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Phuamg phapgiai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu H I N H H Q C K H O N G G I A N T 6 N G HpTP § 1. Q U A N Hfi V U O N G G O C 1 .Jiai durnig thdng vuong goc De chu-ng minh hai duong thang AB va CD vuong goc vol nhau, ta c6 cac each sau Cach 1; Chung minh AB.CD = 0 nen suy ra tam giac S C O vuong tai D Vay cac mat ben ciia hinh chop deu la nhiJng tam giac vuong. Di^n tich xung quanh ciia hinh chop la: Cach 3;Su dung cac ket qua da biet trong hinh hpc phang 2. ^itang thdng vuong goc voi mat phdng. De chung minh duong thang a vuong goc voi mat phang (P) ta thuang di chung minh duong t h i n g a vuong goc voi hai duong thang cat nhau nam trong (P). , Chu y: Neu duong thSng a vuong goc voi mat phang (P) thi duong thang a vuong goc voi moi duong thang nam trong mat phang (P). vudnggoc De chung minh hai mat phang vuong goc, ta chiing minh mat phang nay chua mot duong thang vuong goc voi mat phang kia. Chu y: Neu hai mat phSng cat nhau theo giao tuyen A va vuong goc voi nhau thi mgi duong thang nam trong mat phang nay ma vuong goc voi A thl duong thang do se vuong goc voi mat phang kia. Vi du 2.1.1. Cho hinh chop S.ABCD c6 day ABCD la hinh chii nhat voi dp dai cac canh AB = a, A D = b. Canh ben SA vuong goc voi mat phang day va SA c. Gpi H , K Ian lugt la hinh chieu ciia A len cac duong thang SB, SD. 1) Chung minh rang eac mat ben ciia hinh chop la nhirng tam giac vuong. Tinh di^n tich xung quanh ciia hinh chop theo a,b,c ac + bVa^ +c^ + aVc^ + b^ + be 2) Ta c6: B C l ( S A B ) va A H c ( S A B ) nen A H 1 BC . Mat khac A H 1 SB nen ta c6 A H 1 (SBC) Suy ra A H 1 SC . Chung minh tuong t u ta cung c6 A K 1 SC . Tir do suy ra SC 1 ( A H K ) . 3) Ggi I la giao diem ciia SC voi mat phang (AHK), ta c6 thiet dien ciia hinh chop cat bai mat phang ( A H K ) la tu giac A H I K . Ta CO hai tam giac A H I va A K I la hai tam giac vuong tai H va K. i Do A I 1 SC nen SI.SC - SA^ ^ SI = SuyraH.SI^j,j^SI.BC «C SB Tuong tu: K I = — SB Tir do suy ra BC 1 (SAB), suy ra BC 1 AB hay tam giac SBC vuong tai B Tuong tu, ta chung minh duoc CD ± (SAD) 94 Va^Tb^Tc^ SD AB.AS SB Tuong t v : A K = ,2 bc^ ^^^^^^ a^ + b^+c^) SI.CD 5 ac V(b2 4-c2)(a2 + b 2 + c 2 ) AH = AB 1 BC SC voi nhau. 3) Tinh di^n tich thiet di#n ciia hinh chop cat boi mat phMng ( A H K ) . Do SA 1 (ABCD) nen suy ra SA 1 B C . Lai c6 ABCD la hinh chu nhat nen SA^ Vi SC 1 (AHK) nen H I 1 SC, do do hai tam giac SIH va SBC dong dang T r o n g t a m g i a c v u o n g SAB 1) Ta CO cac tam giac SAB, SAD la nhung tam giac vuong tai A S = i ( S A . A B + SB.BC + S D . C D + S A . A D ) 2) Chung minh rang SC 1 (AHK) JCffigidi. Khang Viet S = S/iSAB + SASBC + S^sCD + SASDA 2 Cdch 2;Chung minh c6 mot mat phang (P) chua AB va vuong goc voi CD 3.Jiai mdt ptidng Cty TNHH MTV DWH ta c6: ac AD.SA AD be = _ V b ^ ^ V^y di^n tich thiet dign can tinh la: abc^ S A H I K = | ( H I . A H + K I . A K ) = (a^ + b^ + c^ ) V ( a ^ + c 2 ) ( b 2 + c 2 ) 95 Phucmg phdp gidi Toiin Hinh hgc theo chuyen de- Nguyen PM Khdnh, Nguyen Tat C t y rmm Thu Vidu 2.1.2. Cho hinh chop deu S.ABC c6 canh day bang a. Goi M , N Ian Nen BN lirgt la trung diem cua SA va SC . Tim dp dai canh ben cua hinh chop, biet: T I T (1) va (2) ta suy ra BN 1 (SHC). 