cua chiing, ta giai theo each 2 nhiT sau: Vi AB//C D ^ AB//(SCD ) £ ah =^ d(AB, SC) = d(AB, (SCD)) = d(A, (SCD)) = A K = III. CA C BA I TOA N V E GO C TRON G KHON G GIA N ,y = mo^^ M A. Ba i toan ve goc giffa hai dif(/ng thang cheo nhau ,? < , |, De giai biii toan nay ta tien hanh theo hai bu^dtc sau day: - Gia su" can xac djnh goc a (hoac ham so lu'dng giac cua goc a) giffa hai dirdng thang cheo nhau d va d'. Chon mot diem A thich hdp tren d. Qua A ve du-dng thang di // d'. Khi do goc c6 dinh A tao bdi d va d, chinh la goc tao bdidvad' . •r,.v%.y'jM'}-^^ ••' "joii iiit l ilnb>, 'till /,„• - Trong mat phang xac djnh bdi d va d|, - - HSi /l.. ; 1 . , /d , bang each diTa vao cac kie'n thiJc cua hinh hoc phang de tinh do Idn cua goc a, hoac tinh ham so Itfdng giac cua A goc a theo yeu cau de bai. ^ _ O day thu'dng la cac bai toan ddn gian ve he thiJc lifdng trong tam giac, hoac la cue bai toan litdng giac cd ban. Cho lang try diJng ABC.A'B' C c6 do dai canh ben b^ng 2a, day la tam giac vuong tai A c6 AB = a, A C = aV3. Hinh chieu vuong goc cua dinh A' tren Boi dia'fng IISG Ilinh hoc khong gian - Phati Hiiy Khdi m a t phan g (ABC ) la Irung die m cua canh BC . Tin h cosin cua goc giiTa ha i dirSng thang AA ' va B' C G o i M la trung die m cua BC , kh i do theo gia '^rv:— C Ihi e t la C O A ' M 1 BC . -i d Tron g (ABC ) qua A ke d // B C (tiJc d //B'C) . , ,,,,, « , G o i a la goc giffa ha i di/dng thang AA ' va / f^i^ff 61 ffi\'iH yjirf? §1 B'C',thia = fAA\d)=:A ^ D o BA C la ta m gia c vuon g ta i A c6 A B = a, A C = aV3 B C = 2a ^ B M = M C = a. B C Tac o A M = — = a 5!r • 2 B => AB M la la m giac de u can h a. G p i H la trung die m cua BM , * ' -s^ ' ^W'^A t h i A H L B M va H B = G M = - . 2 K e M K 1 d (turc M K // HA) . The o djn h l i ba du'iJng vuon g g6c ta c6 A' K 1 d. Dod o A K cos a = cosA'A K = -A A " \.d:]%ii(Mi mi hi: V i AA ' = 2a; A K = H M = - ne n thay va o (1), ta c6 cos a = — = - . 2 2a 4 Vaycos(AA',BC)=- . 4 .;;, ^v;., .., rhi d u 2. (D e thi luye n sinh Da i hoc kho i B ) C ho hin h cho p S.ABC D c6 da y la hin h vuon g canh ban g 2a, S A = a, SB = &S va ma t phang (SAB ) vuon g goc vd i da y (ABCD) . Gp i M , N Ian liTpt lii trun g die m cua AB , BC . Ti m cosin cua goc tao bcli hai du"cJng thang SM , DN . Gia l Ta C O S A = a, SB = aVJ, A B = 2a ^ . ^'"^ ^ * => SA^ + SB^ = AB ^ •'• => AS B la ta m giac vuon g ta i S (tiJc S A 1 SB ) T ' cx/r A B Ta C O S M = = a S A M la ta m giac de u canh a. Cty TNHH MTV DVVH Khnng ViH K e S H ± A B S H ± (ABCD ) (do (SAB ) ± (ABCD) ) va H A = H M = - . 2 Tron g (ABCD ) ti r M k c M P / / D N (P e AD) . Kh i do SM P la gCc tao bd i hai difdng than g S M va DN . Dat a = SMP . - K e S K L MP , the o djn h Ji ba difcJng vuon g goc tacoHKlMP . ,,,, , . Tron g ta m gia c vuon g SMK , ta c6: S M 1 a Detha y AP = -AD = - . • ^ 4 2 Tac o M K = MH.cosHM K =MH . — = - . : , : V. M P 2 CI Tha y v^o (1 ) va c6 cos(SM , DN ) = cos a = Thidu3 . H , M C h o hin h cho p S.ABC D da y li i hin h tho i canh bang Vs, A C = 4 va chic u cao cua hin h cho p la SO = ly/l , d da y A C n B D = O. Gp i M la trun g die m cua SC. Ti m goc giif a ha i diTdng thang S A va BM . R 6 ran g ta c6 M O / / SA , va y 0M B = a la goc giffa S A va BM . " ^'"^ v-", , ^ *i^A:t!")5i^ V i ABC D la hin h thoi , ne n D B 1 AC . M a t khac D B 1 SO (do SO 1 (ABCD) ) => D B 1 (SAC ) => D B 1 OM . j Tron g ta m giac vuon g MO B ta i O, O B ta C O tan a = O M ( 1 ) \o O M = - SA = - VSO^ + OA ^ = - J(2>/2 )^ + 2^ = Vs, , %.cV.4-. 