Một số phương pháp giải toán Hình học theo chuyên đề: Phần 2

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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.

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Nội dung Text: Một số phương pháp giải toán Hình học theo chuyên đề: Phần 2

Phuamg phapgiai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu Cty TNHH MTV DWH Khang Viet nen suy ra tam giac S C O vuong tai D S Vay cac mat ben ciia hinh chop deu la H I N H H Q C K H O N G G I A N T 6 N G HpTP nhiJng tam giac vuong. § 1. Q U A N Hfi V U O N G G O C Di^n tich xung quanh ciia hinh chop la: S = S/iSAB + SASBC + S^sCD + SASDA 1 .Jiai durnig thdng vuong goc = i ( S A . A B + SB.BC + S D . C D + S A . A D ) De chu-ng minh hai duong thang AB va CD vuong goc vol nhau, ta c6 cac each sau ac + bVa^ +c^ + aVc^ + b^ + be Cach 1; Chung minh AB.CD = 0 2 Cdch 2;Chung minh c6 mot mat phang (P) chua AB va vuong goc voi CD 2) Ta c6: B C l ( S A B ) va A H c ( S A B ) Cach 3;Su dung cac ket qua da biet trong hinh hpc phang nen A H 1 BC . Mat khac A H 1 SB 2. ^itang thdng vuong goc voi mat phdng. nen ta c6 A H 1 (SBC) De chung minh duong thang a vuong goc voi mat phang (P) ta thuang di Suy ra A H 1 SC . chung minh duong t h i n g a vuong goc voi hai duong thang cat nhau nam Chung minh tuong t u ta cung c6 A K 1 SC . Tir do suy ra SC 1 ( A H K ) . trong (P). , 3) Ggi I la giao diem ciia SC voi mat phang (AHK), ta c6 thiet dien ciia hinh Chu y: Neu duong thSng a vuong goc voi mat phang (P) thi duong thang a chop cat bai mat phang ( A H K ) la tu giac A H I K . vuong goc voi moi duong thang nam trong mat phang (P). 3.Jiai mdt ptidng vudnggoc Ta CO hai tam giac A H I va A K I la hai tam giac vuong tai H va K. i De chung minh hai mat phang vuong goc, ta chiing minh mat phang nay SA^ Do A I 1 SC nen SI.SC - SA^ ^ SI = chua mot duong thang vuong goc voi mat phang kia. SC Va^Tb^Tc^ Chu y: Neu hai mat phSng cat nhau theo giao tuyen A va vuong goc voi Vi SC 1 (AHK) nen H I 1 SC, do do hai tam giac SIH va SBC dong dang nhau thi mgi duong thang nam trong mat phang nay ma vuong goc voi A thl voi nhau. duong thang do se vuong goc voi mat phang kia. ,2 Vi du 2.1.1. Cho hinh chop S.ABCD c6 day ABCD la hinh chii nhat voi dp SuyraH.SI^j,j^SI.BC bc^ «C SB — SB ^^^^^^ dai cac canh AB = a, A D = b. Canh ben SA vuong goc voi mat phang day va a^ + b^+c^) SA c. Gpi H , K Ian lugt la hinh chieu ciia A len cac duong thang SB, SD. 5 SI.CD ac 1) Chung minh rang eac mat ben ciia hinh chop la nhirng tam giac vuong. Tuong tu: K I = SD V(b2 4-c2)(a2 + b 2 + c 2 ) Tinh di^n tich xung quanh ciia hinh chop theo a,b,c T r o n g t a m g i a c v u o n g SAB ta c6: 2) Chung minh rang SC 1 (AHK) AB.AS ac 3) Tinh di^n tich thiet di#n ciia hinh chop cat boi mat phMng ( A H K ) . AH = SB JCffigidi. 1) Ta CO cac tam giac SAB, SAD la nhung tam giac vuong tai A AD.SA be Tuong t v : A K = = _ Do SA 1 (ABCD) nen suy ra SA 1 B C . Lai c6 ABCD la hinh chu nhat nen AD V b ^ ^ AB 1 BC V^y di^n tich thiet dign can tinh la: Tir do suy ra BC 1 (SAB), suy ra BC 1 AB hay tam giac SBC vuong tai B abc^ Tuong tu, ta chung minh duoc CD ± (SAD) S A H I K = | ( H I . A H + K I . A K ) = (a^ + b^ + c^ ) V ( a ^ + c 2 ) ( b 2 + c 2 ) 94 95
Phucmg phdp gidi Toiin Hinh hgc theo chuyen de- Nguyen PM Khdnh, Nguyen Tat Thu C t y rmm M J V V V V H Khang Viet Vidu 2.1.2. Cho hinh chop deu S.ABC c6 canh day bang a. Goi M , N Ian Nen BN 1 CH (2). 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 2) (BMN) i . (SAC). Gpi E la trung diem ciia BC, jCgigidi. ta suy ra dupe (AME) / /(SHC), Gpi O la tarn ciia day, ta c6 SO 1 (ABC) nen ta c6 dupe BN 1 (AME) Suy ra BN 1 A M (dpcm). va AO = ^ , OE = ^ . D a t SA = h, h > 0 3 6 • . 1) Dat a = AE, b = OS, c = BC. Ta CO a.b = b.c = c.a = 0 aVs va = 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 khi va chi khi OD vuong goc vol AC. gidi. B N = - ! - ( B S + B C ) - - ( ' B E + E6 + OS + BC) = - —a + b + -c Dat OA = OB = OC = OD = R(I) 2\ 2' ^ 3 2 va OA = a,OB = b,OC = c,OD = d . Do BN 1 A M nen ta c6: Ta CO AB = AC = A D nen 5-2 ^-2 3-2 5 3a^ + AM.BN = 0 o - —a - a " + bb"" -—-c~ 7.2 ( =0 _ 1 .,2 ^ Q ^ ^ AAOB = AAOC = AAOD (c - c - c) 9 4 9' 4 6' 2 _ Ar^2 , ^ 2 ^ 3 a ^ ,2 7,2 7a^ 23a ,2 suy ra AOB = AOC = A O D (2), Suy ra SA^ - AO^ + OS^ = — + 4 6 12 6 Tir ( l ) va (2) suy ra a.b = a.c = a.d (3). 2) Goi I la trung diem M N , ta c6 A l l M N . Mat khac ( A M N ) 1 (SBC) nen Gpi M la trung diem cua CD va do AG = 2GM nen ^ A I 1 ( A M N ) . Suy ra A l l SE => ASAE la tarn giac can nen SA = A i ; = — . 3BG = BA + 2BM = BA + BC + BD = OA-OB + OC-OB + OD-OB = a + c + d-3b (4) Vida 2.2.3. Cho hinK chop S.ABCD c6 day ABCD la hinh vuong. Tarn giac Gpi E,F theo t h u t u la trung diem cua AE,BG ta c6 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 . 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 (s). Goi H la trung diem canh A D , suy ra SH 1 A D . T u (4) va ( 5 ) ta c6 36BG.OF = (Za + 3b + c + d)(a - 3b + c + d) , Ma (SAD) 1 (ABCD) nen S H I (ABCD) Suy ra SH 1 BN (1). =7a^ - 9b^ + c % d^ - 18ab + 8ac + 8ad + 2cd . Taco: BN = BC + C N , C H - C D + D H . Theo (3) ta c6 36BG.OF = 2d(c - a) = 2 0 D . A C . 1 Suy ra BN.CH = B C D H + CN.CD = - - BC^ + - CD^ = 0 . Suy ra BG.OF = 0 OD.AC = 0 hay OF 1 BG o O D 1 AC . , ^ 2 2 96