1) A N 1 BM Gpi E la trung diem ciia BC, 2) (BMN) i . (SAC). Viet CH (2). ta suy ra dupe (AME) / /(SHC), jCgigidi. Gpi O la tarn ciia day, ta c6 SO 1 (ABC) va AO = ^ , 3 1 M J V V V V H Khang nen ta c6 dupe BN 1 (AME) Suy ra BN 1 A M (dpcm). OE = ^ . D a t SA = h, h > 0 6 • . 1) Dat a = AE, b = OS, c = BC. Ta CO va a.b = b.c = c.a = aVs = h. 0 =a Vi da 2.1.4. Cho t u dien ABCD c6 AB = AC = A D . Gpi O la diem thoa man OA = OB = OC = OD va G la trpng tarn ciia tam giac ACD, gpi E la trung Ta c6: diem cua BG va F la trung diem ciia AE. Chung minh OF vuong goc voi A M = - ( A B + AS) = i ( A E + EB + A d + OS] = i BG khi va chi khi OD vuong goc vol AC. gidi. B N = - ! - ( B S + B C ) - - ( ' B E + E6 2\ 2' + OS + BC) = - Dat OA = OB = OC = OD = —a + b + -c ^ 3 2 va OA = a,OB = b,OC = c,OD = d . Do BN 1 A M nen ta c6: 5-2 ^-2 R(I) 3-2 AM.BN = 0 o - —a - a " + bb"" -—-c~ ( =0 9 4 ,2 2 _ Ar^2 , ^ 2 ^ 3 a ^ 3a^ + 9' 4 5 7,2 7a^ Suy ra SA^ - AO^ + OS^ = — + 4 6 Ta _ 1 .,2 ^ Q ^ ^ 7.2 AB = AC = A D nen AAOB = AAOC = AAOD (c - c - c) 6' suy ra AOB = AOC = A O D ,2 23a 12 CO 6 Tir ( l ) va (2) suy ra a.b = a.c = a.d 2) Goi I la trung diem M N , ta c6 A l l M N . Mat khac ( A M N ) 1 (SBC) A I 1 ( A M N ) . Suy ra A l l SE => ASAE la tarn giac can nen SA = A i ; = (2), nen — . (3). Gpi M la trung diem cua CD va do AG = 2GM nen 3BG = BA + 2BM = BA + BC + BD = OA-OB + OC-OB + OD-OB = a + c + d-3b Vida 2.2.3. Cho hinK chop S.ABCD c6 day ABCD la hinh vuong. Tarn giac SAD la tarn giac deu va nam trong mat phang vuong goc voi mat day. Gpi M , N Ian lugt la trung diem cua SB, CD. Chung minh rang A M 1 B N . ^ (4) Gpi E,F theo t h u t u la trung diem cua AE,BG ta c6 120F = 6(OA + O E ) = 60A + 3(OB + O G ) = 6 0 A + 30B + 30G = 6 0 A + 30B + OA + 2 0 M = 7 0 A + 30B + OC + OD = 7a + 3b + c + d Goi H la trung diem canh A D , suy ra SH 1 A D . Ma (SAD) 1 (ABCD) nen S H I (ABCD) Suy ra SH 1 BN (1). Taco: BN = BC + C N , C H - C D + D H . 1 Suy ra BN.CH = B C D H + CN.CD = - - BC^ + - CD^ = 0 . ^ 2 2 96 T u (4) va ( 5 ) ta c6 36BG.OF = (Za + 3b + c + d)(a - 3b + c + d) (s). , =7a^ - 9b^ + c % d^ - 18ab + 8ac + 8ad + 2cd . Theo (3) ta c6 36BG.OF = 2d(c - a) = 2 0 D . A C . Suy ra BG.OF = 0 OD.AC = 0 hay OF 1 BG o O D 1 AC . , Phumtg phiipgidi Todn Hinh hoc theo chuyen de- Nguyen Phti Khanh, Nguyen Tat Thu Vidu 2.1.S.Chotudien ABCD c6 ABJ.CD, AC ± BD . Cty mHH du 2.1.6. Cho tam giac deu A B C canh a. Goi D la diem doi xung cua A qua Goi H la true tarn tarn giac BCD. gC. Tren duong thang d 1 ( A B C D ) tai A lay diem S sao cho S D = Chung minh rang: 1) A H 1 (BCD) va A D 1 BC JCffi Gidi. 3) Cac goc xuat phat t u mot dinh ciia hinh chop cung nhon, ciing vuong hoac cung tu. Xffigidi. Gpi I la trung diem ciia B C thi A I 1 B C va I cung la trung diem cua A D . t Tac6<^ B C I A D [BCISD 1) Vi H la true tarn tam giac BCD nen CD 1 B H , CO SA1 IH Dan toi CD 1 A H . SAICB Chung minh tuong tu, ta c6: BD 1 A H . Tu do suy ra A H 1 (BCD) Ta C O A A H I ~ A A D S 2) Ta c6: AB^ + CD^ = AC^ + BD^ o AB^ - AC^ + CD^ - BD^ = 0 S A = V A D ^ f SD^ = ( A B - A C ) ( A B + AC) + (CD-Bb)(CD + BD) = 0 y ra I H - o C B ( A B + A C + C D + B D ) = 0 < » 2 C B . A D - 0 (luon dung) Tuong t u ta cung chung minh dugc cho hai dSng thuc con lai. 3) Xet tai goc A . Ta c6: cosCAD - AC^+AD^-CD^ 2AC.AD AD^ +AB2 - B D ^ Mat khac, ta c6: AB^ + CD^ = AC^ + BD^ = AD^ + BC^ Nen AB^ - BC^ = AD^ - C D ^ AC^ - CD^ = AB^ - BD^ Do do: AB.AC. cos BAC = A C A D , cos CAD = AD.AB. cos DAB Tu do ta suy ra dugc cosBAC, cos CAD, Tu do ta suy ra dpcm. 98 IH AI SD AD \i(^^f a A/6 A I . S D _ ' 2 -'2 AD " Sajl a BC "2" 2 Suy ra BHC - 90*' hay BH 1 H C . Vay BH 1 (SAC) (SAB) 1 (SAC). QBAITAP Bai 2.1.1. Cho hinh chop S.ABC c6 SA = a, SB = b, SC = c va SA, SB, SC C B ( A B + A C ) + C B ( C D + B D ) - 0 cosDAB^ ^ Ma A I = — , A D = 2AI = aV3, B C l (AHD) => B C l A D . 2.AB.AC , SAl(HCB)z^SAlBH. Ta c6: A H 1 BC, D H 1 BC cosBAC = , => B C 1 ( S A D ) =^ B C 1 S A . Dung I H 1 S A , H € S A , khi do ta c6 : AB 1 CD nen ta suy ra CD 1 (ABH) A B ^ + A C ^ - BC^ . Chung minh ( S A B ) 1 ( S A C ) . 2) AB^ + CD^ = AC^ + BD^ = AD^ + BC^ lai MTV DWH Khting Viet cos DAB cung duong, cung am 2) Tinh SH theo a, b, c 3) Chung minh rang: S^^3^=S^3^3+S^3^+S^3^^. ' • v' ' J-Iuang dan gidi ^) Ta CO SA 1 (SBC) =^ SA 1 BC. Mat khac SH 1 BC nen ta suy ra BC 1 (SAH) A H 1 BC . Tuong t u ta cung c6 AB 1 C H , hay H la true tam cua tam giac ABC. 99 Phuang phdp giai Todn Hinh hoc theo chuyen de- Nguyen Phti Khdnh, Nguyen Cty TNHH Tat Thu MTV D W H Khang Vi$t H a i m a t p h i n g ( A ' B D ) va ( M B D ) vuong goc voi nhau ^ AOMA' vuong tai O <=> OM^ + OA"^ = M A ' ' 2 a +• 5b^~ «a2=b2c:>- = l (A'BD) 1 (MBD) khi ^ = l ( K h i d 6 A B C D . A ' B ' C ' D ' la hinh lap phuong). gai 2 . 1 . 3 . Cho hinh chop deu S.ABC, c6 do dai canh day bang a. Goi M, N Ian lu'O't la trung diem ciia cac canh SA, SB. Tinh di^n tich tam giac A M N biet r^ng(AMN)l(SBC). =i(aV+bV+cV) Jiuongddngidi . ^ASAB ^^ASBC ^^ASAC la trung ^ diem cua C C . Xac dinh ti so ^ de hai mat phang ( A ' B D ) va Tu gia thiet ta c6 M N - -!-BC = - ,MN//BC 2 2 =:> I la trung diem ciia SK va M N . Ta CO ASAB = ASAC => hai trung tuyen tuong ung A M = A N => A A M N can tai A => A I 1 M N . ( M B D ) vuong goc voi nhau. (SBC) 1 ( A M N ) Jlit&ng dan gidi GQI O Mat khac la tam cua hinh vuong A B C D . lAA'lBD (SBC)n(AMN) = M N AI Taco B D = ( A ' B D ) n ( M B D ) , A C I B D A I I M N A l l (SBC) =^ A l l SK =^ ASAK can tai A=>SA = AK = '(ACC'A')/^(MBD) taco: S K ^ = S B 2 - B K ^ = Vza^ + b^ VAB^+AD^+AA'^ a' V Taco S^,,^=IUNAI = aVlO 2y '-^. B^i 2 . 