Kiến thức cơ bản và nâng cao hình học 12 NXB đại học sư phạm 2013

lX(f^) =2 Hinh 11 N T Hinh 12 Khong Id khoi da dien / JL Hinh Hinh la kh6'i da dien lOi cu th^ la hinh chop ngu giac ABCDEF Co so dinh D = 6, s6 canh C = 10, s6' mat M = => X(H) = + - 10 = ffinh la khd'i da dien I6i c6 D = 12 so canh C = 18, s6 mat M = 8, X ( H ) = 12+8-18 = i f ffinh la khoi da dien kh6ng I6i, D = 9, C = 14, M = 8, X(H, = 9+8-14 = ffinh khong la kh6'i da dien ffinh khong la kh6'i da dien Hinh 13 Hinh 14 Hinh 15 Khd'i da dien khdng iSi Khoi da dien khdng Idi D = 16,C = 28,M = 14 Hinh 18 Hinh 16 Hinh 17 Khd'i da dien khdng iSi Khd'i da dien khdng loi D = 24.C = 48,M = 24 sau va dan lai khd'i da dien diu: V i du 7: Cho hinh tarn mat diu canh a Tmh th^ tich khoi bat dien deu Chung minh rang tarn cac mat cua hinh bat dien d^u la cac dinh ciia hinh lap phuong Tinh thi tich kh6'i lap phuong nay: Hinh bat dien deu chinh la hai hinh chop tii giac deu chung day ABCD CO dien tich day m6i chop la a^ (ABCD la hinh vu6ng canh a) ASCS' cung la hinh vu6ng canh a nen SS' = a-\/2 Do chieu cao cua m6i Khoi mat deu loai (3,3) chop tu" giac deu la — a ^f2 VSy th^ tich kh6'i mat la: ay/2 V, = 2.^a 1 _ a 'V2 Kh6'i mat diu loai (4,3) Goi tarn ciia mat ben SAB, SAC, SCD, SDA la M , N , P, Q Goi trung die'm ciia AB, BC, CD, DA la M ' , N', F, Q' Ap dung dinh ly Talet ta de 2 AC dang suy M N = NP = PQ = QM = ^ M N ' = M N P Q la hinh vu6ng nen MNPQ la hinh vu6ng canh Khoi mat deu loai (3,4) Do aV2 Chirng minh tuofng tu cac tam M " , N", P", Q" ciia SAB, S'BC, SCD, SDA cung se tao hinh vuong canh - ^ ^ ^ , MNN' M " cung vay suy mat ciia MNPQ.M"N"P"Q" deu la hinh vu6ng Vay MNPQM"N"P"Q" la hinh lap phuong Khi the tich ciia hinh lap phuong la V = MN^ = ^V2 Khoi 12 mat deu loai (5,3) a' ^2 don vi the tich 27 Ti so the tich hinh lap phuong va kh6'i mat d^u la: '6 _ 2V2a^ 27 Khoi 20 mat deu loai (3,5) Hinh 19 9• V2a^- V i du Cho hinh lap phuomg A B C D A ' B ' C ' D ' canh a, bang g6 Got kh6'i lap I phuong de lay khoi mat deu noi tiep no, nghia la dinh ciia khoi mat deu la tam cua mat cua khoi lap phuong n Tinh o the tich kh6'i mat Tinh the tich ph^n g6 bo t—i— tr di \ \ a Do tinh cha't cua hinh lap phuong ta dl C \ y \ \ dang \ chumg minh O M = O N = OP = OQ = O ' M = O ' N = / = QM "~ V2 • = OA A Hinh deu la tam giac deu canh V = 41 The tich kh6'i raV2 \ 21 b Th^ tich hinh lap phucfng la a^ c / ' ^/ = mat ciia khoi OMNPQO' / ' O'P = O'Q = M N = NP = PQ / 4a^ 24 mat diu canh a4i theo v i du se la Hinh 22 The tich g6 bo d i la C BAI T A P V i du Hay phan chia h6p ABCD.A'B'C'D' cac khoi t i i dien, thuc hien theo budc sau: Chia hop ABCD A ' B ' C ' D ' a Hay chiing minh kh6'i da dien c6 cac mat la nhirng hinh da giac c6 so canh la le thi tong s6' cac mat phai la s6' chSn lang tru tam