Phumtg phiipgidi Todn Hinh hoc theo chuyen de- Nguyen Phti Khanh, Nguyen Tat Thu Cty mHH MTV DWH Khting Viet Vidu 2.1.S.Chotudien ABCD c6 ABJ.CD, AC ± BD . 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 Chung minh ( S A B ) 1 ( S A C ) . 2) AB^ + CD^ = AC^ + BD^ = AD^ + BC^ JCffi Gidi. 3) Cac goc xuat phat t u mot dinh ciia hinh chop cung nhon, ciing vuong Gpi I la trung diem ciia B C thi A I 1 B C va I cung la trung diem cua A D . hoac cung tu. B C I A D , , t Tac6 B C 1 ( S A D ) =^ B C 1 S A . Xffigidi. [BCISD ^ ^ 1) Vi H la true tarn tam giac BCD nen CD 1 B H , Dung I H 1 S A , H € S A , khi do ta c6 : lai CO AB 1 CD nen ta suy ra CD 1 (ABH) SA1 IH Dan toi CD 1 A H . SAl(HCB)z^SAlBH. SAICB Chung minh tuong tu, ta c6: BD 1 A H . IH AI Tu do suy ra A H 1 (BCD) Ta C O A A H I ~ A A D S SD AD Ta c6: A H 1 BC, D H 1 BC B C l (AHD) => B C l A D . Ma A I = — , A D = 2AI = aV3, 2) Ta c6: AB^ + CD^ = AC^ + BD^ o AB^ - AC^ + CD^ - BD^ = 0 S A = V A D ^ f SD^ = \i(^^f a A/6 ( A B - A C ) ( A B + AC) + (CD-Bb)(CD + BD) = 0 ^ C B ( A B + A C ) + C B ( C D + B D ) - 0 A I . S D _ ' 2 -'2 a BC y ra I H - AD " Sajl "2" 2 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. Suy ra BHC - 90*' hay BH 1 H C . Vay BH 1 (SAC) (SAB) 1 (SAC). 3) Xet tai goc A . Ta c6: QBAITAP A B ^ + A C ^ - BC^ AC^+AD^-CD^ cosBAC = cosCAD - Bai 2.1.1. Cho hinh chop S.ABC c6 SA = a, SB = b, SC = c va SA, SB, SC 2.AB.AC 2AC.AD
Cty TNHH MTV D W H Khang Vi$t Phuang phdp giai Todn Hinh hoc theo chuyen de- Nguyen Phti Khdnh, Nguyen Tat Thu 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 5b^~ a +• «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). H: Jiuongddngidi i y;/ =i(aV+bV+cV) . Goi K la trung diem ciia BC va I = SK n M N . ^ASAB ^^ASBC ^^ASAC ^ Tu gia thiet ta c6 M N - -!-BC = - ,MN//BC 2 2 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. =:> I la trung diem ciia SK va M N . Goi M la trung diem cua C C . Xac dinh ti so ^ de hai mat phang ( A ' B D ) va 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 Mat khac (SBC)n(AMN) = M N A iJ- - A- - GQI O la tam cua hinh vuong A B C D . Taco B D = ( A ' B D ) n ( M B D ) , AI e (AMN) A C I B D A I I M N (ACC'A')lBD lAA'lBD A l l (SBC) =^ A l l SK =^ ASAK can tai A=>SA = AK = (ACC'A')lBD 3a2 a' a' Vay ^(ACC'A')n(A'BD) = OA' taco: S K ^ = S B 2 - B K ^ = '(ACC'A')/^(MBD) = 0 M SK aVlO do do goc giua hai duong thang O M , O A ' chinh la goc giii-a hai mat phang =>AI = VSA2-SI2 = JSA^- V 2y ( A ' B D ) VT (MBD). Taco S^,,^=IUNAI = '-^. A C VAB^+AD^+AA'^ Vza^ + b^ Ta CO O M = - 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'^ = * e m cua cac canh AD,SC, Goi I la giao diem cua BM, AC. Chung minh \ '^P(SAC) vuong goc voi mp(SMB). Tinh the tich ciia khoi t u di?n ANIB. ' (h 2 5b2 M A ' 2 = A ' C ' 2 + M C ' 2 =a^+h^ + a + l2j 101 inn
Phumig phiip gii'ii Todn Hinh hoc theo chut/en de- Nguyen Phii Khdnh, Nguyen Tat Tttu Cty TNHH MTV DWH Khang Viet J^Iu&ng dan gidi EC'^ = B ' E 2 + B ' C ' 2 - 2 B ' E ' . B ' C ' . C O S 1 2 0 ' ' = — AM BA 1 Ta c6: • AABM ~ AABM AB BC V2 A E ^ + A C ' ^ =EC'2 ^ A E 1 A C ' = > B N 1 A C ' (2). => ABM = BCA A B M + BAC = BCA + BAC=90° 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: AC^ 15a MB 1 (SAC) => (SMB) 1 (SAC). Tu giac H I C C noi tiep => A H . A C = A I . A C AH = 2A C 5 Gpi H la trung diem cua AC f BD-MN^ 7l5a Tu giac B D M N la hinh thang can ta c6 : IJ = ^ ^ N ^ - =^NH//SA=>NH1(ABI) va N H = | s A = - | . Ta CO A I la duong cao cua AABM vuong tai A Di?n tich hinh thang BDMN : S = -IJ(BD + M N ) = ^^^^^. . .1 ;H / i • :• 1 1 1 2 16 AI = AI^ AB^ A M ^ a^ The tich khoi chop A . B D M N : V = i A H . S B n M N ^I'^-^^-^T^' BI = V A B 2 - A I ^ = 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 A E , N la trung diem cua BC. Chung minh M N vuong goc voi BD. 3 2 6 2 3 3 36 Jiu&ng ddn gidi ' ' B a i 2 . 1 . 5 . Cho hinh hop dung ABCD.A'B'C'D' c6 cac canh AB = A D = a, Goi P la trung diem cua SA. ^ AA' = —^ va BAD = 60". Goi M va N Ian lugt la trung diem cua A'D' va Ta CO MP la duong trung binh ciia 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 . =>MP//AD=^MP//NC. Jiit&ng ddn gidi Theo gia thiet ta c6: Va M N = - A D = N C . D' ( 2 B D 1 AC •/ Suy ra MNCP la hinh binh hanh B D I A C (1) • / DBICC y / ^ M N / /CP ^ M N / /(SAC). \> \ : A' I Ta de chung minh dugc BD 1 (SAC) => BD 1 M N trung diem cua E N . a i 2 . 1 . 7 . Cho duong tron (C) duong kinh ABtrong mat phang ( a ) , mot Ta c6: A E / / B N , A E = B N = a. D^ . ' _ \ _ _ \. 7
