Nối tiếp nội dung phần 1 tài liệu Phương pháp giải toán Hình học theo chuyên đề, phần 2 giới thiệu các phương pháp giải toán hình học trong không gian tổng hợp, phương pháp tọa độ trong không gian. Mời các bạn cùng tham khảo nội dung chi tiết.
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 N G HpTP § Q U A N Hfi V U O N G G O C 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 = nen suy 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 hinh hpc phang ^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 nam (P) , Chu y: Neu duong thSng a vuong goc voi mat phang (P) thi duong thang a vuong goc voi moi duong thang nam mat phang (P) vudnggoc De chung minh hai mat phang vuong goc, ta chiing minh mat phang chua mot duong thang vuong goc voi mat phang Chu y: Neu hai mat phSng cat theo giao tuyen A va vuong goc voi thi mgi duong thang nam mat phang ma vuong goc voi A thl duong thang se vuong goc voi mat phang 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 BC Mat khac A H SB nen ta c6 A H (SBC) Suy A H SC Chung minh tuong t u ta cung c6 A K SC Tir suy SC ( 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 SC nen SI.SC - SA^ ^ SI = SuyraH.SI^j,j^SI.BC «C SB Tuong tu: K I = — SB Tir suy BC (SAB), suy BC 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 ac V(b2 4-c2)(a2 + b + c ) AH = AB BC SC voi 3) Tinh di^n tich thiet di#n ciia hinh chop cat boi mat phMng ( A H K ) Do SA (ABCD) nen suy SA B C Lai c6 ABCD la hinh chu nhat nen SA^ Vi SC (AHK) nen H I SC, 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 (AHK) JCffigidi Khang Viet S = S/iSAB + SASBC + S^sCD + SASDA 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 ) ( b + c ) 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 BN (SHC) 1) A N BM Gpi E la trung diem ciia BC, 2) (BMN) i (SAC) Viet CH (2) ta suy dupe (AME) / /(SHC), jCgigidi Gpi O la tarn ciia day, ta c6 SO (ABC) va AO = ^ , M J V V V V H Khang nen ta c6 dupe BN (AME) Suy BN A M (dpcm) OE = ^ D a t SA = h, h > • 1) Dat a = AE, b = OS, c = BC Ta CO va a.b = b.c = c.a = aVs = h =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 va chi 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 ^ va OA = a,OB = b,OC = c,OD = d Do BN A M nen ta c6: 5-2 ^-2 R(I) 3-2 AM.BN = o - —a - a " + bb"" -—-c~ ( =0 ,2 _ Ar^2 , ^ ^ a ^ 3a^ + 9' 7,2 7a^ Suy SA^ - AO^ + OS^ = — + Ta _ ,2 ^ Q ^ ^ 7.2 AB = AC = A D nen AAOB = AAOC = AAOD (c - c - c) 6' suy AOB = AOC = A O D ,2 23a 12 CO Tir ( l ) va (2) suy 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 ) (SBC) A I ( A M N ) Suy A l l SE => ASAE la tarn giac can nen SA = A i ; = (2), nen — (3) Gpi M la trung diem cua CD va 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 mat phang vuong goc voi mat day Gpi M , N Ian lugt la trung diem cua SB, CD Chung minh rang A M 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 ) = A + 30B + 30G = A + 30B + OA + M = A + 30B + OC + OD = 7a + 3b + c + d Goi H la trung diem canh A D , suy SH A D Ma (SAD) (ABCD) nen S H I (ABCD) Suy SH BN (1) Taco: BN = BC + C N , C H - C D + D H Suy BN.CH = B C D H + CN.CD = - - BC^ + - CD^ = ^ 2 96 T u (4) va ( ) 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) = D A C Suy BG.OF = OD.AC = hay OF BG o O D 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 ( A B C D ) tai A lay diem S cho S D = Chung minh rang: 1) A H (BCD) va A D 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 B C va I cung la trung diem cua A D t Tac6- = l (A'BD) (MBD) ^ = l ( K h i d A B C D A ' B ' C ' D ' la hinh lap phuong) gai Cho hinh chop deu S.ABC, c6 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 =:> 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 M N ( M B D ) vuong goc voi (SBC) ( 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 - B K ^ = Vza^ + b^ VAB^+AD^+AA'^ a' V Taco S^,,^=IUNAI = aVlO 2y '-^ B^i 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 ' = A ' C ' + M C ' =a^+h^ + SK =>AI = VSA2-SI2 = JSA^- ( A ' B D ) VT (MBD) inn a' 3a2 = M 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 Cho hinh hgp chu nhat A B C D A ' B ' C ' D ' c 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 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 AB BC V2 => ABM = BCA Cty TNHH MTV DWH Khang Viet EC'^ = B ' E + B ' C ' - B ' E ' B ' C ' C O S ' ' = • AABM ~ AABM — A E ^ + A C ' ^ =EC'2 ^ A E A C ' = > B N A C ' A B M + BAC = BCA + BAC=90° (2) T u (1) va (2) suy ra: A C ( B D M N ) ^ AIB = 90° = > B M AC (1) Goi I , J Ian lugt la trung diem cua B D , M N va H = A C ' n IJ SA (ABCD) ^ SA 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 (SAC) => (SMB) (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 AI^ AB^ 15a 2A C The tich khoi chop A B D M N : V = i A H S B n M N BI = V A B - A I ^ = 7l5a f BD-MN^ Di?n tich hinh thang BDMN : S = -IJ(BD + M N ) = ^^^^^ 16 AI = A M ^ a^ AC^ ;H / i • :• ^I'^-^^-^T^' B a i 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 36 B a i 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 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 A C - Vsa 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 Suy MNCP la hinh binh hanh ^ M N / /CP ^ M N / /(SAC) I Ta de chung minh dugc BD (SAC) => BD M N a i Cho duong tron (C) duong kinh ABtrong mat phang ( a ) , mot 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 FQ, HK / / A C ^ HK ME Suy K H ( M E N ) = > K H 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 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 SB, lai A l l SB (s) va ^ ' A H SB suy SB ( A H K ) CO A l e (a) ^ ^-AIISA S A (a) b + —a - (AD+ X - c + —b = —a + - ^ a DM) b-c X - Tu ta CO A C M N = ^a + b + cj[ b + —a (4) Tu (3), (4) suy A I ( S A B ) => A I A B hay A I la tiep tuyen cua duang X-2 —a + a I tron (C) Bai Cho hinh hop chu nhat ABCD.A,BjCjDj c6 day ABCD la hinh vuong M di dong tren doan AB ( < A M < AB) Lay N thuoc canh A j D j cho A j N = A M Chiing minh M N luon cat va vuong goc voi mot duong thang — b a; -c c +—b = —a + - - b - c ] a a -2 =:x.a + a2-a2=0 VayAC'lMN Bal 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 ( 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 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 cho A N = x AC tai F, ta co F la trung diem cua ME • y^-Q va ME 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 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 ^ = => A C ' I B ' D ' Suy A K ( S B M ) AK ( S B M ) j ; i; : Ati 1) Taco A C ' = a + b + c, B ' D ' = c - b nen MB ( S A M ) , A K c ( S A M ) =:> MB A K AIC(AHK) , Jiu6fng dan giai Tu ( l ) , ( ) suy MB ( S A M ) 3) Taco Vi§t tharig CO dinh HK •SAIMB MB M A Tuong t u Khang Vay 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 (ABCD) => SA O D , them niia O D AC =^ O D (SAC) / ^ " " N, Tir ta c6 OD (SAC) => (SDO) (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 + SO = VOA^ + A S ^ = ,.