1 . 4 . Cho hinh chop S.ABCDco day la hinh chu nhat, AB = a, = >/2a, SA = a va vuong goc voi mp(ABCD). Goi M, N Ian lugt la trung OA'^ = AO^ + AA'^ = M A ' 2 = A ' C ' 2 + M C ' 2 =a^+h^ + SK =>AI = VSA2-SI2 = JSA^- ( A ' B D ) VT (MBD). inn a' 3a2 = 0 M do do goc giua hai duong thang O M , O A ' chinh la goc giii-a hai mat phang A C iJ- - A- - (ACC'A')lBD ^(ACC'A')n(A'BD) = OA' Ta CO O M = - A e (AMN) (ACC'A')lBD Vay i y;/ Goi K la trung diem ciia BC va I = SK n M N . B a i 2 . 1 . 2 . Cho hinh hgp chu nhat A B C D . A ' B ' C ' D ' c 6 A B = A D = a , A A ' = b. Goi M H: * e m cua cac canh AD,SC, Goi I la giao diem cua BM, AC. Chung minh (h \ l2j 2 a + 5b2 '^P(SAC) vuong goc voi mp(SMB). Tinh the tich ciia khoi t u di?n ANIB. ' 101 Phumig phiip gii'ii Todn Hinh hoc theo chut/en de- Nguyen Phii Khdnh, Nguyen Tat Tttu J^Iu&ng dan gidi Ta c6: AM BA 1 AB BC V2 => ABM = BCA Cty TNHH MTV DWH Khang Viet EC'^ = B ' E 2 + B ' C ' 2 - 2 B ' E ' . B ' C ' . C O S 1 2 0 ' ' = • AABM ~ AABM — A E ^ + A C ' ^ =EC'2 ^ A E 1 A C ' = > B N 1 A C ' A B M + BAC = BCA + BAC=90° (2). T u (1) va (2) suy ra: A C 1 ( B D M N ) . ^ AIB = 90° = > B M 1 AC (1) Goi I , J Ian lugt la trung diem cua B D , M N va H = A C ' n IJ SA 1 (ABCD) ^ SA 1 BM (2) Ta CO A H la duong cao ciia hinh chop A . B D M N Tu (1) va (2) suy ra: Tu giac H I C C noi tiep => A H . A C = A I . A C MB 1 (SAC) => (SMB) 1 (SAC). AH = Gpi H la trung diem cua AC =^NH//SA=>NH1(ABI) Tu giac B D M N la hinh thang can ta c6 : IJ = ^ ^ N ^ - va N H = | s A = - | . Ta CO A I la duong cao cua AABM vuong tai A 1 1 1 AI^ AB^ 15a 2A C 5 The tich khoi chop A . B D M N : V = i A H . S B n M N BI = V A B 2 - A I ^ = 7l5a f BD-MN^ Di?n tich hinh thang BDMN : S = -IJ(BD + M N ) = ^^^^^. 2 16 AI = A M ^ a^ AC^ . .1 ;H / i • :• ^I'^-^^-^T^' B a i 2 . 1 . 6 . Cho hinh chop t u giac deu S.ABCD c6 day la hinh vuong canh a. Gpi E la diem doi xung ciia D qua trung diem cua SA, M la trung diem cua T h e t i c h h i d i e n A N I B : VA N I B 3 2 6 2 3 3 36 B a i 2 . 1 . 5 . Cho hinh hop dung ABCD.A'B'C'D' c6 cac canh AB = A D = a, AA' = —^ va BAD = 60". Goi M va N Ian lugt la trung diem cua A'D' va B D 1 AC DBICC B D I A C ( •/ • / (1) y : A' trung diem cua E N . Ta c6: A E / / B N , A E = B N = a. nen suy ra A C - Vsa . 2 3a^ A C ' ^ = A C ^ + C C ' ^ =3a^ + \> \ D^ Theo gia thiet ta c6 A B D deu 15a^ A / . ' _ \ _ _ \. Ta CO MP la duong trung binh ciia Va M N = - A D = N C . 2 Suy ra MNCP la hinh binh hanh ^ M N / /CP ^ M N / /(SAC). I Ta de chung minh dugc BD 1 (SAC) => BD 1 M N a i 2 . 1 . 