Phuviig phdp giai Todti Hinh hQC theo chuyun lie- Nguyen Phu Khanh, Nguyen Tat Thu Cty TNHHMTV DVVIi Khang Vi§t 3) Goi I la giao diem cua H K va M B . Chung minh A I la tiep tuyen ciia duang I^$t khac K H 1 FQ, HK / / A C ^ HK 1 ME . tron (C). Suy ra K H 1 ( M E N ) = > K H 1 M N . Jiuang dan giai Vay khi M thay doi thi duong thang M N luon vuong goc va cat duong SAl(a) tharig CO dinh HK. l)Ta CO •SAIMB (l) gal 2.1-9' Cho hinh lap phuong A B C D . A ' B ' C ^ D ' canh a. Tren cac canh DC MBc(a)' va BB' lay cac diem M va N sao cho M D = NB = x (0 < x < a). Chung minh rang: Lai CO MB 1 M A (2) 1)AC'1B'D' 2)AC'1MN. (t/c goc chan nua duong tron) Jiu6fng dan giai , : Ati j ; i; Tu ( l ) , ( 2 ) suy ra MB 1 ( S A M ) . Dat A A ' = a,AB = b , A D = c. . 2) Ta C O A K I S M , 1) Taco A C ' = a + b + c, B ' D ' = c - b nen ., f / v - MB 1 ( S A M ) , A K c ( S A M ) =:> MB 1 A K . 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 ) . 2) M N = A N - A M = (AB + B N ) - (AD+ DM) AK 1 ( S B M ) Tuong t u =^ A K 1 SB, lai CO A H 1 SB suy ra SB 1 ( A H K ) . X - " SB c ( S B M ) b + —a c + —b = —a + 1 - ^ b-c a AIC(AHK) A l e (a) X - 3) Taco ; A l l SB (s) va ^ ^-AIISA (4). Tu do ta CO A C M N = ^a + b + cj[ b + —a c +—b = —a + 1 - - b - c ] [SBI(AHK) ^ ' S A 1 (a) a a 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 2 -2 —a + 1 — b -c =:x.a + 1-- a2-a2=0. tron (C). a I a; Bai 2 . 1 . 8 . Cho hinh hop chu nhat ABCD.A,BjCjDj c6 day ABCD la hinh 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 vuong. M di dong tren doan AB ( 0 < A M < AB). Lay N thuoc canh A j D j sao D, A B = 2a, A D = DC = a, SA 1 ( A B C D ) va SA = a. cho A j N = A M . Chiing minh M N luon cat va vuong goc voi mot duong thang CO dinh khi M thay doi. 1) (a) la mat phang chua SD va vuong goc vol (SAC). Xac djnh va tinh di^n Jiuang ddn giai N tich thiet d i f n cua (a) voi hinh chop S.ABCD . 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 It Mat phang (p) di qua M N va vuong goc vol ( S A D ) . Xac djnh va tinh di?n va ME 1 AC hch thiet di^n cua hinh chop cat bai (p). Do A M = A,N=>AE = AiN=i>NE//AAi. Goi I la trung diem cua M N , ve F I Jiuang ddn giai cat A , C , tai Q , ta c6 I la trung diem ^) Goi E la trung diem cua canh ABva O la giao diem cua AC va DE thi 'H doan F Q D •^DCE la hinh vuong c6 tam la O. Goi K, H Ian luot la trung diem cac Ta CO SA 1 (ABCD) => SA 1 O D , them niia O D 1 AC =^ O D 1 (SAC). / ^ " " N, doan thang AAi, CCi suy ra K, I , H Tir do ta c6 OD 1 (SAC) => (SDO) 1 (SAC). •* thang hang. B 104 105
Phiwug phiip giai Todn Hinh hgc theo chuyen da - Nguyen Phii Khdnh, Nguyen Tat Thu Cty TNHH MTV DWH Khang Viet Vay (SDO) chinh la mat phang ( a ) . § 2. G O C Thiet dien ciia hinh chop v6i mat phang (a) la tam giac SDE . I Goc giua luii duang thdng cheo nhau. ' ' \ E)e tinh goc giiia hai duong thang cheo nhau a va b ta c6 cac each sau Ta CO SO = VOA^ + A S ^ = ,.| + =aj|. Cdch l:T\m goc giira hai duong thang a, b bang each chon mot diem O 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 = . hai duong thSng a va b. u •]>;( Chu ^'De tinh goc nay ta thuong su dung dinh l i cosin trong tam giac: ABI(SAD) 2) Taco b^-c^-a^ cosA = • 2 be Me(p)n(SAB) Cdch2:Tur[ hai vec to chi phuong u ^ U j ciia hai duong thSng a, b Vay ABc:(SAB) (p) n ( S A B ) = M Q / / A B , Q e S B . U1.U2 AB//(p) Khi do goc giua hai duong thang a, b xac dinh boi cos(a,b) Ne(p)n(ABCD) Tuong t u , AB c (ABCD) Chu y: DG tinh Uj.Uj, Uj , U2 ta chon ba vec to a, b, c khong dong AB//(p) phang ma c6 the tinh duoc do dai va goc giOa ehiing, sau do bieu thj cac vec ^ (p) n ( A B C D ) = N P / / A B , P e B C . to Uj,U2 qua cac vec to a, b, c roi thue hien cac tinh toan. Thiet dien la i\x giac M N P Q . 2. Goc giua duang thdng vai mat phang NP//AB De xac djnh goc giiia duong thclng a va mat Do NP//MQ (l) MQ//AB' phang (a) ta thuc hien theo cac budc sau: MNC(SAD) • Tim giao diem O = a n (a) Lai CO A B I M N (2) ABI(SAD) ' • Dung hinh chieu A ' ciia mot T u ( l ) / ( 2 ) suy ra (tu giac MNPQ la hinh thang vuong tai M va N . diem A e a xuong (a) • Goc A O A ' = (p chinh la goc Do do SMNPQ = | ( N P + M Q ) M N . giua duong thang a va ( a ) . a 2 4x^ M N = VAM2+AN2 = — +x = , MQ = - A B - a 4 2 2 • De dung hinh chieu A' cua diem A tren (a) ta ehgn mot duong thang NP D N ,^^__^J_±A-_2[.-.) b 1 (a) khi do A A V / b . AB DA DA • De tinh goc (p ta su dung he thuc luQ'ng trong tam giac vuong AOAA'. c ^Ini \? + 4 ? (3a-2x)V?+4? ll^goai ra neu khong xac djnh goc cp thi ta c6 the tinh goc giira duong thang a V ^ y SMNPQ = - ( 2 ( a - X) + a) = ^ . 106 107
Phucmg fthtip gidi Todn Hhth hoc theo chuyen de - Nguyen Phii Khdnh, Nguyen Tat Thu Cty TNHH MTV DWH Khang Vift u.n • Dvng H N 1 A => M N 1 A . va mat phang (a) theo cong thiic sincp = trong do u la VTCP ciia a con phuang phap nay c6 nghia la tim hai duong thang nam trong hai mat 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) . Chu ^."Cho hinh chop S.A]A2...A^ c6 duong cao SH . De xac dinh goc giua 3. Goc giua hai mat phang matphSng (SAjAj) ( i , j e {l,2,...,n}; i j ) voi mat day ta lam nhu sau: De tinh goc giua hai mat (a) • Tu H ve H K 1 A j A j , K e A^A^ , .,, . p h i n g (a) va (p) ta c6 the thuc hien theo mot trong cac each sau: • Khi do SKH la goc giua hai mat phang (SAjAj) va mat day. Cdch l : T i m hai duong thang 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 mat phang (a) va ( p ) . /a hai duong Xgigidi. thang a,b chinh la goc giiia hai mat phang (a) va (p). Cdch J.-Ggi I la trung diem ciia AC . fa 1 ( a ) bl(p) .((a),(p))^(a,b). ^^^'^C/CD"^(^H"^)- Dat M I N = a 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 a CD a . ..^ aVs IM = = —,IN = = - , M N = — do goc giiia hai mat phang (a) va (p) xac dinh boi cos(p = 2 2 2 2 2 Theo djnh l i cosin, ta c6 Cdch S.-Su dung cong thuc hinh chieu S' = Scos(p, t u do de tinh coscp thi ta can tinh S va S'. IM^+IN^-MN^ U cos a - Cdch 4: Xac dinh cu the goc giua hai mat phSng roi su dung hf thuc lugng 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 /(p) 'h2: COS(AB,CD) = COS(IM,IN) = IM IN • Tim cac giao tuyen a (y) n (a), b = (y) n (p), / . M k h i d 6 : ( ( ^ 0 ^ ) = (Xb). \ M N = IN - IM MN^ = (IN - IM)^ = M I ^ + IN^ - 2IN.IM; b) • Tim giao tuyen A = (a) n (p) ^ ^ IM^+IN^-MN^ a^ IM.IN INIM . =- = ; COS(AB,CD)= COS(IM,IN) 8 '1 IM IN •LayME{p). ^ 7 \ " V y V Dung hinh chieu H cua M tren (a) V^iy ( A B , C D ) = 6 0 ' ' . 108 109