2 2 2 ^ . • c6nOB=7BC^-OC^~==M)^-2^=l. ^^'^^-^^^r-'^'--^^^ 55 Bdi dicdiig IISG Hinh hoc khdng gian - Phan Iluy Khdi Tit do thay vao (1), la c6 tana = 1 a = 30". ^' Vay ^ SA~SM J = 30" '''' •'^''^ '^j^-' ''^^'•^ '0';JHA> »nr<-fT T hi du 4. Cho hinh chop tarn giac S.ABC day la tarn giac vuong can ABC tai B, irong do BA = BC = 2a va SA vuong goc vdi day (ABC). Bic t rang SB lao v di day AB C goc 60". Ti m goc giifa hai diTdng thang AB va SC. . . • i • Gia i Ta CO SBA la goc giffa SB va (ABC) ncn SBA = 60" => SA = AB.tan60" = 2ci^ . Trong (ABC) difng hinh vuong ABCD V i DC // A B => SCD la goc giffa hai dffting thang A B va DC. Dat SCD = a , . / nof i Tac6C D = 2a, ,,, , ,„'_ii SD = VSA^ +AD ^ = y](2ayl3f +{2a)^ = 4a. Ta CO DC ± A D => SD 1 DC (djnh l i ba dffcJng vuong goc). '^^^ ,M2)?.{W SD 4a Trong tarn giac vuong SDC, ta c6 tana = DC 2a = 2 Vay a = (AB, AC) = arctan2. B. Bai toiin ve g()c siuTa duf&ng thang va mat phang va goc giffa hai mat phang Si phu-ang phiip giai cac bai loan nay dffa IriTc tiep vao djnh nghla goc giffa dffdng thang va mat phang va goc giffa hai mat phang da dffcJc trinh bay k l IffOng trong sach giao khoa hinh hoc Idp 11. Thid u 1. Cho hinh hop chff nhat ABCDA'B'C'D ' day la hinh vuong canh a, canh ben A A ' = b. Goi M la trung diem cija CC . Ti m ty so - de (A'BD) va (MBD) la b hai mat phang vuong goc vdi nhau (tffc goc giffa hai miit phang tren bang 90"). Gia i Gia sijr A C n B D = O Ta CO (MBD) n (A'BD) = BD Dc tha'y M D = MB , A' D = A'B , nen ta c6 M O 1DB , A'0 1 DB • Vay A'O M la goc giffa hai mat phang (A'BD) va (MBD ) Cly TNHII MTV DVVH Khnncj Vic, Tit do (A'BD ) -L (MBD ) <=> A'O M = 90" o A'M - = A'O^ + OM ' (1) • .'^/n:- iC ) Ar! D6 thay Iheo dinh l i Pitago ta co: isWjj. yi^jj y&r/ hip r.! i •«:! \2 OM^ = OC ' + MC ' = + l^/^ A'0 ^ = A'A^ + OA^ = b^ + A'M ^ = A'C'^ + C'M^ = (d^/lf + Thay (2) (3) (4) vao(l)v a c6 2 4 2 (3) '.QZ :• (; = 2a +-(4 ) 3 2 ^3^^ ^^g^^, . . - OA?.) 1 :m A'O M =90"o— + — + b V — = 2aV- 2 4 2 I H A <:r:: iQD?,) 1 (1 / <^a^ = b'c > - = 1 (doa>0;b>0 ) b Thidu2 . l...>tif>Miv i,d/\ lUo UKJ ooji, iii i4>!A ' Cho hinh lap phffdng ABCD.A'B'C'D ' canh a. Ti m so do cua goc tao bSi haimat phang (B'AC) va (D'AC). Ta CO (B'AC) n (D'AC) = AC. Gia i Gia sff AC n BD = O. '3 A . Do D' A = D'C = B'A = B'C = aV2 . (ct day a la canh cua hinh lap phffUng), ta CO B'O 1 AC , D'0 1 AC. Tff do B'OD ' la goc giffa hai mat phang. Dat B^ ' =a . G oi O' = A' C nB'D ' => B00 ' = 0'0D ' = B'OD' = - . 2 Ta CO 00 ' = a; B'O' = • •'- ) vt/ it')','' => tan — = 2 00 ' 2 ^ a 2 cv 7 ~ arctan — 2 i.dq Hun ::>••; = a = 2arctan V2 2 • BSi dKCtiif] IISG Hinh hoc kh6ng girui - Phan Huy Khdi Nhdii xet: Bang each hoan loan ti/cJng tif, ta c6 ke l qua sau: ((BA'C);(DA'C))-60 " ' v6i chu y ta qui xXdc goc giffa hai mtTit ph^ng la g6c < 90*'. T h i dij 3. Trong mat phang (P) cho tam giac ABC vuong tai C, A B = 2a, CAB = 60". Doan SA = a va vuong goc v<3i (P). Goi a la goc tao bdi hai mat phang (SAB) va (SBC). Tinh sina. ' ifv • ; 'i :.\. Gia i Ke A H 1 SC, A K 1 SB. J \ (HGSC , KeSB ) Do BC 1 AC, BC ± SA (vi SA 1 (P)) =^ BC 1 (SAC) => (SBC) 1 (SAC). V i (SBC) n (SAC) = SC, nen A H 1 SC => A H 1 (SCB) => A H 1HK . Ta C O A K 1 SB H K 1 SB (dinh l i ba diTcfng vuong goc) ,0 SC = a%/2. , Trong tam giac vuong SAC, thi SA.AC = AH . SC a.a aV2 A H = (3) aV2 ~ 2 • Trong tam giac vuong SAB, thi AK.SB = SA.AB '•O'A A H 'I (2) A K = a.2a 2a^f5 j' M ' " Thay (3) (4) vao (2) va c6 sina = (4) aV2 2 2a^/5 1 N/IO rhf du 4. Trong mat phiing (P) cho hinh vuong ABC D canh a. Doan SA co djnh vuong goc vdi (P) tai A. M , N Ian iu-cn la hai diem di dong trcn canh BC v^ CD. Dat B M = u, D N = v. Chrfng minh rang a(u + v) + uv = a^ la dieu kicn can va du dc hai mtlt phiing (SAM), (SAN) tao vdi nhau mot goc 45". Cty TNHH MTV DWJI Khang Viet Gia i ;>rt"> Ta C O (SAM) r -i (SAN) = SA. Do SA ± (ABCD) => A M 1 SA, A N 1 SA => MA N 1^ goc tao bdi hai mat phlng (SAM) va (SAN). , ,^ . / i > \ Dat DA N = ft. MA B = 3 . ' ' / A;^ _ ^ _X K hi do MA N = 45" <=> a + p = 45 otan( a + P)= l D ^ tana + tanP ^ ^ ^ ^ ^' -"'''^ I-ta n a tan (3 - uv tsd it rinilMrf- ... , 1-^ 2 ) Ml. a(u + v) + UV = a^. i Do la dpcm. T hi du 5. Cho hinh lap phiTdng ABCDA'B'C'D ' canh a. Goi E, F va M Ian liTOt la trung diem cua AD , AB , CC . Goi (p la goc giffa hai mat phang (ABCD) va (EFM). Tinh coscp. Gia i B' Ta c6 (EFM) n (ABCD) = EF. • ' 03 C Gia suT A C n FE = 1.1 ^>mui'i'i\' -1 M V i FE//B D nen AC l BD r i =>FE1AC . 