7 . Cho duong tron (C) duong kinh ABtrong mat phang ( a ) , mot 7 MP//AD=^MP//NC. Jiit&ng ddn gidi D' Goi P la trung diem cua SA. tarn giac EAD A ' B ' . Chung minh A C ' I ( B D M N ) va tinh the tich khoi chop A . B D M N . Theo gia thiet ta c6: A E , N la trung diem cua BC. Chung minh M N vuong goc voi BD. Jiu&ng ddn gidi ' ' i.r-.:^/ 103 Phuviig phdp giai Todti Hinh hQC theo chuyun lie- Nguyen Phu Khanh, Nguyen Cty TNHHMTV Tat Thu 3) Goi I la giao diem cua H K va M B . Chung minh A I la tiep tuyen ciia duang tron (C). SAl(a) CO Lai CO MBc(a)' I^$t khac K H 1 FQ, HK / / A C ^ HK 1 ME . Suy ra K H 1 ( M E N ) = > K H 1 M N . (l) gal 2.1-9' Cho hinh lap phuong A B C D . A ' B ' C ^ D ' canh a. Tren cac canh DC va BB' lay cac diem M va N sao cho M D = NB = x (0 < x < a). Chung minh rang: (2) 1)AC'1B'D' 2)AC'1MN. (t/c goc chan nua duong tron) Dat A A ' = a,AB = b , A D = c. 2) Ta C O A K I S M , " SB c ( S B M ) ; [SBI(AHK) 2) M N = A N - A M = (AB + B N ) =^ A K 1 SB, lai A l l SB (s) va ^ ' A H 1 SB suy ra SB 1 ( A H K ) . CO A l e (a) ^ ^-AIISA S A 1 (a) b + —a - (AD+ X - c + —b = —a + 1 - ^ a DM) b-c X - Tu do ta CO A C M N = ^a + b + cj[ b + —a (4). Tu (3), (4) suy ra A I 1 ( S A B ) => A I 1 A B hay A I la tiep tuyen cua duang X-2 —a + 1 a I tron (C). Bai 2 . 1 . 8 . Cho hinh hop chu nhat ABCD.A,BjCjDj c6 day ABCD la hinh vuong. M di dong tren doan AB ( 0 < A M < AB). Lay N thuoc canh A j D j sao cho A j N = A M . Chiing minh M N luon cat va vuong goc voi mot duong thang — b a; 2 -c c +—b = —a + 1 - - b - c ] a a -2 =:x.a + 1-- a2-a2=0. VayAC'lMN. Bal 2 . 1 . 1 0 . Cho hinh chop S . A B C D c6 day A B C D la hinh thang vuong tai A D, A B = 2a, A D = DC = a, SA 1 ( A B C D ) va SA = a. 1) (a) la mat phang chua SD va vuong goc vol (SAC). Xac djnh va tinh di^n CO dinh khi M thay doi. Jiuang ddn giai tich thiet d i f n cua (a) voi hinh chop S.ABCD . N Qua M ve M E / / B D ( E e A D ) cat 2) Goi M la trung diem cua S A , N la diem thuoc canh A D sao cho A N = x . AC tai F, ta co F la trung diem cua ME • y^-Q va ME 1 AC Mat phang (p) di qua M N va vuong goc vol ( S A D ) . Xac djnh va tinh di?n It hch thiet di^n cua hinh chop cat bai (p). Do A M = A,N=>AE = AiN=i>NE//AAi. Jiuang ddn giai Goi I la trung diem cua M N , ve F I cat A , C , tai Q , ta c6 I la trung diem 'H doan F Q Goi K, H Ian luot la trung diem cac doan thang AAi, CCi suy ra K, I , H B thang hang. 104 ., f / v - A C ' . B ' D ' = (a + b + c ) ( c - b ) = a ( c - b ) + c^ - b ^ =a^ - a ^ = 0 => A C ' I B ' D ' . Suy ra A K 1 ( S B M ) . AK 1 ( S B M ) j ; i; : Ati . 1) Taco A C ' = a + b + c, B ' D ' = c - b nen MB 1 ( S A M ) , A K c ( S A M ) =:> MB 1 A K . AIC(AHK) , Jiu6fng dan giai Tu ( l ) , ( 2 ) suy ra MB 1 ( S A M ) . 