Cty TNHH MTVDWH Khang Viet Phumig phapgiai foan Hinh hoc theo chmjen dc - Nguyen Pliii Khdnh, Nguyen Td't Thu Ap dving djnh li Co sin cho tam giac A M E , ta c6: 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 AM^+ME^-AE^ 57 cos AME = • trung diem canh AD. Mat phang (SCD) tao voi day mot goc 60". Goi M la 2.MA.ME 474181 trung diem doan CD. Tinh c6 sin cua goc giua hai duang thang A M va SC. 57 Vay cos(AM,SC) = ^ - ^ . £gl gidi. Goi H la trung diem cua AD, ta c6 SH 1 (ABCD). X>i dii 2.2.3. Cho hinh chop S.ABC c6 SA 1 (ABC), SA = a, AB = a, AC = 2a, Ta CO ABCH la hinh vuong canh a. 5 A C = 120°. Tinh c6 sin ciia goc giiia hai mat phang: Tit H ve UK 1 CD , K € CD . 1) (ABC) va (SBC) 2) (SAC) va (SBC). Suy ra CD±(SHK), nen SKH la 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 =>HK = CI CD 1) Gpi K la hinh chieu cua A len BC, ta c6 BC 1 (SAK) nen SKA la goc giua Do do SH = HKtan60" = hai mat phang (SBC) va mat phang (ABC). 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. A K . B C = A B . A C . s i n l 2 0 ' ' {=2SAABC) Taco M E / / S C , E F / / S H , VTO . ^ A K = ^ = ^ S K = VSA2 + A K 2 = ^ 7 7 EF = | S H = va ( A M ^ S C ) = ( A M > I E ) . A K VSO Suy ra cos SKA = Gpi I la hinh chieu cua C len AD, ta c6: SK 10 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 Suy ra tam giac SMC la hinh chieu cua tam giac SBC len mat phSng (SAC) Nen M E = i s C = i V c H 2 + S H 2 = i : ^ , A F = 1 A D = ^ S 2 2 20 4 4 'ASBC (Theo cong thuc hinh chieu, ta c6: coscp = - ^ ^ ^ voi (p la goc giua hai m^t Suy ra A E ^ = A F ^ + FE^ = — . tiang (SAC) va (SBC) Taco: M F l A D = * M F = l c i = - , suy ra AM = N / A F ^ T M F ^ = . 2 2 10 Tac6:S^BC=-SK.Bc4.^.a7^ = - 110
Cty TNHHMTV DWHKhang Viet Phumtgphapgiai Todn Hinh hgc theo chuyen de- Nguyen Phii KItdnh, Nguyen Tat Thu Do B A M = 60" => A M = ABcos60° = ^ S^^MP = ^SA.MC = Taco HB^ = 0 H 2 + 0 B 2 = 2 • -ASMC 2 4 4 , 8 2 8 , 2^ 3 Vhy cos(p = .^5 27To ' Xet A M H N c6 M N = ^ = M H = N H tan60° = ^ . 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 cos 60° 1 ^ Isfl 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). G(?i I la trung diem cua OB, J la trung diem ciia SO thi M J / /IN va MJ = I N . Xgigidi. GQi K = I J n M N = > J K = |lJ Taco: A' va MJ 1 ( S B D ) :=> UK] la goc giiia M N va (SBD) . / BC^ = AB^ + AC^ - 2AB.ACcosl20° = 7a^ Ta CO i f = 1 0 ^ + 0 12 I =>BC = aV7 2 ra72^ > 15a' = MH^+Ol2 = I ^2a 1 A ' M ^ = A'C'^ + C M ^ =4a2 +53^ =9a^ + 4 J = > A ' M = 3a / i > a BM^ = BC^ + CM^ = 7a^ + 5a^ = 12a^ . IJ = a72 va IK = 2 BM = 2aV3 ^ a^|i A'B^ = A ' A ^ + AB^ = 21a2 = A ' M ^ + BM^ tn — MJ 4 Suy ra tam giac A ' M B vuong tai M , (p - JK a72 2 suy ra S^^'MB = ^ M A ' . M B - Sa^Vs Vaygoc giira M N va (SBD) la (p = a r c t a n i . 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: Cdch2;TaCO M N = i(sC + A B ) = |(sO + OC + A O + O B ) - ~ ( S O COSCP = | M B C . ^ ^ 1 ^= ((iKiX^^)). r>^2 5a' ^ABMA' 3a^V3 ^ Suy ra MN^ = ifSO^ + AC^ + OB^) = - =:>MN = - , ' 44\V " 4 / 4^ 2 2V Vi du 2.2.5. Cho hinh chop S.ABCD c6 day la hinh vuong canh a, O la tam J cua day, SO 1 ( A B C D ) ; M , N Ian lugt la trung diem cua SA, CD. Biet goc MN.n Taco (p la goc giua M N va (SBD) nen sin(p = ( n la vec to c6 gia giua M N voi ( A B C D ) bang 60° . Tinh goc giira M N va ( S B D ) . MN n JCffi Gidi Jong goc voi ( S B D ) ). Cdch l:Ke M H / /SO,H e O A . ["^(2 X SO - • Do A C 1 (SBD) ner\n n = A C , t u do ta c6 [MH/ZSO ACIBD ^ ' DoL^ . „_,^MHI(ABCD) [SO 1 ( A B C D ) ^ ^ -(SO + AC+OB)AC 2a sincp = suy ra N H la hinh chieu aia M N tren ( A B C D ) => M N H chinh la goc giiia duong thSng M N voi ( A B C D ) . 113 112
I'jiianin pluipgiai Todn Hinh hoc theo chuyett de- Nguyen Phu Khdnh, Nguyen Tat Thu Cty TNHHMTV UVVII Khang Vi?t Do goc giua duang thang M N va ( A B C D ) bang 60*^ nen va CHlBD=>Cni.(BDCi). C,H CjH.CiI CC? b^ _ 2b^ S MN.S6 \so^ ^ a < arcsin^ => maxa = arcsin- khi a = b . . 1 3 3 3 Vay max a - arcsin— khi a = b . 3 Vi du 2.2.7. Cho hinh chop S . A B C D c6 day A B C D la nua luc giac deu npi =b tiep trong duong tron duong kinh AB = 2a; canh ben SA vuong goc vm Cach 2: Dat CB = x,CD - y,CCi =z=> = a. dayvaSA = aV3. B D i = - x + y + z, BDj =^x^ + y^ + z^ =V2a^+b^ 1) Tinh goc giCra hai mat phang ( S A D ) va ( S B C ) . Goi H la hinh chieu cua C tren C j l thi C H 1 C j l 2) Tinh goc giua hai mat phSng ( S B C ) va ( S C D ) . 114 115