1 " J ( t Tff do theo djnh li ba dffdng vuong goc suy ra M I 1 EF. D Vay MI C chinh la goc tao bdi hai mat phang (MEF) va (ABCD), nen MI C = cp. Trong tam giac vuong MIC , ta cd coscp = If^'>'.' ' • W ^ ^ 3 3 / - • " • •'•p « ' tiilOO DoIC'=-A C = -aV2 , con !:-J Ml=VMe + IC ^=J ^ + ^=^ ^ ''''' ' 4 8 2^2 ' V$y tff (I ) suy ra coscp a^/^T 11 2N/2 Doi diCchif) IISG Illnh hoc khong gian - Phan Huy Khdi Thi du 6. Cho lang tru du-ng ABCA'B'C day la tarn giac can BAC dinh A, c6 goc BAC= 60". Goi M la trung diem ciia AA*. Gia sur (MBC ) tao vdi day goc p. Bie't rang BMC la tarn giac vuong. Tinh goc p. ' • • • GiM Trong (AA'CC ) gia suf C M n AC = E ^ =>(MBC')n(ABC) = BE. •(wi Si . ¥\/'~i4 Ta CO hai lam giac vuong A'M C va MAE bang nhau => AE = A'C . AE = AC - AB " ' ' =^ EBC la lam giac vuong lai B, ttfc la EB 1BC. A' Thco dinh li ba dtfdng vuong goc thi C B 1 BE. C Do do CB C la goc tao bdi (M'BC) vdi (ABC) =^ CB C = p ' • Theo tren la c6 ME = M C ,^^.,y^, ^•''(vi AAME = A A'MC ) ' ; , Dc Iha'y trong lam giac vuong thi . B'M=—=M C "'^^^ , Txi do BM C la lam giac can dinh M. Tuf gia thiet BM C la tarn giac vuong nen suy ra no phai vuong tai M BE = BC . (1 ) v. . j i i * 'V' Gia sur AB = AC = a. Do BAC = 60" =^ EEC = 30". Trong lam giac vuong EBC, ta c6 EB = EC. cosBEC = 2acos30' ' 5I)1jT = 2a ^ = aV3. (2) Ta CO BC = EC sin BEC = 2asin30" = a BC a U3 3i) ,3IM U(^>'''y^ -'ii ' =>B C = cos|3 cos3 (3) ;>C1 Tir (2) (3) suy ra aTs = ^ cos 3 <^cos(3 = - ^ o |3 = arccos Thi du 7. Cho hinh chop S.ABCD c6 day la hinh thang vuong vdi AB // CD, AB = 2a, CD = a va duTcfng cao AD = a. Gia suf SA vuong goc vdi (ABCD) va ,. SA = aV2. Tinh goc giiTa hai mat phang (SBC) va (SCD). An Cty TNIIIIMTV DWH Khang Viet Gia i Ta CO (SBC) n (SCD) = SC. ""'^ ' Trong hinh thang vuong ABCD, aieoa fi; tCf gia thiet suy ra AC 1 CD, t'i^'^^.^ vay tiJC do theo djnh li ba du'dng ^ p,^ ^y;^ vuong goc taco: SC I CD. ••>.:-— Vi AB 1 BC SB 1 BC (dinh li ba du'dng vuong goc) Trong (SBC) ke BH 1 SC (H e SC). Trong (SCD) tir N ke HK // CD. Do CD 1 SC HK 1 SC, vay BHK la goc giffa hai mat phang (SBC) va (SCD) (i-Mr ' ^ : - Ta CO SB = aV3, SC = (a72)V2 = 2a . Trong lam giac vuong SBC, ta c6 ISA ) i 1 B H ' SB' =>BH^= ^ BC^ 3a2 a^ 3eL^ ( 1 ) '-h SB' 3a' iA i !K :u r;' 3a 'H I i^f |Ta CO CD = aV2; SB^ = SH.SC => SH = |Ap dung dinh li Talet ta CO: "" ' IK SH SH.CD y-^^ ^ SC 2a 2 < J HK = HK = 372a (2) *CD SC SC 2a v,^ • • ui. . ..,.^,i4.Jurv~ott) Ijl ^^aiA;!' )S thaj BD = V4a^+a2 = asf5. .a5iAh3i^) Prong lam giac BSC, theo djnh li ham so" cosin, ta c6: B D' = SB' + SD' - 2SB.SD.C0SBSD " ' >5a' = 3a' + 9a' - 2.aV3.aN/6.cosBSD 2 ^ 6>y^cosBSD:=4 => cosBSD = 3V^ (3) 3a Pa CO SK SH 2 3 3^_ 3 rr - - — = — = — = - =^SK= -SD = -aV6;) ? r ].u] , • SD SC 2a 4 4 4 ^ ' Trong tam giac SBK , theo dinh li ham so cosin la c6 , 1, « B K ' = SB ' + SK'-2SB.SK.COSBSK . { Al?iuilMl' i Boi (Inong HSG Iluih hoc khoiuj (jinn - Phnn Iluij Khni TO (3) suy ra BK^ = 3a^ + ^a ^ - 2aV^.^a^^.^ = 1 ^ _ 3a^ = ^ . 16 4 3 8 o Bay gid ap dung dinh li ham so cosin Irong tarn giac BH K ta c6: - i '-')«: j; : BK^ = BH^ + HK'-2BH.HKcosBH K " 27a ,2 3a^ 9a^ ^ aV3 3V2a . /Jil t 1 t,lrS, -2. - 8 -.cosBHK r I U I I I 3^6 cosBHK = - cosBHK = V6 4 • 4 V ay (SBC),(SCD) = arccos V6 Thi du 8. Cho hinh vuong ABC D va tarn giac deu SAB canh a d Irong hai mat phang vuong goc vdi nhau. Goi 1 la trung dicm canh AB . 