| Khang Viet ' ' E)e tinh goc giiia hai duong thang cheo a va b ta c6 cac each sau \ CO DWH § 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 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 hai duong thang) voi a va b Goc giua hai duong thang a', b' chinh la goc giij-a D E ( S A C ) => D E A O ^ S^^^ = |sO.DE = ^ a ^ a V z = 2) Taco hai duong thSng a va b •]>;( Chu ^'De tinh goc ta thuong su dung dinh l i cosin tam giac: ABI(SAD) b^-c^-a^ cosA = • 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 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 = - ( ( a 106 ta chon ba vec to a, b, c khong dong ' a — +x = = \? + ? - 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^ , MQ = - A B - a • De dung hinh chieu A' cua diem A tren (a) ta ehgn mot duong thang b (a) 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 SMNPQ = | ( N P + M Q ) M N NP D N Uj.Uj, Goc giua duang thdng vai mat phang T u ( l ) / ( ) suy (tu giac MNPQ la hinh thang vuong tai M va N M N = VAM2+AN2 tinh phang ma c6 the tinh duoc dai va goc giOa ehiing, sau 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 tam giac vuong (3a-2x)V?+4? = ^ AOAA' ll^goai 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 u la VTCP ciia a phuang phap c6 nghia la tim hai duong thang nam 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 SKH la goc giua hai mat phang (SAjAj) va mat day Cdch l : T i m hai duong thang fa ( a ) j ) voi mat day ta lam nhu sau: • Tu H ve H K A j A j , K e A^A^ hien theo mot cac each sau: mat phang (a) va ( p ) • Dvng H N A => M N A Chu ^."Cho hinh chop S.A]A2 A^ c6 duong cao SH De xac dinh goc giua 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) Xet tarn giac I M N c6 AB IM = goc giiia hai mat phang (a) va (p) xac dinh boi cos(p = a CD = —,IN = a ^ aVs = - , M N = — 2 2 Theo djnh l i cosin, ta c6 Cdch S.-Su dung cong thuc hinh chieu S' = Scos(p, t u 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 tarn giac de tinh Ta thuang xac djnh goc giiia hai mat phMng theo mot hai each sau: => M I N = 120° suy ( A B , C D ) = 60° a) • Tim giao tuyen A = (a) n (p) IM.IN • Chgn mat phang (y) A • Tim cac giao tuyen a /(p) (y) n (a), b = (y) n (p), / k h i d : ( ( ^ ^ ) = (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: ^ \ " V y V IM^+IN^-MN^ a^ INIM == V^iy ( A B , C D ) = ' ' ; 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 (ABCD) Ta CO ABCH la hinh vuong canh a Tit H ve UK 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 (ABC), SA = a, AB = a, AC = 2a, A C = 120° Tinh c6 sin ciia goc giiia hai mat phang: 1) (ABC) va (SBC) Suy CD±(SHK), nen SKH la 2) (SAC) va (SBC) goc giOa mat phang (SCD) voi mat day, 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 (SAK) nen SKA la goc giua hai mat phang (SBC) va mat phang (ABC) 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 cos SKA = Suy A E ^ = A F ^ + FE^ = 20 — A K VSO SK 10 Suy tam giac SMC la hinh chieu cua tam giac SBC len mat phSng (SAC) A F = S A D = ^ 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 AM = N / A F ^ T M F ^ 2 110 VTO 2) Gpi M la hlnh chieu cua B len AC, ta c6 B M (SAC) IH = A H - A I - A H - B C = - r ^ C H = ^ ' ^ 2 Nen M E = i s C = i V c H + S H = i : ^ , = VSA2 + A K = ^ va ( A M ^ S C ) = ( A M > I E ) Gpi I la hinh chieu cua C len AD, ta c6: A B A C s i n l ' ' {=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 = Phii KItdnh, Nguyen ^ S^^MP = ^SA.MC = • -ASMC Tat Cty Thu Taco HB^ = H + B = 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^ = M H = N H tan60° = ^ ^ = MH^+Ol2 = 15a' > + = > 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 / I ^2a J / i > a a^|i Suy tam giac A ' M B vuong tai M , suy S^^'MB = ^ M A ' M B - Sa^Vs ra72^ MJ tn — (p - JK A'B^ = A ' A ^ + AB^ = 21a2 = A ' M ^ + BM^ a72 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^^)) ^ Vi du 2.2.5 Cho hinh chop S.ABCD c6 day la hinh vuong canh a, O la tam cua day, SO ( 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 ( S B D ) :=> UK] la goc giiia M N va (SBD) Ta CO i f = ^ + 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 MN^ = ifSO^ + AC^ + OB^) = ' 44\V " / 4^ => M N H chinh la goc giiia duong thSng M N voi ( A B C D ) =:>MN = - , 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 (SBD) ner\n n = A C , t u ta c6 ACIBD ^ ' -(SO + AC+OB)AC ^ suy N H la hinh chieu aia M N tren ( A B C D ) 112 Viet GQi K = I J n M N = > J K = |lJ A' [SO ( A B C D ) DWHKhang ^5 27To ' COSCP = | M B C ^ ^ ^ABMA' 3a^V3 , W 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 TNHHMTV sincp = 2a 113 Suy mat phSng (P) luon di qua M 1 Cly TNllIl MTV DWH Kliniig Vict n v2 2J' BAI TAP 2) Gia sir mat cau (S) ngoai tie'p liinh chop OABC c6 phimng trinh: + + gdi 3.4.1 Lap phuong trinh mat cau (S) biet - 2mx - 2ny - p z + q = 1) Mat cau (S) c6 tam I(l;2;3) ban kinh R = Vs Thay toa cac diem A, B, C, O vao ta c6: a -2ma+q=0 -2nb + q = c^-2cp + q = p=0 2) Mat cau (S) c6 tam nam tren Ox va di qua A ( l ; ; l ) , B(3;l;-2) a b m=-; n = - 3) Mat cau (S) eo tam 1(3;-2; 4) va tie'p xuc v6i mp(P): 2x - y + 2z + = 4) Mat cau (S) di qua C(2;-4;3) va cac hinh chieu cua C len ba true toa dp p = f; q = o 5) Mat cau (S) c6 tam nam tren mp(Oxy) va di qua M(1;0;2),N(-2;1;1), va p(-i;-i;i)- Suy tam cua mat cau (S) la I (a h c l2'2'2 Jiu&ng ddn gidi 1) Phuong trinh mat cau (S): (x -1)^ + (y - 2)^ + (z - 3)^ = Va ban kinh R = lO = i Va^ + b^ + c^ 2) Goi I la tam mat cau Vi I e Ox Suy tam 1(2; 0;0) va ban kinh R^ = IB^ = Ta cung CO hai BDT tuang t u : b ^ + — > — ; c^+ — > — ^ • 4b 4c Cong ba BDT lai v o l ta dxxgc: (I \ - + - + Va b c) Vay phuong trinh mat cau (S): (x - 2)^ + y^ + z^ = 3) Vi mat cau (S) eo tam 1(3;-2;4) va tie'p xiic voi mp(P) Suy R = d(I,(P)) = D5ng thiic xay o a = b = c = - Vay R = ^ Stp ab + bc + ca + V a V b V c V ^ I bV+cV) Suy C i (2; 0; 0), C2 (0; -4; 0), C3 (0; 0; 3) Giasu (S):x2 +y2 + z2 - a x - b y - c z + d - Do (S) d i qua C C p C j / ^ a nen ta c6 h$ phuong trinh \+4 29 a^b^ + b^c^ + c^a^ > i(ab + be + ca)^ Nen r < - — abc ^ 2abc + Aabc V3 -4a + d = - 2(N/3 + 1) 8b + d = -16 V3' Mat khae: a^b^ + b^c^ + c^a^ < (ab + be + ca)^ = (2abc)2 +{z-^f 4) Goi C p C j X a Ian lugt la hinh chieu ciia C len cac true toa dp Ox,Oy,Oz Tu gia thiet, suy ab + be + ca = 2abc Ma 20 Vay phuong trinh mat cau (S):(x -3)^ + (y + if Di^n ti'ch toan phan ciia t u dien: S^p = i ( a b + be + ca + Va^b^ + abc 12.3 + + 2.4 + 41 ' 3) Ta CO the tich ciia t u dien OABC la: V Q ^ B ^ = - a b c 3V l(x;0;0) Ta CO l A ^ = IB^ o (x - ) + 2^ +1^ = (x - 3)2 +1^ + (-2)^ x = Ap dung BDT Cosi ta c6: a^ + — + — > — a^ + — >27 8a 8a 4a Suy r = r > abc 4abc -6c + d = -9 a= u b= a= l d + 16 d +9 b = -2 ôã c= -2d = Vay phuong trinh m | t cau (S): x^ + y^ + d =0 - - 2x + 4y - 3z = Vi tam I cua mat cau nam tren mp(Oxy) nen l(x;y;0) 284 • Phumig phdp giai Todn Hinh hoc theo chuyen di- Nguyen Phu Khanh, Nguyen Cty TNHH T&'t Thu Ta Ta c6: IM2=IN^ Va R2 = I M ^ = r-6x + 2y = l 10 y=5 CO IM(1; 6; 5) nen [ I M , u ^ J= MTV DWH Khang Viqt (1; - 4; 5), do ^ x - y = -3 UO ^(-1)2+I2+I2 "A- 109 20 Vi mat cau cat A' tai hai diem A,B nen ban kfnh mat cau dugc xac dinh f Vay phuong trinh mat cau (S): | ~ ~ r 44 f ''-5; +z theo cong thuc : R = d (I, A') + 109 =• 20 AB^' = 14 + 36 = 50 Vay mat cau can tim c6 phuong trinh la: (x -1)2 + (y - 3)2 + (z -5)2 = 50 Bai 3.4.2 Lap phuang trinh mat cau S(I,R) biet 3) Duong thang d' qua diem N ( - ; - ; ) va c6 u^ = ( l ; l ; - ) la VTCP 1) Mat cau c6 tam thuoc duong thang A : ~ ^ = = va tiep xiic v6i Vi tam mat cau I € d nen 1(2 +1; - ; + 2t) mat phang (a^): 3x + 2y + z - = va mat phang ( a j ) : 2x + 3y + z = Taco M I = (t + l ; - t - l ; t - l ) , N I = (4 + t ; - t ; - l + 2t) nen ' N I , U d - ] = ( t - ; t + 15;2t + 2) 2) Mat cau c6 tam 1(1;3;5) va cat A': ^ ^ = ^ ^ = -^ tai hai diem A,B Vi mat cau qua diem M va tiep xiic voi d ' nen M I - d(I,d') = R c h o A B = 12 3) Mat cau c6 tam thuQc duong thang d : ~Y~^^~~y~' M(l;l;4) Do M I - < ^ x+2 y +2 z-4 va tiep xuc voi d : = = ^ 1 - Jiitang ddn gidi t= -l o 18(6t2 + 3) = 44t2 + 160t + 278 Mat cau tiep xiic voi hai mat phang (a^) va (a2) nen : d ( l ( a i ) ) = d(I,(a2)) = R « =1 Voi t = - l thi 1(1; 1;1), R = nen phuong trinh mat cau Suy |3(-2 + t) + 2(l + t ) - l - t - 2^+12 2(-2 + t) + 3(l + t ) - l - t V22+3^+12 •6t-ll = 7t-2 t = -9 t - l l = 2-7t t= l (x-l)2+(y-l)2+(z-l)2=9 Voi t = - thi I i l ; - ^ ; 2 , 'x • Neu t = thi I ( - l ; 3; - 3),R = -y= nen phucmg trinh m^t cau ^ Ij 3N/34 nen phuong trinh mat cau { 7^ 153 + y + - + (z-10)^ = V 2) Bai 3.4.3 Trong khong giian Oxyz, cho mat cau (S) c6 phuang trinh (x + l ) + ( y _ ) + ( z + 3)2^25 14 • Neu t = -9 thi I ( - l l ; - ; 17),R = Ud'J i -J J(2t - 7)2 + (6t +15)2 + (2t + 2)2 Hay V(2t -1)2 + 2(1 +1)2 = ^ / ' ^ 1^+1^+4^ 1) V i t a m l e A nen I(-2 +1; + t ; - - t ) t - l l | = |7t-2 [NI, x2 + y2 + z2 + 2x - 4y + 6z + m = 65 nen phuang trinh m|it cau 4225 (x +11)^ + (y +17)2+(z-17)2 = 14 2) Duang thSng A' qua diem M(2; -3; 0) va c6 vec ta chi phuang la u ^ , ( - l ; 1; 1) Tim m cho: 1) Mat cau tiep xiic voi mat ph4ng (P): x - 2y + 2z - = 2) Mat cau cat mat phSng (Q) :2x - y - 2z +1 = theo giao tuyen la mpt duong 14 tron CO dien tich bang 4n % 287 Phuotig phlip giiii Toiiit Hiiih hoc theo chtiyen ile- Nguyen Phii Khdiih, 3) M a t cau cat d u o n g thang A : — — = tai hai d i e m phan b i r t A , B = ' ^ Ngiii/eii Till Tim cho tarn giac TAB v u o n g ( I la tarn mat cau) Cty TNHH 1) Ta CO d ( I , ( a ) ) = + + + 71 V22+2^+12 Vigf = < R , suy ( a ) cat m a t cau (S) theo H l a h i n h chieu cua I len m a t phang ( a ) , suy p h u o n g t r i n h cua H I la: D i e u k i e n ton tai mat cau 14 - m > x = l + 2t M a t cau (S) c6 tarn I ( - ; 2; - ) va ban k i n h R = V l - m ' y = l + 2t 1) K h o a n g each t u tarn mat cau den mat phang la z =l +t x = l + 2t ^2+(_2)2+22 V i mat cau tiep xiic v o i mat phang nen R = m = -2 Toa d o d i e m I la n g h i e m ciia h ^ Vay gia t i i can t i m cua m la m - - , -i|;14tlL = l Vl^+(-2)2+22 Vay tam H V i d u o n g t r o n giao t u y e n c6 d i $ n tich la An nen c6 ban k i n h r = , d o ban k i n h cua m a t cau la: R^ = + d^{l, (Q)) o R = o l - m = < : : > m - Vay gia t r j can t i m la: m = y ^ l + 2t x=y= z =l+t -1 2x+2y+z+7=0 z - _5 _ \ "3' 3' x =l+t 2) Ta CO A B = (2; ; - ) nen p h u o n g t r i n h d u o n g thang A B : y = - l + 3t y = 2-2t 3) G o i H ( - l - t; 2t; - t) la h i n h chieu cua 2; - 3) tren A Ta CO i H ( - t ; 2t - 2; - t) va u ^ ( - l ; 2; - 2) nen IHlAd(I,d) = G