Phuaitg phiipgiiii loan Iliiili lioc Ihco chuyen de - Nguyen Vhii Khiinh, Nguyen Tat Thu Cty TNHH MTV DWH Khang Viet Xffi gidi. 1) Goi I = A D n BC thi SI = ( S A D ) n (SBC) . Vay ((SBC),(SCD)) = arccos Vio B D I A D , , X?i du 2.2.8. Cho tam giac ABC c6 AB = 3a , duong cao CH = a va A H = a => B D ± (SAD) => B D 1 SI . BDISA ^ ^ nam trong mat phang (P). Tren cac duong thang vuong goc voi (P) ke tu Dung D E 1 SI, E e SI khi do ( B D E ) 1 SI. A,B,C Ian luot lay cac diem A',B',C' tuong ung nam ve mot phia ciia (p) Do do BED la goc giQ-a hai mat phang ( S A D ) va (SBC). sao cho A A j = 3a,BBj = 2a,CCj = a . Tinh dien tich tam giac A ' B ' C . Do day ABCD la nua luc giac deu nen lAB = IBA = 60° => AIBA deu. Xffi gidi. 3a^ Vi vay A I = A B = 2a, SI = T S A ^ T A F = J(aSf + {laf = a77 . Ta CO SABC = DE DI T^r: SA De thay ASAI ~ ADEI ^ - - Vi CH 1 A B , C H = a, A H = a => AC = a>y2 SA SI a77 77 DE = — 77 = a,V va BAC = 45". BD 1 (SAD) => BD 1 DE . Trong tarn giac vuong BDE ta c6 : Goi I = B ' C ' n B C , J = A ' C ' n A C . 4 t a n B ^ = -DE ^ =^ = V7 Taco C C ' - i B B ' ^ B C = CI 2 =>BED = a^ctan^/7 C C = iAA-=^CJ = l A C = ^ . 3 2 2 Vay ((SAD),(SBC)) = arctan ^7 Xet ABCH ta c6 BC^ = BH^ + CH^ = Sa^ ^ BC = aVs 2) Dung A P 1 S H , P € S H . Mat khac AB^ = CA^ + CB^ - 2CA.ABcosC Do C D l ( S A H ) r ^ A P l C D ^ A P l ( S C D ) . CA^+CB^-AB^ 1 Tuong tu, dung A Q 1 SC,Q e SC =>C0SC=:- 2CA.CB 10 thi AQ 1 (SBC). ft.'!,- 26a' Xet AICJ ta c6 IJ^ = C P +CJ^ -2CI.CJcosICJ = Do do PAQ = ((SBC),(SCD)). Trong tam giac S A H ta c6 : Ke duong cao C K cua A I C K , do C C 1 (iCj) nen C ' K 1 I J . 1 1 ^ 1 1 ' 1 -+ • Vay C " ^ chinh la goc giua hai mat phang ( A B C ) va ( A ' B ' C ' ) n e n Ap2 AS^ AH^ •'ABC = S A . B ' C ' ' ^ o s C ' K C . T a CO Sjcj = | S A B C D l thay ASAC vuong can tai A nen AQ = ^ S C = = ^ , 3a^ _ 2Sicj _ 2 _ _ 3a Mat khac S^q = ^IJ.CK => CK = Do A P 1 (SCD) ^ A P 1 P Q . IJ V26a N/26 ' Trong AAPQ c6 cosAPQ = AP _ ^is^^ VlO APQ = arccos Vio CC'^ a 726 Xet AC'CK taco tanC'KC = AQ aVe 5 CK ~ Ja^ V26 116 117
Phuwig phap giai Toan Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen TS^ Thu Cty TNHH MTV DVVU Khang Vi$t 1 A M = I s B = IVSA^ + A B ^ = .1, > ' Ma l + tan^C'KC = •cosC'KC = cos^C'KC V35 '1 ,1 • 'ABC AE = A N = VAB2+AN2-2AB.AN.COS120° = ^ Vay SABC = S A ' B ' C C O S C ' K C => S^.B^C = " cosC'KC 2 '• Goi I la trung diem A B , CaBAITAP Bai 2.2.1. Cho hinh chop S.ABC c6 day ABC la tarn giac can tai A, AB = a ^H|uy ra M I / / S A , M I = ^SA = ^ , va SA 1 (ABC). Mat phang (SBC) tao voi day mot goc 60" . Goi M , N Ian luot 2 2(AE2 + B E 2 ) - A B ^ Sa^ la trung diem cua SB va AC. Tinh : EI = ;; —r- 1) Co sin ciia goc giiia hai mat phang (SAC) va (SBC) 4 4 15a^ 2) Co sin ciia goc giiia hai duong thang A M va B N . E M ^ - E I ^ +Ml2 = 16 Jiicang dan giai AM^+AE^-ME^ 5 Gpi K l a trung diem BC, suy ra A K 1 BC BCl(SAK) Suy ra cos M A E = = ^ . Vay cos(AM,BN) = ^ . , 7,:[ 2.AM.AE Do do SKA la goc giiia hai mat phSng (SBC) va (ABC), hay SKA = 60° Bai 2.2.2. Cho hinh chop S.ABC c6 SA = SB = SC = a va BC = aV2 . Tinh goc 1V3 giiia hai duong t h i n g AB va SC . Ta c6: B A K = 60° A K = AB.cos60° = - , B K = AB.sin60° = BC-aVs. 2 Jlicong ddn giai ' ! v Trong tam giac S A K , ta c6: Goi M , N , P Ian luot la trung diem cua SA,SB, A C , khi do M N / / AB ^ SA = A K . t a n 6 0 ° = ^ nen (AB,SC) = ( M N , S C ) . 2 . , MN^+MP^-NP^ / X =>SK = V S A ' + A K 2 =a I Dat (p = N M P , trong tam giac M N P c6: coscp = 2MN M P 1) Goi B ' la hinh chieu ciia B len A C , suy ra B B ' l ( S A C ) nen tam Ta CO M N = M P = - , A B ^ + A C ^ = BC^ => A A B C vuong tai A , v i vay giac S B ' C la hinh chieu cua tam giac SBC len mat phang (SAC). PB2.AP2+AC2=^,PS2=^. 4 4 ^ Do do coscp = SASB'C Trong tam giac PBS theo cong thuc tinh duong trung tuyen ta c6 'ASBC 5a^ 3a^ Taco: S,,SBC =^SK.BC = ^ , pj^j2 _ P B ^ + P S ^ SB^ ^ 4 ^ 4 a^^3a^ ~ 2 4 ' 2 4 4 ' ,v:> Do BAB' = 60° =^ AB' = ABcos60° = - ^ CB' = — . 1 'Thay M N , M P , N P vao ( l ) t a d u g c cos(p = = 120°. 2 2 /Sil. , 1 aVs 3a SaVs 3 NenS^SB,c=-SA.CB' = ^. ^ ^ — . Vay cos(p = - . Vay (AB, S C ) = ( M N , SC) = 60°. Bai 2.2.3. Cho lang tru ABC.A' B' C' c6 day ABC la tam giac can AB = AC = a, 2) Trong mat phSng (ABC) dung hinh binh hanh A N B E , suy ra AE / /BN. BAC = 120°va AB' vuong goc voi day ( A ' B ' C ) . Goi M , N Ian lugt la trung Do do ( A M , BN) = ( A M , A E ) . ^iem cac canh C C va A ' B ' , mat phang ( A A ' C ) tao voi m^t phang (ABC) Ta c6: I 119 118