1. Ti m goc giffa SA, SB, SC, SD vdi (ABCD) 'n' . I (' hJ ",fKt l i 2. Ti m goc giffa SI va (SCD) 3. Ti m goc giiJa SC, SD va (SAB) , ^ , Gia i 1. Ta C O SI l A B 3 ^ SI 1 (ABCD) (do (SAB) 1 (ABCD)) SA C O hinh chieu la A I tren (ABCD), nen SAI la goc giffa SA va (ABCD) (SA^(ABCD)) = SAI = 6()". Tirdng lir (SB,(ABCD)) = SBI = 60", (SQ(ABCD)) = SCI. (I ) Ta C O IC = T a' a ' -75 I i aVI _si __2r _Vr 5 -^•aa: tan SCI = — = — f = IC a>/5 Vaytir(l)c 6 (SC,(ABCD)) = arctan \/i5 Ti/dng tif do SDI = SCI (SD,(ABCD)) = arctan 5 r .tf , = •'*JJ:?. 2. Goi J la trung dicm cua DC, thi IJ 1 DC => DC 1 (SIJ) (ket hdp vdi DC 1 SI) => (SDC) ± (SIJ). Do (SDC) n (SIJ) = SJ, nen neu ke I H 1S J => I H 1 (SDC) ^ ISH-(SUSDC) ) A O CtijTNHlIMTV DVVH Khnng ViH Trong tarn giac SIJ, ta c6 tan ISH = tan ISJ = — = — ^ - SI aV3 3 • (SI,(SDC)) = arctan 2 ^ 3. Ta C O D A 1 AB , D A 1 SI (do SI 1 (ABCD)) =i> D A 1 (SAB) SA la hinh chieu cua SD tren (SAB) => (DS,(SAB)) = DSA . ''^ Ta C O DSA la tam giac vuong can dinh A vdi canh SA = D A = a , => DSA = 45". Vay (sb,(SAB)) = 45" Tifdng tir (SC,(SAB)) = 45^. ' " ^ VI I. ' .uj.ni i 'uu 111 1 • 0 J 1 I V . Sir DVN G PHl/CfNG PHA P TQ A DQ GIA I CA C BA I TOA N V E KHOAN G CAC H V A GOC TRON G KHON G GIA N >' a n Trong nhieu tru"dng hdp neu c6 the dura vao mot he true toa do Decac vuong goc Oxyz mot each thich hdp, thi nhieu bai loan ve tim khoang each va xac dinh goc trong khong gian se c6 mot Idi giai ddn gian. Trong muc nliy ta se xet nhi?ng bai toan nhu'vay. Tri/dc het nhac lai mot so kien thtfc can diing den trong muc nay. i'• ' - Trong khong gian cho vectd M N vdi M = (x,; y, ; z,), N - {xj, yi, Z2 ) ihi M N =(x2-Xi;y2-yi;z2-Z|) . • - Doda i cua vectd u (ui; U2 ; U3 ) di/dc xac dinh nhu'sau: = yU | +U 2 +U 3 . I'd H,32 - Cho hai vectd ii = (ui; U2 ; Uj) ; v = (vf, V2 ; V3) . Tich vo hu'dng cua u , v diTdc k i hieu u . v va dtfdc xac djnh nhu" sau: u . V = C0S (U,V ) = U|V | + U2V2 + U3V3. - Goi a giCfa hai vectd u = (ui; U2 ; U3 ) va v - (vi ; V2 ; V3 ) du'dc xac dinh: cosa = • UiV , +U2V2+U3V 3 7uf+ u^+u2 .^v f +v^+v ^ Tir do suy ra u± v c >U|V | + U2V2 + U3V3 = 0. - Cho hai v6ctd u = (ui; U2 ; U3 ) va v = (vi ; Vj ; V3) . Khi do tich c6 hiTdng cua hai vectd n , V la mot vectd (difdc k i hi$u la [ia.v], va ta c6 u,v " 2 « 3 U3 u, U , U 2 V 2 V 3 V3 V , 9 V l V 2 63 {/if iliiniif/ use, llhih hoc khoiuj ()i d(SA;BC) = a-^763 21 + 63 1 36 36 3 aV63 _ 6 496 (£) (s; IN/42 'a thu lai ket qua giai b&ng phiTdng phap hinh hoc khong gian thuan tiiy! (xem thi du 1, mue B, II chiTdng 2). Thidu2 . F„h-. L Cho hinh chop tarn gidc S.ABC day \k tam giac vuong can tai B, trong do ltoL\ = BC = 2a. Gia suT hai mat phang (SAB) va (SAC) ciing vuong goc vdi day (ABC). Goi M la trung diem cua AB. Mat phing qua SM va song song vdi BC cat AC tai N. Biet r^ng hai mat phang (SBC) va (ABC) tao vdi nhau g6c 60". Tim khoang each giCfa hai diTdng thang AB va SN theo a. m'}}'.) Ta ed SA = (SAB) n (SAC) 'SAl(ABC) . 'I , 1 ((SBC), (ABC)) = 60" SBA = 60" I':*' S ~ A ;(0 ;0 ;0) -.^ H rir B ke Bz // SA => Bz 1 (ABC). ^ DiTng he true toa do Bxyz ^. p \ (xem hinh ve) Trong he true nay ta c6: B = (0; 0; 0); A = (0; 2a; 0); S = (0; 2a; 2a73 ); N = (a; a; 0) (do SA = AB tan SBA = 2aV3; do MN // BC nen N la trung diem cua AC) Ta CO AB = (0; - 2a; 0); SN = (a; - a; - 2aV3) 65 lioi cUCftiuj IISG Ilinh hoc khong (jinn - Plum IIuij Khdi AB.SN .AN Ttfdotaco: d(AB, SN) = d^AB, SN (1) AB, SN -2a 0 0 0 0 -2a Ta lai c6 AB,SN — - a -l&S 1 > a - a AN = (a;-a;0). ,,,;,}, : Thay (2) (3) vao (1) va c6 ; i j;;?^. 