giac ABC.A'B'C va b Trong khoi da dien ne'u m6i dinh la dinh chung ciia mot s6' le mat thi CBD.CD'B' tdng so cac dinh cua no phai la mot s6' chan Chia m6i lang tru t i i dien va m6t hinh chop t i i giac Chia kh6'i lap phuomg khoi tii dien bang Chia hinh chop t i i giac tii dien Vay se c6 hinh t i i dien Chirng minh tam ciia cac mat cua tir dien deu lai la dinh ciia mot t i i dien d^u T i m t i s6' the tich cua kh6'i tii dien mdi va cu Cho khd'i tii dien ABCD, E va F iSn luot la trung di^m cua AB va CD Hai mat phang (ABF) va (CDE) chia kh6'i tii dien ABCD khoi tii dien Ke ten cac khoi tii dien va chiing minh th^ tich khd'i tii ditn bang nhau, neu ABCD la khoi tii dien deu thi kh6'i tu didn trdn c6 bang khong? Cho chop S.ABC c6 ducmg cao SA = a Day la tarn giac vu6ng can c6 AB = BC = a Goi B' la trung diim SB, C la chan duomg cao A C cua ASAC a Tinh the tich cua khoi chop S.ABC b Chiing minh SC vu6ng goc A B ' C I c Tinh the tich cua kh6'i chop S.AB'C Hay chi each chia mot khoi tii dien hai kh6'i tii dien cho th^ tich ciia hai khoi tii dien c6 ti so bang — > cho tnrdrc n Cho kh6'i lang tru ABC.A'B'C c6 day la tarn giac deu canh a, dinh A' each deu dinh A, B, C canh ben A A ' tao vdri mat day goc 60° a Tinh the tich khoi lang tru b Chiing minh mat ben BCC'B' la mdt hinh chu nhat c Tinh tong dien tich cac mat ben ciia lang tru (goi la dien tich xung quanh ciia lang tru) 12 Hay tim th^ tich kh6'i h6p n6u d6 dai canh ben la a, dien tich hai mat cheo la S, va Sj, goc giffa mat cheo la a 13 Cho S.ABCD la chop deu, khoang each tir A den mat phang (SBC) la 2a, goc giua mat ben va mat day la a Tinh the tich V cua khoi chop, v6i a bang bao nhieu thi V c6 gia tri nho nhat? 14 Cho tii dien ABCD, khoang each giCra AB va CD la a, a la goc giiia hai ducmg thang Chiing minh VABCD = - AB.CD.a.sir\ a 15 Tinh the tich kh6'i tii dien ABCD biet AB = CD = a, AC = BD = b va A D = BC = c 16 Cho khoi lap phucmg ABCD.A'B'C'D' Cac diem E va F Hn luot la trung diem cua C B ' va C D ' Dung thie't dien ciia lap phuong bi cat bcri (AEF) Tinh ti s6' the tich hai ph^n ciia kh6'i lap phucmg mat phang (AEF) cat 17 Cho chop SABCD c6 day la hinh binh hanh Goi B', D' lan luot la trung diem ciia SB, SD Mat phang (AB'D') cat SC tai C Tim ti s6' the tich hai khd'i chop SABCD' va SABCD 18 Cho tii dien ABCD c6 didm O nam tir dien va each deu mat cua tii dien m6t khoang d Goi hA, hg, he, ho la khoang each tilt cac dinh den mat doi dien Chiing minh: Cho khoi chop SABC Tren canh SA, SB, SC lay diem A', B', C khac d \&v S Goi V la the tich chop SABC, V la thd tich chop S'A'B'C ^ V SA SB SC Chung minh — = — V SA' SB' SC Cho khoi lang tru diing ABC.A'B'C c6 day la tam giac vuong tai A, ACB = 60° AC = b BC tao v6i (AA'CC) goc 30° Tinh d6 dai A C va tinh the tich V cua khoi lang tru da