Phuoitg phcip giiii Todii Hinh hoc theo chuyen de - Nguyen Phii Khdnh, Nguyen Tat Thu Cty TNHH MTV DWH Khang Vi?t mgt goc 30*^. Tinh the tich khoi lang try A B C . A ' B ' C va c6 sin cua goc giiia suy ra AB ± (SKH) => A K H la goc hai duang thang A M va C ' N . » L ijja hai mat phSng (SAB) va mat day Jiicong dan gidi ^gnSKH = 60° Ta c6: B C ^ = A B ^ + A C ^ - 2 A B A C cos A = 3a^ BC = aS Taco: H K = | A B = y Goi K la hinh chieu cua B ' len A ' C , suy ra A ' C 1 ( A B ' K ) J 3a V3 Do do A K B ' = ((A • B' C'), (A A ' C •)) = 30° . ,SH = HK.tan60° = Trongtamgiac A ' K B ' c6 K A ^ ' = 6 0 " , A ' B ' = a Gpi I la giao diem cua A M voi BC. Ve IJ / /SB, J e SD, ta CO (AM, SB) = (AI, IJ) Nen B ' K = A ' B ' s i n 6 0 ' ' - — . ^ Vi I la trpng tam tam giac ABC . 2... layjl ;| Suy ra A B ' = B'K.tan30° = | nen A I = - A M = 3 3 The tich khoi lang try: SB DB 3 V = AB'.S AABC 8 =^JI=-SB = ^ V B H 2 + S H 2 - ^ ' Gpi E la trung diem cua AB', 3 3 2 suyra M E / / C ' N , Ve J P / / S H , P e B D ^ J P = - S H = aV3.^ nen ( C ' N , A M ) - ( E M , A M ) 3 Vi A B ' 1 C ' N = > A E 1 E M Ve P Q I A D , Q € A D = > P Q = i A B - | , AQ = | A D = - y 6— 6"'^ 6 " 3 =>(C'N,AM) = AME 101al_,^T2 ..2...,2 209a^ Suy ra AP^=AQ^+PQ^ -r^Af =AP2+JP^=- Ta c6: 36 36 AE = i A B ' = ^ ; E M ^ = C N ^ . 2 ( C ' B ' - . C ' A ' - ) - A ' B ' Ap dung dinh l i Co sin cho tam giac AIJ, ta c6: 2 4 4 AI^+lf-AJ^_ 1 COS AIJ = • AX^2 A c 2 r : w 2 29a^ . . . aV29 2AI.IJ N/34 A M = AE + EM = => A M = . li 16 4 1 'ay cos(SB,AM) = - y = -—- ME v 34 V^y cosAME = — = 2„ ^ MA 29 V Bal 2.2.5. Cho t i i di?n A B C D c6 A B = b, A C - c, A D = d doi mpt vuong goc. Bdl 2.2.4. Cho hinh chop S.ABCD c6 day ABCD la hinh chii nhat AB = a, Gpi a,p,y Ian luat la goc giua mat phang ( B C D ) v o i cac mat phang A D = 2a . Gpi O la tam ciia day, SO = SD . Mat phang (SBD) vuong goc voi mat (ACD),(ABD),(ABC). phang day, mat phSng (SAB) tao voi day mot goc 60°. Gpi M la trung diem doan BC. T i n h c6 sin cua goc giiia hai duang thang A M va SB . 1) Chung minh cos^ a + cos^ P + cos^ y = 1. Jiuang dan gidi 2) Tinh SBCD theo khi a - 30°, p = 45°,y = 60°. Gpi H la trung diem cua O D , suy ra SH 1 BD => SH 1 (ABCD) Jiuang dan gidi Ve H K X AB, K 6 AB . 1) Cach 1 121 120
Phucmg phdp giai Todn Hinh hoc theo chuyen de- Nguyen Phii Khdnh, Nguyen Tat Thu Cty TNHH MTV DWH Khang Viet Ke duong cao A H ciia tarn giac A C D , b^c^d^ 1 AC b.AH cd ,08 } , fAB bV+c^d^+dV _ do • AB±(ACD)=>ABICD. cosa = AH b^c^ + c^d^ + d^b^ VbV+c^d^+dV [ A B J_ A D b^c^d^ Vay ( A B H ) I C D va C D la giao tuyen B Tuong t u : cua hai mat phang ( A C D ) va ( B C D ) c.AH bd d.AH be nen a = A H B . cosp = ,cosy = AB b AH VbV+c^d^+dV AH VbV+c^d^+dV Ta c6: tan a = AH A H ' Suy ra cos^ a + cos^ p + cos^ y = 1 1 1 1 1 ma •+- 2) Su dung cong thuc hinh chieu AH^ A C ^ A D ^ c^ d^ Gpi H la hinh chieu cua A tren ( B C D ) . nen tana = Truoc tien ta chung minh tam giac B C D nhon. , , ^ cd Khong giam tong quat, gia su B Ion nhat. B4 / -1 I 2 1 7 c^d^ Mat khac 1 + tan a = — => cos a = b V + c ^ d ^ + d V Taco C D ^ - A C ^ + A D ^ ^ c ^ + d ^ \l cos a Tuong hf CB^ = b^ + c^DB^ = b^ + d^ b^d^ 2„2 b^c Tuong ty ta c6 : cos P = , COS Ap d\ing djnh l i cosin cho ABCD ta c6: bV+c^d^+dV bV+c^d^+dV Tir do suy ra cos^ a + cos^ p + cos^ y = 1 . BC2 + BD2 - CD2 (b^ + ) + (b^ + d2) - (C2 + d^) cosB = Cach 2. G(?i H la hinh chieu cua A tren (BCD) va I la trung diem cua C D . 2BC.BD Dat AB = b,AC = c , A D - d = b. = d. 2b^ . > 0 do do B nhon, hay tam giac BCD nhon. BH.BI = BA2=b2 b^fc^+dM 2 De thay A H 1 ( B C D ) va c^d^ =k IH.IB = lA^ = IH c^d^ AHICD c^+d^ Ta CO B H 1 C D , tuong t u ta c6 CH 1 BD t u do suy ra H la AB 1 CD 1, Suy ra A H = — A B + A I , t^Vc tam ciia ABCD, ma ABCD nhpn nen H thupc mien trong tam giac BCD . ^ 1+k 1+k Do do SpcD = SHBC + SHBD + SHCD = ^ABC ^osy + S^go cosp + S^CD . IC AC^ c^ — d^ ma — = = — => A I = AC + CD lu ^nO l . j . r O 1 J onO bc + 72bd + Vscd ID A D 2 d2 c2+d2 c2+d2 = -bccos60" + - b d c o s 4 5 " +-cdcos30" = —• cd bV+dV 4 nen A H = AB + - AC + AB 2 2 2 bV+c^d^+dV 2u2 c2+d2 bV+c^d^+d^b c^+d^ fi 2.2.6. Cho hinh chop S.ABCD c6 day la hinh vuong canh a, tam O va c^d^ ^ d^b^ b^c^ 1 ( A B C D ) .Mat phSng (a) d i qua A va vuong goc voi SC cat hinh chop b+ c+ *heo mpt thiet dien c6 di^n tich S^^ = - a ^ . Tinh goc giCra duong thang SC va bV+c^d^+dV bV+cV+dV bV+c^d^+dV "JJ min nmi-i Lai CO b,c,d Ian luot la cac vec to vuong goc voi cac mat phang (ACD)/ •^^t p h i n g (ABCD). .aixmiiffrn IM! ni'uig "yog ( A B D ) , ( A C B ) . T u do ta c6: 122 123
Phumtg phdp giai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, NguySn Tat Thu Cty TNHH MTV DWH Khang Viet Jiuang dan gidi JIu&ng ddn gidi Gia su (a) cat cac canh SB,SC,SD Ian lugt tai cac diem H , J, K. Goi H la trung diem ciia BC. Do AABC can tai A nen A H 1 BC va A C H = 30" Do 1 ^ ^ - ^ ^ ° = : ^ B D 1 ( S A C ) ^ B D 1 S C ma (a) 1SC => (a)//BD . IBDIAC ^ ' ^ ^ ^ ' Ta c6: A H = AC. sin ACH = a. sin 30° = |-, B BD c (SBD) BC = 2BH = 2.acos30'' = aVs Vay BD//(a) =^KH//BD^HK1(SAC)=^HK1AJ, (SBD)n(a) =H K = > I B ' 2 = = I C ' 2 + B ' C ' 2 = ^ + 3a2=.13^' 4 4 d o d o SAHJK = 2 ^ K . A I . A A ' B ' B la hinh vuong canh a 5a ^ Do S O 1 ( A B C D ) => O C la hinh chieu cua S C tren ( A B C D ) suy ra nen A B ' = aVi, A I ^ = I C ^ + A C ^ = ( S C , ( A B C D ) ) = Sc6 = (p. . A l 2 + A B ' 2 = ^ + 2 a 2 = l ^ = IB'2 Taco AJ = ACsin(p = aV2sin(p; SO = OCtan(p = ^ ^ t a n ( p . • A A B ' I vuong tai A . ASOC~ASJI ^sTj = SCO = (p=>AIO = SU = (p. a^x/To Taco: S ^ B . , = - A I . A B ' = ^ Txx do ta CO O I = OAcot9 = cotcp. 