4a^73 __2aV39 j d(AB;SN) = aV4 8 + 4 13 " Ta Ihu lai kc't qua bang cdch suT dung phiTdng phap hinh hoc khong gian .'X thuan liiy dc giai thi du nay (xcm thi du 2, muc B, II chiTdng 2) Thidu3. •i.j.'i/. ' Cho lang tru diJng ABC.A'B'C day Ih tarn gidc vuong c6 BA = BC = a; canh ben AA' = ayfz. Goi M la trung diem cua BC. Tinh khoang edch giiJa hai 3' dircJng thang A M va B'C. Giai ^ ^ Di/ng he true toa do Bxyz (xem hinh ve). TCr gia thict suy ra trong he true tpa dp nay, ta eo: B = (0; 0; 0); A = (0; a; 0) 2 ; C = (a; 0; 0) M = -;0; 0 Tirdoco A M = J ;-a; 0 2 A M, B'C .AC (1) 66 Taco d(AM,B'C) = d(AM,B'c ) = A M, B'C R6 rang A M, B'C -a 0 fl 0 -a>y2 Ctij TNIIII MTV DWn Khang Viet 0 i a — —a 2 ; 2 -aV2 'I a 0 a^^;i^;a ^ AC = (a; -a; 0) Thay (2) (3) vao (1) va c6: d(AM,B'C) = - ............... (2) (3)"^ a_N/2 _^^l2_^aV 7 7 Ta thu lai ket qua giai bang phep tinh suT dung hinh hoc khong gian thuan tuy (xcm thi du 3, muc B, II chiWng 2). Cho hinh chop tiJ gidc deu S.ABCD ctinh day bling a. Goi E la diem ddi xu-ng cua D qua trung diem cua SA. Goi M, N tiTdng xSng la trung diem cua AE va BC. Tim khoang each theo a gii^a hai diTdng thang M N va AC. Giai • Goi O la tam cua day. Xet he true toa do Oxyz (xem hinh ve) ' ' Bat SO = h. Trong he true tpa do nay ta c6: O = (0; 0; 0);C = fa^/2 ;0;0 B = D = A = ; S = (0; 0; h) 2 ;0;0 Gpi I la trung diem cua SA, thi I = . '1,11, ' •^;0;i l 4 2 67 Bdi diedng HSG TRnh hoc khong gian - Phan IIiiij Khdi Do do E C O to a do la E = a^/2 aV2 ^ ; ; h , lu- do M = Ta C O N = ax/2 HS/T. ^ ^ . Do do MN = 4 ' ' 2 MN,AC .NC Taco: d(MN, AC) = d^MN, AC MN.AC aV2 aV2 2 ' 4 '2 ' ; AC = (a-Jl; 0; (1) De thay 2 MN.AC 0 0 NC = aV2 aV2 ;0 4 4 3aV2 2 4 0 aV2 (3) 3a72 0 4 aV2 0 a^h a^^ 0;-^; 0 . (2) >i«ri or!3 Thay(2)(3)vao(l)vacd: d(MN, AC) = ^y - = ^ . ' • • 2 Ta thu lai ket qua khi giai bai tren bang phiTdng phap hinh hoc khong gian thuan tuy (xem thi du 4, muc B, II chiMng 2). -^.^ j.fj., ^} Q J^^Q Nhdn xet: Dai liTdng d(MN, AC) khong phii thuoc vao h.y,,,, , t;^ ThiduS. • •" ' ' -r, , «•' ••)'-itn* Cho hinh chop S.ABCD day ABCD la hinh vuong canh a. Goi M va N Ian Itfdt la trung diem cua cac canh AB va AD. Gia suT H la giao diem cua CN va DM. Biet SH = aVs vti vuong goc vdi day (ABCD). Tim khoang each giifa hai dudng thang DM va SC theo a. ' z Giai Cty TNIIIIMTVDVVH Khang ViH Trong hinh vuong ABCD, dc thay CN L SM. Do SH L (ABCD), ncn difng he true toa do Hxyz (xem hinh ve) ^ Ta C O NC = b + = ^ =^ HC = DC. cosSCH = a". ^ = -1-='^ V 4, 2 NC aVs 5 r I H D = VDC^-HC ^ ^a^ - = , ^ ( .J ..^ .„ : , . => HM = DM - DH = a75 -dS 3a^/5 2 5 10 ,r ;;-'V...] TiJf do suy ra trong he toa do noi tren, ta cck H = (0; 0; 0); 2a VS 0 i "jo \ C = -;0;0 ; M = 10 ; S = (0;0;aNy3); D = 5 'Way DM = 2 va SC = '2aV5 5 ;0;-a^/3 Vir. I (?;(); . DM.SC aVs 0 0 -a>/3 0 0 2a 75 -a 73 5 0 aV5 ' 2asf5 5 0 iTaco d(DM,SC) = dpM,SC = DM,SC .MC DM,SC . (2) ..fiu! ill (Xjli/-. : |Do MC = 2aV5. 3aV5.Q 5 10 , nen tCfCl) (2) suy ra rd(DM,SC) = a-''^/3 2aV57 19 pTa thu lai ket qua giai thi du tren bang phu'dng phap hinh hoc khong gian uan tuy (xem thi du 8, muc B, II, chiTcJng 2). ^ ' idu 6. . •' :': • Cho hinh lap phiTdng ABCDA'B'C'D' cjinh bang 1. Goi M, N Ian liTcJt la trung diem cua AB va CD. Tim khoang each giiTa hai difdng A'C va MN. 69 Hoi (lii-^ \ \ N B D y A'CMN 1 0 0 0 0 1 = (1;0;1). ,of:;>;;;.o*-y -4u\ivi! m «(w A'CM N .CN Laico : d(A'C,MN ) = d(A'C,MN j = A'CM N 1 ( I ) • Do C N = 1 ; 0 ; 0 2 ncntir(l)c6:d(A'CMN ) = ^ = — . ' ' V 2 4 Ta thu hii kct qua gici i v i du Iron bang phiTdng phap hlnh hoc khong gian Ihuan tuy (xem thi du 5, muc B, II , chUdng 2 ) :\ ThiduV . , t. • ,i ^ J' V',::.' , Cho hinh chop ti? giac vS.ABCD day la hinh thoi canh AB = Ts, diTdng cheo AC = 4; SO = lyfl vii vuong goc vdi day, d day O la giao diem cua AC va B D. Goi M lii Irung diem ciia SC. Tim khoang each giffa hai du^dng thang SA va BM . Gia i V i ABC D la hinh thoi ncn AC 1 BD, do vay difng he true toa do Oxyz (xem hinh ve). Trong he true toa do nay, ta co: O = (0;0;0) ; A = (0;-2;C)); S = (();();2N/2]; B = (!;();()) C = (0; 2; 0) ; M = ((); 1; N/2) ' (do O B = VAB ^ -AO ^ = = \) / i^-^cf -1- - -\ Tird o S A = (();-2;-2V2) ; B M = (-1;1 ; V2 ) r Ct;j TNHII MTV Dl^ll Khnng Viet SA,B M - 2 -2 N / 2 -2%/2 0 0 - 2 SA,B M 