3) Taco Vi§t tharig CO dinh HK. •SAIMB MB 1 M A Tuong t u Khang Vay khi M thay doi thi duong thang M N luon vuong goc va cat duong Jiuang dan giai l)Ta DVVIi ^) Goi E la trung diem cua canh ABva O la giao diem cua AC va DE thi D •^DCE la hinh vuong c6 tam la O. Ta CO SA 1 (ABCD) => SA 1 O D , them niia O D 1 AC =^ O D 1 (SAC). / ^ " " N, Tir do ta c6 OD 1 (SAC) => (SDO) 1 (SAC). •* 105 Phiwug phiip giai Todn Hinh hgc theo chuyen da - Nguyen Phii Khdnh, Nguyen Cty TNHH Tat Thu Vay (SDO) chinh la mat phang ( a ) . I Goc giua luii duang thdng cheo nhau. + SO = VOA^ + A S ^ = ,.| Khang Viet ' ' E)e tinh goc giiia hai duong thang cheo nhau a va b ta c6 cac each sau \ CO DWH § 2. G O C Thiet dien ciia hinh chop v6i mat phang (a) la tam giac SDE . Ta MTV Cdch l:T\m goc giira hai duong thang a, b bang each chon mot diem O =aj|. thich hop (O thuong nam tren mot trong hai duong thang). T u O dung cac B C = D E = aN/2 duong thang a b ' , Ian lugt song song (c6 the trung neu O nam tren mot trong hai duong thang) voi a va b. Goc giua hai duong thang a', b' chinh la goc giij-a do D E 1 ( S A C ) => D E 1 A O ^ S^^^ = |sO.DE = ^ . a ^ . a V z = 2) Taco . hai duong thSng a va b. •]>;( Chu ^'De tinh goc nay ta thuong su dung dinh l i cosin trong tam giac: ABI(SAD) b^-c^-a^ cosA = • 2 be Me(p)n(SAB) Cdch2:Tur[ ABc:(SAB) Vay u hai vec to chi phuong u ^ U j ciia hai duong thSng a, b (p) n ( S A B ) = M Q / / A B , Q e S B . AB//(p) Khi do goc giua hai duong thang a, b xac dinh boi cos(a,b) U1.U2 Ne(p)n(ABCD) Tuong t u , AB c Chu y: (ABCD) AB//(p) ^ to (p) n ( A B C D ) = N P / / A B , P e B C . Lai NP//AB MQ//AB' NP//MQ MNC(SAD) CO ABI(SAD) AB c A B I M N V ^ y SMNPQ = - ( 2 ( a 106 ta chon ba vec to a, b, c khong dong ' a 2 — +x = 4 = \? + 4 ? - X) + a) Uj,U2 qua cac vec to a, b, c roi thue hien cac tinh toan. • Tim giao diem O = a n (a) (2) • Dung hinh chieu A ' ciia mot diem A e a xuong (a) • Goc A O A ' = (p chinh la goc giua duong thang a va ( a ) . 4x^ 2 , MQ = - A B - a 2 • De dung hinh chieu A' cua diem A tren (a) ta ehgn mot duong thang b 1 (a) khi do A A V / b . DA ^Ini U2 phang (a) ta thuc hien theo cac budc sau: ,^^__^J_±A-_2[.-.) DA Uj , De xac djnh goc giiia duong thclng a va mat (l) Do do SMNPQ = | ( N P + M Q ) M N . NP D N Uj.Uj, 2. Goc giua duang thdng vai mat phang T u ( l ) / ( 2 ) suy ra (tu giac MNPQ la hinh thang vuong tai M va N . M N = VAM2+AN2 tinh phang ma c6 the tinh duoc do dai va goc giOa ehiing, sau do bieu thj cac vec Thiet dien la i\x giac M N P Q . Do DG • De tinh goc (p ta su dung he thuc luQ'ng trong tam giac vuong (3a-2x)V?+4? = ^ . AOAA'. ll^goai ra neu khong xac djnh goc cp thi ta c6 the tinh goc giira duong thang a 107 Phucmg fthtip gidi Todn Hhth hoc theo chuyen de - Nguyen Phii Khdnh, Nguyen Tat Thu va mat phang (a) theo cong thiic sincp = u.n trong do u la VTCP ciia a con phuang phap nay c6 nghia la tim hai duong thang nam trong hai mat matphSng (SAjAj) ( i , j e {l,2,...,n}; i mat (a) p h i n g (a) va (p) ta c6 the thuc . Vidu 2.2.1. Cho t i i dien ABCD. Ggi M , N Ian lugt la trung diem ciia BC va a, b Ian lugt vuong goc vai hai /a hai duong Xgigidi. thang a,b chinh la goc giiia hai mat