2 GQI 9 = ( ( A B C ) , ( A B ' I ) ) COS9 = = 10 a-j2. HK SI _ - ^ _ 0 £ ^ j _ cotm Bai 2.2.8. Cho lang tru A B C A ' B ' C , c6 day ABC la tam giac deu canh a, va 2 - = l-cot2 BC SO SO tanq) A'A = A ' B = A ' C = a ^ . Tinh the tich khoi lang tru A B C . A ' B ' C theo a va • K H = B D ( l - c o t 2 9 ) = aN/2(l-cot2(p). D goc giua hai mat p h i n g (ABB' A ' ) va (ABC). Jiu6mg dan gidi Vay S^HJK = ^ H K . A I = a^/2sin(p.a^/2|l-cot^ (pj = 2a^ sin(p|l-cot^9J Gpi H la hinh chieu cua A tren (ABC) Vi A ' A = A ' B = A ' C n e n H A = HB = HC Tie gia thiet suy ra 2a^ sincp^l - cot^ tpj = ia^ o 4sin^ 9 - sincp - 2 = 0 2 Suy ra H la tam tam giac deu ABC. • 1 + N/33, , „ .71 .. „ l + ^/33 Gpi I , J Ian lugt la trung diem cua sin(p = - ( d o 00) (p = arcsin- 8 2 8 BC,AB. , ^ I + V33 Vay goc giira duong thang SC va mat phang (ABCDj la (p = a r c s i n — - — Taco A ' J = 7 A A ' ^ - A J ^ 8 A Bdi 2.2.7. Cho hinh lang tru dung A B C . A ' B ' C c6 day ABC la tam giac can 7a^ 2} a tai A voi AB = AC = a va goc 6 X 0 = 120", canh ben BB' = a. Goi I la trung 1 12 >/3 diem cua C C . Chung minh rang tam giac A B ' I vuong tai A . Tinh c6 sin cii^ ,_\ ^a4l =>A'H =VA'J2-HJ2 =1 goc giira hai mat phang (ABC) va ( A B ' I ) . ~3 3 2 6 124 125
Phumtg phdp gidi Toan Hinh hgc theo chuySn dJ-Nguyht Phu Khdnh, Nguyen Tat Thu Cty TNHH MTV DWH Khang Vi?t a a^Vs a^^/3 Jiicong dan gidi The tich khoi lang tru A B C . A ' B ' C la: V = A ' R S ^ B C = 2 • 4 8 Goi H la hinh chieu cua S tren AB, suy ra SH ± (ABCD). Do do SH la A'JIAB (jaong cao ciia hinh chop S.BMDN. Ta c6: SA^ + SB^ = a^ + 3a^ = AB^ Vi •(A'Cj)lAB^A'JC CJ1 AB AB . A S A B vuong tai S => S M = .0.' =a . chinh la goc giua hai mat phang ( A B B ' A ' ) va ( A B C ) . Do do tam giac SAM deu, suy ra SH = a Taco: tanA'JC = — = - 4 - = V3=> A'JC = 60' Di?n tich t u giac B M D N la: JH aVs ARr-n - SBMDN ~ - ^^ABCD 2a Vay goc giira hai mat phang (ABB'A') va ( A B C ) bang 60° . The tich khoi chop S.BMDN: Bai 2,2.9. Cho lang tru ABC.A'B'C c6 dQ dai canh ben bang 2a, day ABC la a^Vi V--SH.SBMDN (dvtt). tam giac vuong tai A, AB = a, AC = aV3 va hinh chieu vuong goc cua dinh A ' tren mat phang (ABC) la trung diem ciia canh BC. Tinh theo a the tich khoi |KeME//DN (E€ AD) =^ AE = -^. chop A'.ABC va tinh cosin cua goc giiia hai duong thang AA', B ' C Jiuang dan gidi 9t cp la goc giiia hai duong thang S M va D N . Ta c6: S M , M E = cp. . G(?i H la trung diem BC A ' H 1 (ABC) 'heo dinh ly ba duong vuong goc ta c6: S A 1 A E va A H = i B C = iVa^T3a^ = a Puy-ra S E = V s A ^ + A E 2 ME V A M ^ + AE^ = ^ •J. Ha fM 2 2 Do do A ' H ^ = A ' A 2 - A H 2 =3a2 a 5ME can tai E nen SME = 9 va coscp = • 2 Vi = > A ' H = aV3 Vay VA'.ABC=^A'H.SAABC Bai 2.2.11. Cho t u dien ABCD c6 cac mat (ABC) va (ABD) la cac tam giac deu = ^(dvtt). c^nh a, cac mat (ACD) va (BCD) vuong goc voi nhau. Hay tinh theo a the tich khoi t u dien ABCD va tinh so do cua goc giiia hai duong thang A D va BC. Trong tam giac vuong A'B'H c6 Jiuang dan gidi HB' = V A ' B ' 2 + A ' H ^ = 2a Gpi M , N , I Ian lug't la cac trung diem cac nen tam giac B'BH can tai B'. Dat 9 la goc giiia hai duong t h i n g A A ' va B'C c ^ h CD, AB, BD => VABCD = 2V.AMCD- thi: (p = B ' B H . Vay cos(p = ; ^•\_ 2.2a~4" A B I B N •AB1(BCN)=^AB1MN Bdi 2.2.10. Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh 2a, A B I C N " ' ' g SA = a, SB = aVs va mat phSng (SAB) vuong goc vol mat phSng day. Goi M/ Do A A C D can tai A A M 1 C D => A M 1 ( B C D ) N Ian lu9t la trung diem ciia cac canh AB, BC. Tinh theo a the tich cua khoi AB a chop S.BMDN va tinh cosin cua goc giiia hai duong thang SM, D N . ; => A M 1 B M AAMB vuong tai M => M N = 2 2 4W 126 127
Phuang phdp gidi Todn Hinh hgc theo chuyen de- Nguyen Phi't Khdnh, Nguyen Tat Thu Cty TNHH MTV DWH Khang Viet § 3. K H O A N G C A C H => D M = V N D ^ - = - ^ =^ ^ S^CND = M N . M D = ^ ^ , . r i c. , De tinh dugc khoang t u diem M den mat phSng (a) ta c6 cac each sau: Vay the tich t i i di§n A B C D : VABCD = 2VANCD = 2 . | A N - S ^ C D = • Cdch l:Xac djnh hinh chieu vuong goc H ciia M len ( a ) . Ta CO AMNE la tarn giac deu => M E N = 60*^. De xac dinh dugc v i tri hinh chieu H ta c6 mpt so' luu y sau: . ^ Do ^''^^ (AD^BC) = ( M J M ) = 60° . . Neu CO d 1 (a) thi M H / / d . ( h . l ) , , , llMZ/BC • Chpn (p) chua diem M va (P) 1 (a), roi xac djnh giao tuyen A = ( a ) n (p). Bai 2.2.12. Cho t u dien ABCD c6 AC = A D = aN/2 , BC = BD = a, khoang each Trong (P) dung M H 1 A M H 1 ( a ) . (h.2) tir B den mat phang (ACD) bang . Tinh goc giua hai mat phang (ACD) va • Neu trong (a) c6 hai diem A, B sao cho M A = MB thi trong (a) ke duong trung true d cua doan AB, roi trong m p ( M , d ) dyng M H 1 d. (BCD). Biet the c u a k h o i t i i d i ^ n ABCD bang . K h i d o M H l ( a ) (h.3) Jiuang d&n gidi That vay, gpi I la trung diem ciia AB. Do M A = MB nen AMAB can tai Goi E la trung diem cua CD, ke B H 1 A E . M M I 1 AB c ( a ) . Lai c6 AB 1 d => AB 1 m p ( M , d ) => AB 1 M H . Ta CO AACD can tai A nen CD 1 AE . Tuong t u CD ± BE Suy ra CD 1 (ABE) =^ CD 1 BH Vay|^"^^^^MHl(a). [MHld ^ ' Ma BH 1 AE BH 1 (ACD) BH = . Goi a = ((ACD),(BCD). 1 a^JlE M M The tich ciia khoi t u di|n ABCD la V = - B H . S ^ c D = Mat khac: AE^ + DE^ = 2a^ => AE^,DE^ la hai nghiem ciia phuang trinh : 1«L n '•'A''' h.l 2 o 2 u 5a2 -4 h.2 h.3 • Neu trong (a) c6 mpt diem A va mpt duong th3ng d khong d i qua A sao ViDEDE = ^,AE.4^ / VVV cho M A 1 d thi trong (a) ke duong thang d ' d i qua A va d' I d , roi trong Xet ABED vuong tai E m p ( M , d ' ) k e M H l d ' = > M H l ( a ) (h.4) That v a y , do d 1 d ' va d 1 M A d 1 mp(M,d') d1 MH. \ nen B EU^VV B BDD ^2- D E ^E =2 =^^ - D Lai CO M H l d ' ^ M H l m p ( d , d ' ) = ( a ) . BH Xet ABHE vuong tai H nen sina = — = => a • Neu trong (a) c6 cac diem A^, A2,...,Aj, (n > 3) Vay goc giua hai mp(ACD) va (BCD) la a = 45°. Ma M A j = M A j =... = MA„ hoac cac duong thang MAi,MA2,...,MA„ tao 129 128