cho: 10 Cho A ^ hop ABCD.A'B'C'D' c6 tat ca cac =BAD = A ^ canh la a, cac goc = a (a < 90°) Hay tinh th^ tich cua h6p I I Cho h6p ABCD.A'B'C'D' c6 day ABCD la hinh chii nhat canh la a, b, hai mat ben (ABB'A') va (ADD'A') tao vdi day ABCD goc 45° va 60° Tinh th^ rich ciia h6p n^u canh ben A A ' la c 1 1 + — hg + — hp + hp 19 Cho chop S.ABC, M la mot diem nam day ABC, cac du5ng thang qua M song song vdfi SA, SB, SC Mn luot cat (BCS), (CAS), (ABS) tai A', B', C Chung mmh —ALSCS Vs^c ^ ^ SA SA ^ SB khong doi SC 20 Cho khd'i chop S.ABCD c6 day ABCD la hinh binh hanh Mat phang (P) cat cac canh ben tai K, L, M , N Chiing minh VgABCD = SA SC SB SD va — + =— + — SK SM SL SN VSACD = ^SABD = VSBCD- 21 Cho khdi chop S.ABCD c6 day ABCD la hinh chff nhat, canh ben SA vu6ng goc vdi day, mat phang (a) qua A va vuong goc SC cat SB, SC, SDaB',C',D' a Chiing minh AB'C'D' c6 goc dd'i dien la vudng b Chiing minh S chay trfen du6ng thing vu6ng goc vol day tai A thi (AB'C'D') lu6n di qua mot du5ng thing cd dinh va cac di^m A, B, B', C, C, D, D cung each m6t d'dm c6 dinh m6t khoang kh6ng d6i c Gia sur goc giua SC va (SAB) la x Tinh th^ tich cua chop S.AB'C'D' va S.ABCD bid't AB = BC 22 Cho tii dien ABCD: a Chiing minh neu chSn ducmg cao H cua tur dien xua't phat tiif A triing vdfi true tarn tam giac BCD va AB vuOng goc AC thi AC vu6ng goc AD va AD vuong goc vdfi AB b Gia sit BC = CD = DB, AB = AC = AD H la chdn duomg vudng goc tir A de'n (BCD), J la chan ducmg vu6ng goc tiir H xudng AD Dat AH = h, HJ = d Tinh th^ tich tii dien theo d va h c Chiing minh neu AABC va AABD c6 dien tich bang thi ducmg vu6ng goc Chung cua AB va CD di qua trung di^m cua CD 23 Cho hinh chop diu day la da giac diu n canh, canh day Ik a a Tinh th^ tich va didn tich xung quanh neu goc giiia canh bSn va day la a b tinh th^ tich va dien tich xung quanh n^u g6c gitta mat ben wk diy la]3 24 Cho hinh chop cut d6u day la da giac d^u n canh, canh day la a, b (a > b) a Tinh the tich va dien tich xung quanh chop cut ne'u goc giiia canh ben va day la a b Tinh th^ tich va dien tich xung quanh chop cut n^u goc giiia mat ben va day la p 25 Day hinh chop SABCD la hinh chff nhat, c6 AB = a, AD = b, SA vu6ng goc day va SA = 2a La'y M e SA vdi AM = x (0 < x < 2a) a (MBC) cat hinh chop theo thie't dien gi? Tun dien tich thiet dien a'y b Xac dinh x d^ (MBC) chia hinh chop hai phdn c6 th^ tich bang (DH Y Duoc Thanh ph6' H6 Chi Minh, nam hoc 1996, chucmg tiinh phan ban; 26 Cho tii dien ABCD, chiing minh: a Cac ducmg thang noi m6i dinh vdi tam mat doi diien dong quy tai m6t die'm G b Cac hinh chop dinh G c6 day la cac mat ciia tii dien c6 the tich bang 27 Cho tii dien SABC c6 cac goc phang d dinh S deu vuong a Chiing minh V3 SABC ^ SSAB + SSBC + ^SACb Cho SA = a, SB + SC = k, SB = x Tinh the tich tii dien