1 4i 4i -1 -1 1 (();2V2;-2 ) (i) SA.B M . A B Laico : d(SA, BM ) = d(sA, BM ) = SA,B M (2) V i AB = (1;2;0), nen tiir(l) suy ra d(SA,BM ) = 4V2 2V6 # + 4 3 Ta thu lai ke't qua giai v i du tren bang phU'dng phap suf dung hinh hoc khong gian thuan tuy (xem thi dii 6, muc B, II , ehu'dng 2) Thi du 8. Cho hinh lap phu-dng ABCDA,B|C|D| canh a. Ti m khoang each giffa hai diTdng thang A|B va B|D. i {' , , , „ • • • Gia i • \l ' • DiTng he true toa dp Axyz (xem hinh ve). Khidolaco : A = (0; 0; 0), A, = (0; 0; a) B = (a;0;0), B, =(a ; 0 ;a), D = (0; a; 0). Tiifdo: A , B = (a;();-a ; B ,D = (-a;a;-a ) X V ay A|B,B,D '0 - a a - a - a a —a —a a 0 - a a = (a^2a^a2). (1). . |[A,B,B,b .BD Laico : d(A,B,B|D) = d(A,B,B, D = Do B D = (-a;a;0), n6n tiT (1) (2) suy ra: a^ aV6 A,B,B, D (2) (0 III d(A,B„B,D|) = ; ' - aVl+4+ 1 6 Ta thu \&'\t qua giai thi du tren bang phifctng phap hinh hoc khong gian thuan tuy (xem thi du 9, muc B, II , chtfdng 2) T hi du 9. Cho hinh lap phiTdng ABCDA|B,C|Di canh a. Go i M , N , P Ian WcJt la trung diem cua BBi,CD,A|D|. Ti m khoang each giffa hai diTcfng lhang B()l diCcnig HSG mnh hoc khdng gian - Ph(ui liny Khdi Giai Difng he true toa do Axyz (xem hinh ve) Khi do la c6: , ^ ,; A = (0; 0; 0); B = (a; 0; 0), H^^l B,=(a;();a); C = (a;a;0), D = (0; a; 0); A, = (0; 0; a), * D| =((); a; a); C| = (a; a; a). TCr do ta c6: a a a a; 0; - ; N = - ; a; 0 ; P = 0;-; a i 2, 2 ) 2 of; 17 8!!; ilu p I'XA i, ii l iiff) i-'l Ttrdo ta c6: MP = a a 2 2 va C,N = —a;—; — --;0;- a qfi' fffiifi odD .8 t;b . MP,C N a a a - a - a a 2 2 2 a ; a 2 0 —a —a ~2 " 2 0 a^ 5a2 a^ (1) Lai CO d(MP,CN ) = d(MP,CN j = MP,C N .MN MP,C N ,fO;0';0) = A •Mi iil V i M N = a a " 2 "''" 2 ncn tir(l) , (2) c6: 3 5a^ a-^ 8 9a 3 3aV30 d(MP,CN ) = - V4 16 16 ^/30 20 Ta thu lai kct quci giai thi du trcn bang phu'dng phap hinh hoc khong gian thuan tuy (xem thi du 10, muc B, II , chufdng 2) T h i du 10, Cho lang tru diJug ABC.A'B' C c6 do dai canh la 2a, day la tarn giac vuong lai A c6 A B = a; AC = a%/3. Hinh chieu vuong goc cua dinh A' tren mat phang (ABC) lii trung diem cua canh BC. Ti m cosin cua goc giiJa hai dirdngthang AA'v a B'C . Gia i Ta CO AB = a, AC = i\yf3 => BC = 2a => A M = B M = AB = a. • Ke A H 1 BC => H M = HB = - . 2 Kc Mx 1 BC (tiJc M x // AH ) ; I liUili Cty TNHH MTV DWH Khnng Viet Difng he true toa do Mxyz (xem hinh ve). Khi do ta c6 ayj3 a M = (0; 0; 0), A = -;0 2 2 A' = (O; 0; 2^3) , B - (0; -a ; 0), C = (0; a: 0) (viA'M = VA'A^-AM ^ =V4a ^-a ^ = a^^) B ^-^j- l — --^ Ta c6: (AA' , B'C ) = (AA' , BC ) => cos(AA', B'C ) = cos(AA', BC ) = COS(AA^,BC AA'.BC A A ' BC V i AA ' = ayfi a ;-;aV 3 ; BC = (0;2a;0) 2 2 Thay lai vao (1) va c6 cos(AA', B'C') = • a' 1 ''' I 4 • ^ 4 - + 3a^2a Ta thu lai ket qua linh bang phUdng phap suf dung thuan luy hinh hoc khong gian (xem thi du 1, miic A, III , chu'cfng 2). . ,v- ' Jti^v/ ^ ot . ,<,.• T hf du 11. Cho hinh chop S.ABCD c6 day ABC D la hinh vuong canh la 2a, SA = a, SB = aVs va (SAB) vuong goc vdi day (ABCD). Goi M , N Ian lu-cft la trung diem cua AB , BC. Tim co,sin cua goc giffa hai du'dng thang SM, DN . Gia i Ta CO (SAB) 1 (ABCD ) ma (SAB) n (ABCD) = AB , nen neu ke SH 1 AB, thi SH vuong goc vdi (ABCD). • Ta CO SA = a, SB = aVS; A B = 2a ' => ASB = 90" => SM = M A = M B = a (d day M la trung diem cua AB) BSi ditdng HSG innh hoc khoiuj (jian - Phnn IIuij Khdi Trong he true niiy ta c6: H = (0; 0; 0), M = 0;-;0 2 Do SAM la tarn giac deu canh a => SH = aV3 S = 0; 0; aVJ Ta CO B = f 3a f 3a ^ 3a ^ 0;—;0 2a;--; 0 ; c = 2a;—;0 =>N = a; ; 0 ; D = 2a;--; 0 2 2 , 2 2 V i the SM = fQ.a.aV^ 2 2 ; DN = (-a;2a;0) Tifdo cos(SM,DN) = • i I = cos SM,DN SM.DN SM DN 1 V5 /a^ 3a^ ri , , 2 75 Ta thu lai ke't qua giai thi du tren bang phu'dng phap hinh hoc khong gian thuan tuy (xem thi du 2, muc A, 111 chUOng 2). Thi du 12. Cho hinh chop S.ABCD day la