Phitang phdp gidi Todn Hinh liQC theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu Cty TNHH MTV DWH Khang Viet voi (a) cac goc bang nhau thi hinh chie'u ciia M tren (a) chinh la tarn duong , Dung O H 1 b . tron ngoai tiep da giac A j A j . - . A ^ . Doan O H chinh la doan vuong goc chung ciia a va b. • Neu trong (a) c6 cac diem A j , A2,..., A„ (n > 3) ma cac mat phang Qfich 2;Dung mat phang (a) di qua a va song song voi b , khi do : (MAIA2),(MA2A3), ...,(MAJ,AJ) thi hinh chie'u ciia M la tam duong tron noi d(a,b) = d(a,(a)) = d(M,(a)) voi M la diem bat ki thuoc ( a ) . tiep da giac A j A j - A ^ . Q^h 3;Dung hai mat phang (a) di qua a va song song voi b, (P) di qua b va • Mot ket qua c6 nhieu ung dung de tinh khoang each tir mpt diem den mat song song voi a. Khi do: d(a,b) = d((a),((3)). phang doi v o i t u dien vuong (tuang t u nhu he thuc lug-ng trong tam giac Cck;/i 4;Phuong phap vec to: vuong) la: A M = xAB Neu t u di^n OABC c6 OA, OB, OC doi mot vuong goc va c6 duong cao O H CN = yCD M N la doan vuong goc chung ciia AB va CD khi va chi khi 1 1 1 1 thi + r+ MN.AB = 0 OH^ OA^ OB^ OC MN.CD = 0 Cdch 2: Su dyng cong thuc the tich: Xet mpt hinh chop c6 M la dinh, day nam 3V Vi du 2.3.1. Cho hinh chop S.ABC c6 S A l ( A B C ) ; AB = a, AC = 2a, trong mat phang (a). Khi do: d(M,(a)) - • BAC = 120°. Mat phang (SBC) tao voi mat phang (ABC) mot goc 60° . Tinh: 1) Khoang each t u A den mat p h i n g (SBC) 2) Khoang each t u B den (SAC). JUgigidi. Ap dyng dinh If c6 sin, ta eo: fsC^ = AB^ +AC^ -2.AB.AC.cosBAC = 7a2 BC = a77 ipi K la hinh chie'u ciia A len BC Cdch 3: Chuyen viec tinh khoang each tir M ve tinh khoang each t u diem N de suy ra B C l ( S A K ) ne n SKA la goc tinh hon bang each su dung eae ket qua sau: giiia hai mat phang (SBC) va (ABC). • Neu M N / / ( a ) thi d ( M , ( a ) ) = d ( N , ( a ) ) . Suy ra SKA = 60° . Ta c6: • N e u M N n ( a ) = {l} t h i d ( M , ( a ) ) = ^ . d ( N , ( a ) ) . SAABC = | A K . B C -1 A B A C , sin 120° = 2. JChodng cdch giua hai duong thdng cheo nhau De tinh khoang each giiia hai duong t h i n g cheo nhau ta c6 the dung mpt =>AK = >SA = AK.tan60°= trong cac each sau: Cdch i : D y n g doan vuong goc chung M N cua a va b . K h i do d(a,b) = M N . ^) Gpi H la hinh chie'u ciia A len SK. Chii y:Neu a l b thi ta dung doan vuong goc chung cua a va b nhu sau Do BC 1 (SAK) r:> BC 1 A H => A H 1 (SBC) • Dyng mat phang (a) ehua b va vuong goc voi a. SA.AK 3aV7 Suy ra d(A,(SBC)) = A H = • Tim giao diem O = a n ( a ) . VSA2+AK2 14 130 131
Phucmgphdp gidi Todtt Hinh hqc theo chuyen de- Nguyen Phu Khanh, Nguyen Tai Thu Cty TNHH MTV DWH Khang Viet Goi I la tam ciia hinh vuong ABB'A'. 1 1 3a a^S 3^721 3a2 2)Tac6: Vg ^ B C = gSA-S^ABC = g - ^ - 2 , S ASAC = - S A . A C = AD'c:(AB'D') 14 7? Ta c6: BD//(AB'D') 3Vs. ABC ^ aV3 Suy ra d(B,(SAC)) = => d(AD',BD) = d(BD,(AB'D')) SASAC 2 Vidu 2.3.2. Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai A = d(B,(AB' D')) = d(A',(AB' D')) • va B, AB = BC = a, AD = 2a . Tarn giac SAD la tarn giac deu va nam trong Vi A ' A B ' D ' la tii dien vuong dinh A' mat phang vuong goc voi day. Goi H la hinh chieu vuong goc ciia A len SB. Tinh khoang each tu H den mat phSng (SCD). nen d(A',(AB'D')) = 3 G(?i H la trung diem ciia A D , Suy ta d(AD',BD) = Hinhl suy ra S H 1 A D ^ SH1 (ABCD). Cflc^^.Sii dung phuong phap vec to i> GQI E la giao diem ciia A B va C D , GQI MN la doan vuong goc chung ciia A D ' va BD voi M e A D ' , N e B D suy ra B la trung diem ciia A E . Dat AB = x,AD = y , A A ' = z=> x = y = z = a,xy = yz = zx = 0 i ' Ta c6: HS AD^ = y + z ^ AM = kAD^ = k (y + z), DB = X - y => D N = m (X - y j. d(H,(SCD))- d(B,(SCD)) BS T^co MN = A N - A M = AD + D N - A M = mx + ( l - k - m ) y ^ i .d(A,(SCD)) 2'AE Vi M N l D B : ^ M N . D B = O o ( m x + ( l - k - m ) y + k z ) ( x - y ) = 0 1 AD d(H,(SCD)) I o 2m + k -1 = 0 . 4 AH [Tuong tu MN.AD' = 0 o 1 - m - 2 k = 0, tu do ta c6 h^ = id(H,(SCD)) 2m + k = 1 1 m = k = - . Ve H F I C D , H K I S F , m + 2k = l 3 suyra C D 1 { S H F ) E Eyay MN = - x + - v — z CD 1 H K => H K 1 (SCD) => d(H,(SCD)) = HK 3 3^ 3 Ta CO tam giac CHD la tam giac vuong can tai H nen F la trung diem c ^ CD iV3 ' => MN = MN [x +y +z J = 1 ayjl Do do HF = — CD = . SH la duong cao tairv giac deu caiUi 2a Cdch3.(Hinh 2) nen SH = a73 Chon ( D C B ' A ' ) vuong goc voi AD' HF.SH S t r u n g diem O ciia A D ' . Suy ra HK = ^^.Vay d(H,(SCD)).^, N/HF^+SH^ 7 Goi I la tam ciia hinh vuong BCC'B' B I I C B ' va B I I C D Vi du 2.3.3. Cho hinh lap phuong A B C D . A ' B ' C ' D ' canh a. Tinh khoang each giija hai duong thang A D ' va BD . ^ nen B I l ( D C B ' A ' ) tu do D I la hinh JCgigidi. *ie'ucua DB len ( D C B ' A ' ) . Hinh 2 Cdchl: (Hinh 1) 133 132