phang (a) va (p). bl(p) , .,, • Khi do SKH la goc giua hai mat phang (SAjAj) va mat day. Cdch l : T i m hai duong thang fa 1 ( a ) j ) voi mat day ta lam nhu sau: • Tu H ve H K 1 A j A j , K e A^A^ hien theo mot trong cac each sau: mat phang (a) va ( p ) . • Dvng H N 1 A => M N 1 A . Chu ^."Cho hinh chop S.A]A2...A^ c6 duong cao SH . De xac dinh goc giua 3. Goc giua hai mat phang giua hai Khang Vift phang (a)'(P) va vuong goc voi giao tuyen A tai mgt diem tren giao tuyen. n la vec ta c6 gia vuong goc vol (a) . De tinh goc Cty TNHH MTV DWH Cdch J.-Ggi I la trung diem ciia AC . ^^^'^C/CD"^(^H"^)Dat M I N = a .((a),(p))^(a,b). Cach 2;Tim hai vec to nj,n2 c6 gia Ian lugt vuong goc vai (a) va (p) khi Xet tarn giac I M N c6 ... AB IM = do goc giiia hai mat phang (a) va (p) xac dinh boi cos(p = a CD = —,IN = a . ..^ aVs = - , M N = — 2 2 2 2 Theo djnh l i cosin, ta c6 2 Cdch S.-Su dung cong thuc hinh chieu S' = Scos(p, t u do de tinh coscp thi ta can tinh S va S'. Cdch 4: Xac dinh cu the goc giua hai mat phSng roi su dung hf thuc lugng cos a - IM^+IN^-MN^ U 2IM.IN trong tarn giac de tinh. Ta thuang xac djnh goc giiia hai mat phMng theo mot trong hai each sau: => M I N = 120° suy ra ( A B , C D ) = 60°. a) • Tim giao tuyen A = (a) n (p) IM.IN • Chgn mat phang (y) 1 A • Tim cac giao tuyen a /(p) (y) n (a), b = (y) n (p), / k h i d 6 : ( ( ^ 0 ^ ) = (Xb). Dung hinh chieu H cua M tren (a) 108 ^ ^ COS(AB,CD) = COS(IM,IN) = IM IN . M M N = IN - IM MN^ = (IN - IM)^ = M I ^ + IN^ - 2IN.IM; \ b) • Tim giao tuyen A = (a) n (p) •LayME{p). 'h2: ^ 7 \ " V y V IM^+IN^-MN^ a^ INIM . == V^iy ( A B , C D ) = 6 0 ' ' . 8 ; COS(AB,CD)= COS(IM,IN) IM.IN IM IN '1 109 Cty TNHH MTVDWH Phumig phapgiai foan Hinh hoc theo chmjen dc - Nguyen Pliii Khdnh, Nguyen Td't Thu Vidu 2.2.2. Cho hlnh chop S.ABCD c6 day ABCD la hinh thang vuong tai A va B, AB = BC = a, AD = 3a. Hinh chieu cua S len mat phang day trung voi trung diem canh AD. Mat phang (SCD) tao voi day mot goc 60". Goi M la trung diem doan CD. Tinh c6 sin cua goc giua hai duang thang A M va SC. £gl gidi. Goi H la trung diem cua AD, ta c6 SH 1 (ABCD). Ta CO ABCH la hinh vuong canh a. Tit H ve UK 1 CD , K € CD . Khang Viet Ap dving djnh li Co sin cho tam giac A M E , ta c6: AM^+ME^-AE^ 57 2.MA.ME 474181 cos AME = • 57 Vay cos(AM,SC) = ^ - ^ . X>i dii 2.2.3. Cho hinh chop S.ABC c6 SA 1 (ABC), SA = a, AB = a, AC = 2a, 5 A C = 120°. Tinh c6 sin ciia goc giiia hai mat phang: 1) (ABC) va (SBC) Suy ra CD±(SHK), nen SKH la 2) (SAC) va (SBC). goc giOa mat phang (SCD) voi mat day, do do SKH = 60''. Ta c6: CD = 7AI3^ + (AD - BCf = . ^• Goi I la hinh chieu cua C len AD. Do tam giac CID dong dang voi tam giac HKD nen suy ra: HK HD CI CD =>HK = 1) Gpi K la hinh chieu cua A len BC, ta c6 BC 1 (SAK) nen SKA la goc giua hai mat phang (SBC) va mat phang (ABC). Do do SH = HKtan60" = Taco: BC^ = AB^ + AC^-2AB.AC.cosBAC = 73^ =^ BC = aV7 . Gpi