hinh thoi canh bang , AC = 4 va chieu cao cua hinh chop la SO = 2N/2, d day AC n BD = O. Goi M la trung diem cua SC. Tim g6c giffa hai dUctng thang SA va BM. iM'*'>..iA iiub^.;,',! Giai : ru ^M' wjj p ;v'.Mi ;i,ydri ;1''. , Vi AC 1 BD, SO 1 (ABCD), nen chon he true toa do Oxyz nhiThinh ve. Ta CO OC = 2, OB = ^(^yfsf -2^ = 1, nen trong he true n^y ta c6: O = (0; 0; 0); S = (O; 0; 2V2) " A = (0;-2;0);B = (0;0; 0) C = (0;2;0), do d6M = (0;l;72 Tiirdo S A = (0;-2;-2N/2); , BM = (-l;l;^^ ) Vithe co.s(SA,BM) = cos ^ = ^ => (SA, BM) = 30". V4 + 8V 1 + I + 2 2V3.2 2 HA Cty TNHII MTV DV\'II Khang Viet Ta thu lai ket qua bang each giai vi du tren bang phtfcfng phap hmh hoc khong gian thuan tiiy (xem thi du 3, muc A, III chiTdng 2) Xhi du 13. Cho hinh chop S.ABC day lii tam giac yuong tai B (BA = BC = 2a) va SA vuong goc vdi day (ABC). Biet rang SB tao \d i day goc 6O". Tim goc gifra hai difc^ng thang AB va SC. , Giai Ta CO SBA = 6O" => SA = 2a.tan60" = 2aV3. ' ' i ' i ' " Difng hinh vuong ABCD va xet he true toa do Axyz (xem hinh ve) Trong he true niiy ta c6 A = (0; 0; 0); S = (0;();2a73) B = (0; 2a; 0); C = (2a; 2a; 0) AB = (0; 2a; 0); SC = (2a; 2a; - 2a73) : Taco: cos(AB,SC) = cos AB, SC AB.SC 4a^ ^/5 AB SC 2a.V4a^ +4a^ +12a^ Vay (AB, SC) = arccos S Ta thu lai ket qusi giai vi du tren bang phu'dng phap hinh hoc khong gian thuiin tuy (xem thi du 4, muc A, III chu'cfng 2). «(, Thi du 14. Cho hinh hop chi? nhat ABCDA'B'C'D' day la hinh vuong canh a, canh ben AA' = b. Goi M la trung diem cua CC. Tim ty so - de (A'BD) va (MBD) lii hai mat phfing vuong goc vdi nhau. Giai Dyng he true toa do Dxyz nhi/hinh ve. Trong he true niiy ta c6: D = (0; 0; 0); A' = (a; 0; b); B = (a; a; 0); C = (0; a; 0); C = (0; a; b) M = 2 Matphang(A'BD)c6 vcc-tdphapla: ii, DA', DB (1) 75 Bdi diCdiig IISG Ilinh hoc khong cjian - I'han Ihiij Khdi M a t phang (MBD ) c6 vecld phap la: DM,D B ifiii v (TSiiS K I yi ' y-Tacd : DA ' = (a;0;b); a--'ii!' I T f;iifi A' DB = (a; a; 0); D M = 2 VithetiJf(l)(2)c6 : n, = n , = 0 b a 0 0 b a 0 a 0 n n a 0 a a = (-ab;ab;a^), b - 0 0 i\ 2 Ta c6: (A'BD ) 1 (MDB) n, i . n , .tA; = :>;(();j;£ ;0) = 0 ; (0 ;ij£ ;0,) = ffA - Y <=> Hi.nj = 0 o — a^b^ a^b^ 2 2 o-a V + a' = 0 : dD B'T f a^ = 0 It n 1 oaV-b' ) = 0. ' (3) Do a > 0, b > 0, ncn (3) o a = b <=> — = 1. b V ay - = 1 la dieu kien can va du dc hai mat phang (A'BD ) va (MBD ) la b vuong goc vd i nhau. ¥*> «f «"H «« Thidu 15 * .(i - snWod-i 111 ,A •juiti^i^ ut, iril-mox) vol niiurii Cho hinh hip phiMng ABCDA'B'C'D' . Tim goc giiJa hai mat phang (BA'C) va (DA'C). y i JTii ( i J i.uj ifri,!-"/luij ft M ;ut * .tj A A it jx - Cin TNIiJI MTVDWIIKhang Viet M a t phang (DA'C) c6 vec td chi phu-dng la: = A'C , D C (2) Ta c6: A' C = (-a ; a; - a); DC = (-a; 0; 0) va BC = (0; a; 0). " Dod6tir(l)(2)c6 : *S a —a —a —a \ n, = n2 - a a a 0 0 0 » 0 a a —a —a —a —a a 0 0 0 —a 1 - a 0 = (a^0;-a2) , = (0;a^a^). Theo qui U'dc neu goi a la goc giiJa hai mat phang (BA'C) va (DA'C) thi a<90 " va tCfdo ta c6: < ii fin );•:»> H / _ _ •',>. imn -a a^ 1 cosa = cos(n|,n2 ) a = 60". "2 "1 "2 V ay (BA'C),(DA'C) = 60 . Thi du 16. Cho hinh chop S.ABCD co day la hinh thang vuong vdi A B // CD, AB = 2a; CD = a va diTdng cao A D = a. Gia su' SA vuong goc vdi (ABCD) va SA = iisjl. Tim goc giiJa hai mat phang (SBC) va (SCD). Giai Du"ng he true Axyz nhu'hinh ve. Trong he true toa do nay ta cd: A = (0; 0; 0); S = fO;0;aV2' B = (0; 2a; 0); C = (a; a; 0) a72 (va neu goi M la trung diem cua AB, thiCMlAB), D = (a;0;0). A M a t phang (SBC) cd vec td phap la: SB, SC (1) M a t phang (SDC) cd vec td phap la:* „ , . J, a (2) —i-ir " n-, = SD.SC _ cr» cn /"T» ••- \--~\\' I .1 Ta cd: SB = (O; 2a; - aN/2); SC = (a; a; -aV2); SD = (a; 0; -aV2 ' V ay tir (1) (2) cd: ,^. , ^ _ j i ^ ^_ _ , 2a -aV2 -aV2 0 0 2a n, = a -aV2 -aV2 a a a = (-a^^/2;-a2^^;-2a^ ) 77 Bdi dudng HSG