theo a, k, x va xac dinh SB, SC de the tich tii dien SABC lorn nhat (DH Quoc gia Thanh ph6' Ho Chi Minh, nam hoc 1996) 28 Cho hinh chop tii giac d^u SABCD c6 tat ca cac canh bang a a Tinh the tich cua no b Tinh khoang each tii tam day de'n cac mat ben (DH Da Nang, khdi D, nam 1997) 29 Cho hinh chop OABC vdi OA, OB, OC vu6ng goc vdi tutng doi m6t va OA = a, OB = b, OC = c a Ke OH vu6ng goc vdi mat phang ABC Chiing minh H la true tam tam giac ABC b Cho H la true tam tam giac ABC Chiing minh OH vuong goc mat phang ABC c Tinh dien tich tam giac ABC theo a, b, c d Chiing minh: a^ tgA = b^tgB = e^tgC '' (DH Ngoai Ngfl Ha N6i, 1997, theo phan ban) 30 Cho hinh h6p ehff nhat ABCD.A'B'C'D' c6 A'A = a, AB = b, AD = e Tinh thd tich tii dien ACB'D theo a, b, c (Hoc vien Quan he qu6c te' nam 1997) 31 AB la du5ng vuong goc chung ciia hai dudfng thing cheo x, y Lay A e X, B e y, AB c6' dinh va AB = d Me x, N e y, M, N thay d6i va AM = m, BN = n (m, n > 0) Gia sir c6 m^+ n^ = k > 0, k khong doi a Xac dinh m, n d^ d6 dai doan thang MN dat gia tri Idn nhat, nho nhat b Trong trilcmg x vu6ng y va mn ^ 0, hay xac dinh m, n theo k va d d^ the tich tii dien A D M N dat gia tri iom nha't va tinh gia tri (DH Qudc gia Ha N6i, nam 1997, khoi A) 32 Cho tarn giac ABC can dinh A Mot dilm M thay ddi tren duofng thang vu6ng goc v6i mp (ABC) tai A (M Chuang II A) M A T NON;M A T T R U ,M A T C A U a Tim quy tich tarn G va true tam H cua tam giac MBC b Goi O la true tam tam giac ABC Hay xac dinh vi tri cua M de the tich tur dien OHBC dat gia tri Idn nha't I M A T NON, HINH NON, K H O I NON (DH Quoe gia Ha Noi, nam 1997, kh6'i B) 33 Cho hinh chop tii giac d6u S.ABCD c6 day ABCD la hinh vu6ng canh a va SA = SB = SC = SD = a a Tinh dien tich toan ph^n va the tich ciia hinh chop theo a A LY T H U Y E T C A N NH6 Su tao m^t tron xoay Trong khong gian cho mat phang a, chiia ducmg thang A va dudng r b Tinh cosin cua goc nhi diSn [(&45), (&4Z)) Khi quay mat phang a xung quanh ducmg thang A thi tap hop cac diem cua (DH Su pham TP Ho Chi Minh, khoi D - E - 2000) duofng r tao nen mot mat tron xoay nhan ducmg thang A lam true Ducmg 34 Cho hinh chop d^u SABCD Day ABCD la hinh vu6ng c6 canh bang 2a Canh ben SA = aVs Mat phang (P) di qua AB va vuong goc v6i mat phang (SCD) (P) Ian lugft cat SC va SD tai C va D' a Tinh dien tich ciia tii giac ABCD' b Tinh the tich ciia hinh da dien A B C D O ' C (Dai hoc Nong nghiep I - Kh6'i A - 2000) r sinh mat tron xoay nen dugfc goi la ducmg sinh cua mat tron xoay Tinh chat ciia mdt tron xoay * Ne'u cat mat tron xoay bed mot mat phang vuong goc vdi true A thi giao tuyen la m6t ducmg tron c6 tam nam tren true A * M6i diem M thu6c mat tron xoay d^u nam tren mot ducmg tron thu6c mat tron xoay va c6 tam tren true A (Cho nen ngudi ta noi mat tron xoay