E, F Ian \ugt la trung diem cua cac doan thing SD va HD. Taco M E / / S C , E F / / S H , EF = | S H = A K . B C = . ^ A K = ^ = ^ S K Suy ra cos SKA = Suy ra 2 A E ^ = A F ^ + FE^ = 20 — 7 A K VSO SK 10 Suy ra tam giac SMC la hinh chieu cua tam giac SBC len mat phSng (SAC) A F = S 1 A D = ^ 4 4 'ASBC (Theo cong thuc hinh chieu, ta c6: coscp = - ^ ^ ^ voi (p la goc giua hai m^t . Taco: M F l A D = * M F = l c i = - , suy ra AM = N / A F ^ T M F ^ 2 2 110 VTO 2) Gpi M la hlnh chieu cua B len AC, ta c6 B M 1 (SAC) IH = A H - A I - A H - B C = - r ^ C H = ^ ' ^ 2 2 Nen M E = i s C = i V c H 2 + S H 2 = i : ^ , = VSA2 + A K 2 = ^ 7 va ( A M ^ S C ) = ( A M > I E ) . Gpi I la hinh chieu cua C len AD, ta c6: 2 A B . A C . s i n l 2 0 ' ' {=2SAABC) tiang (SAC) va (SBC) = . 10 Tac6:S^BC=-SK.Bc4.^.a7^ = - Phumtgphapgiai Todn Hinh hgc theo chuyen de- Nguyen Do B A M = 60" => A M = ABcos60° = Vhy cos(p = 2 Phii KItdnh, Nguyen ^ S^^MP = ^SA.MC = • -ASMC 2 Tat Cty Thu 4 Taco HB^ = 0 H 2 + 0 B 2 = va BAC = 120" . Goi M la trung diem cua canh C C . Tinh c6 sin cua goc giug hai mat phang (BMA') vai (ABC). Xgigidi. Taco: BC^ = AB^ + AC^ - 2AB.ACcosl20° = 7a^ Xet A M H N c6 M N = , 2^ 1 2 = M H = N H tan60° = ^ ^ = MH^+Ol2 = 15a' > 1 + = > A ' M = 3a . IJ = a72 va IK = BM^ = BC^ + CM^ = 7a^ + 5a^ = 12a^ ^ Ta CO tam giac ABC la hinh chieu cua tam giac B M A ' len mat phang (ABC), nen ap dyng cong thuc hinh chieu ta c6: I 4 2 / I ^2a J / i > a 2 a^|i Suy ra tam giac A ' M B vuong tai M , suy ra S^^'MB = ^ M A ' . M B - Sa^Vs ra72^ MJ tn — (p - JK A'B^ = A ' A ^ + AB^ = 21a2 = A ' M ^ + BM^ 4 a72 2 Vaygoc giira M N va (SBD) la (p = a r c t a n i . Cdch2;TaCO M N = i(sC + A B ) = |(sO + OC + A O + O B ) - ~ ( S O ^= ((iKiX^^)). 1 ^ Vi du 2.2.5. Cho hinh chop S.ABCD c6 day la hinh vuong canh a, O la tam cua day, SO 1 ( A B C D ) ; M , N Ian lugt la trung diem cua SA, CD. Biet goc giua M N voi ( A B C D ) bang 60° . Tinh goc giira M N va ( S B D ) . JCffi Gidi M H / /SO,H e O A . [MH/ZSO DoL^ . „_,^MHI(ABCD) ^ . Isfl va MJ 1 ( S B D ) :=> UK] la goc giiia M N va (SBD) . Ta CO i f = 1 0 ^ + 0 12 A ' M ^ = A'C'^ + C M ^ =4a2 +53^ =9a^ BM = 2aV3 ^ cos 60° G(?i I la trung diem cua OB, J la trung diem ciia SO thi M J / /IN va MJ = I N . =>BC = aV7 Suy ra MN^ = ifSO^ + AC^ + OB^) = ' 44\V " 4 / 4^ => M N H chinh la goc giiia duong thSng M N voi ( A B C D ) . =:>MN = - , 2 Taco (p la goc giua M N va (SBD) nen sin(p = MN.n r>^2 5a' 2V J ( n la vec to c6 gia MN n Jong goc voi ( S B D ) ). ["^(2 X SO • Do <^ => A C 1 (SBD) ner\n n = A C , t u do ta c6 ACIBD ^ ' -(SO + AC+OB)AC ^ suy ra N H la hinh chieu aia M N tren ( A B C D ) 112 8 Viet GQi K = I J n M N = > J K = |lJ A' [SO 1 ( A B C D ) 2 DWHKhang .^5 27To ' COSCP = | M B C . ^ ^ ^ABMA' 3a^V3 8 , 3 W 2 . 2 . 4 . Cho l a n g t r u dung A B C A ' B ' C c6 AB = a; AC = 2a; A A ' = 2aV5 Cdch l:Ke 4 TNHHMTV sincp = 2a 113