Hinh hoc khdng gian - Phan Iluy Khcli 0 -ax/2 -aV2 a a 0 " 2 = a -a72 a a =-(a^V2;0;a2) Vay neu goi a la goc giffa hai mat phang (SBC) va (SCD), Ihi Iheo qui \S6c a<90" , ncnlaco : COS a = cos(H,,ri2) n ,. -2a'*+()-2a' 4 S -* " 2 a y 2 + 2 + 4.aV2 + 0 + l -d^ll.S 3 ' Ta thu hii kc t qua giai vi du trcn bang phu'dng phap hinh hoc khong gian {.,; Ihuan tiiy (xcm thi du 7, muc B, III , chu'dng 2) Ta nhan thay vt^i v i du ntiy phu'dng phap silrdung toa do la gon gang hcfn. T hi du 17. Cho hinh vuong ABC D va tam giac dcu SAB canh a d trong hai mat phang vuong goc vdi nhau. Goi I la trung diem cua canh AB . Ti m goc giu'a Siv a mat phang (SCD). • ;»V'''''' • I -Giai r ,.--uc . Tac o SIl(ABCD) . ,C Goi J la trung diem CD, Ihi IJ 1 AB. S V i the diTng he true toa dp Ixyz nhif hinh ve. /i \ Ta c6: SI = tijf do trong he true toa do / [ \ 2 nay, thi I = (0; 0; 0); S = 0;0; 2 ; D = C = a 4 ; 0 ' Ma t phin g (SCD) CO vec td phap la: n = SC,SD Ta c6: SC = a a\/3 ; SD = a ax/s (1) , vay tir(I)c6 : n = a a\/3 2 2~ - 2 2 2 a 2 a a 2 a a ~2 ;0;-a ' U i c6 SI = 0;0; - Gpi a la gdc ^iiJa SI va mat ph^ng (SCD) thi Ctfj TNimMTVDVVIl Khnng Vw S in a = cos^SI, ri (2) Ta CO cos SI,n Sl.i i 2x/7 SI „2 /3 7^ 2N/7 Vay tu" (2) .suy ra a = arcsin . CAa j'. " Theo each giai bang phiTcfng phap hinh hoc khong gian thuan tiiy, ta co: a = arc tan 2V3 De y rang tana = => cota = — sina = 1 + cot a > sina = 1 2V7 I + - Ta thu lai hai ke't qua nhiT nhau (xem thi du 8, miic B, III , chiTcJng 2) ' T hi du 18. (D 6 thi tuyen sinh Da i hoc khoi D ) Trong khong gian vdi h0 toa do Oxyz cho hinh lang Iru durng ABCAiBiCi .Biet A(a; 0; 0), B(-a ; 0; 0), C(0; 1; 0), B|(-a; 0; b) vcJi a > 0, b > 0. 1. Ti m khoang each giffa hai difcfng thang BiC va ACi theo a va b. 2. Cho a, b thay ddi nhiTng luon thoa man a + b = 4. Ti m a, b de khoang eachgiij-a B,C va AC, la Idn nhat. Gia i 1. Ta c6: A, = (a; 0; b), C, = (0; l;b) . Theo cong thuTc tinh khoang each giCfa hai dirdng ih^ng ta c6: \ B,C, AC, .CC, d(B,C,AC,) = B.CAC , Do B,C = (a; 1; - b); AC, = (-a; 1; b) ; CCi = (O; 0; b) => B,C,AC 1 - b 1 b - b a b - a a 1 -a 1 = (2b;0;2a). Thayvao(I)v a c6: d(B|C,AC,) = 2b ab V4b^+4a^ 7a^+b^ (2) 79l Boi dialiif) nSCi U'lnh hoc klumg (/inn - Phaii Iluy Khdi 2. TO (2) va ap dung bat dang thtfc Cosi, ta c6: d (B,C, AC,) <-VS ^ <-^^y^ = ^^ (do a + b - 4). Vay max d(BiC, AC,) = V 2 o a = b = 2. Nhqn xet: Trong cac thi du 1 - 17 bai toan ra diTdi dang hinh hoc khong gian, nhtfng khi giai ta dung phiMng phap toa do, con trong bai tren de bai ra du^di dang hinh hoc toa dp nen vice suT dung phiTdng ph^p tpa dp de giai bai toan nay la hdp li. ^ Binh luan: Qua cac thi du 1 - 18, ta rut ra phiTdng phap giai ckc hki toan ve khoang each va goc trong hinh hoc khong gian bang phUdng phap toa dp nhu" sau: , - Lap mot he true tpa dp thich hdp vdi dau bai. , , , , ^ , - Tim tpa dp cua cdc diem, cdc vec td can thiet. - Sur dung cac cong thiJc tUdng iJng da biet de tinh cac dai luTdng theo yeu cau dau bai. Can nhan manh rang, viec xay diTng he true tpa dp la quan trpng nhat vi no dam bao cho viec tinh toan d cac b^dc tiep theo la ddn gian hay phiJc tap phu thupc vao viec lifa chpn he true tpa dp ban dau. ;"j ,j| V. TH E TIC H CU A KH6I D A DIE N ^y^^^ 5^ ^5,^ Lta aoditt anrnl' Bai toan tinh the tich cua khoi da dien la mot trong nhffng chu de thiet yg'u cua chiTdng trinh hinh hpc dtfdc giang day trong chiTdng Irinh d nha triTdng trung ^.^ hpc pho thong. Npi dung nay luon luon du'dc de cap den trong de thi tuyen sinh mon toan vao cac triTdng Dai hpc va Cao dang 6 cac khoi A, B va D. T OM TA T LI THUYE T - The tich hinh chop: , 8. V = ^Sh, fiSi«j if: (5 day V la the tich, S la dien tich day con h \k ehieu cao cua hinh chop. ^ - The tich hinh la ngtru: S day V la the tich, S la dien tich day i d con h la chieu cao cua lang try. i %--< :),H)I)