la tap hgfp cac ducmg tron nam tren cac mat phang vuong goc vdfi ducmg thang A c6' dinh va c6 tam nam tren ducmg thang A ) Mat non tron xoay Dinh nghia: Cho ducmg thang d va A cat tai O tao goc cp vdfi < < 90" Khi quay ducmg thang d xung quanh true A cho goc cp khOng thay ddi thi tao mat non tron xoay (goi tat la mat non) (h.23) O goi la dinh ciia mat non va goc is dinh bang 2(p, d goi la ducmg sinh cua mat non + 4t ^ , ITe "Tg" 120 (A) y = - 6t , khu t ta co: z = -l-8t Do chinh la phuang trinh chinh tac cua ducmg thang (A) Dap.anBdung 121 M(3; 6; -7), N(-5; 2; 3), Q la trung di^m cua MN thi toa d6 cua Q = (-l;4;-2) QM = (4;2;-5) Dudng thang QM di qua diem Q va M, nen phuofng trinh ducmg thang QM la: x-3 y-t6 z + x + y-2 z = —f— = va — — = — — = —r•4 f ^ - X = Dap an D dung j 122 Ducmg thang A i di qua diem M(l; 7; 3) va c6 vecta chi phuofng M=(2;l;4) Ducmg thang A' di qua diem M'(6; - ; -2) va c6 vecta chi phuong MM' = (5;-8;-: 5) u = (3;-2; 1) Bieuthiic [u , u'] MM' = (9; 10;-7).(5;-8;-5) = 9.5+10.(-8)+7.5 = => A va A' dong phang, matkhac [ Z , u ' ] = (9;10;-7) Vay A va A' cat DapanAdiing ' 123.a)Tac6: IA + L ? = 2IE C + ID = 2IF T^O b)C6: IA+ IB +IC +ID =2(IE + IF)=0 MA +MB =2 M^ MC+ MD = MF MA+ MB +MC +MD = 2(ME + 124 1) Do G, la tam cua A BCD nen MF)^4.MI (1) G,B +G,C +G,D = 2) Ta da bid't trpng tam ciia tii dien chia m6i trung tuyen ciia tii dien theo ti s6' 3:1 (trung tuy6h ciia tii dien la doan thing n6'i dinh va tam cua mat d6'i dien) nen GA =30^ (2) Tiir(l)tac6: G.G + GB +G,G +GC +G,G +G,D = r = - , vay I = ( - ; - ; - ) 3 3 1 3) Dudng thang vuong goc vdi mp(ABC) c6 vecta chi phuong (1; —; - ) vay ducmg thang di qua I vuong goc vdi mp(ABC) c6 pt: 127 1) Di^m A e O x , B e O y => mp O A B chinh la mat x=- phang Oxy nen c6 pt z = Tucmg tu pt mat phang =0, y Mat 1 z = - + -t 3 =0 phang (ABC) chan tren ba true tea d6 tai ba Tat nhien diem J doi xiing vdi I qua mp(ABC) ciing thuoc ducmg thang diem A, B, C ndn pt long quat cua mp(ABC): + t y ^ l t (OBC), ( O C A ) ^ lucrt la: X , ,,1 Gia su J ( - + 1 1 ; - + - / o ; - + - / „ )• Hinh 79 - + — + - = (pt doan chan) ^ 2) Goi r la ban kinh hinh c^u npi tiep tii dien O A B C K h i r chinh la khoang each tiir I den cac mat phang (OAB), (OBC), ( O A C ) va (ABC) Goi M la trung diem ciia IJ thl M: 1 ,3 1 ° 1 "3 Vay toa d6 ciia I(r; r; r) Khoang each tir O de'n mp(ABC): «;n 151 Dang chinh tac: « = , , 13 1 , -.V,yJ(-;-;-) _ X _ ^ ^ -1 ""iT'"! b) mp Oxy c6 pt: z = c6: cos^ = 128 l ) D s : 15x- l l y - z + = 2) Viet PT mp (P) qua A va vu6ng goc vdri (d) Mp (P) nhan vecto chi phirong u = (3; 4; 1) cua (d) lam vecto phap tuye'n, nen PT mp (P): 3(x - 1) + 4(y - 2) -f- z - 12 = 130 I ) c ( d ) f 2x — v — 11 = ^ ^ y = -ll+2x x-y-z+5=0 Datx = t nen CO PT (d): Toa giao diem B cua (d) va (P) la nghiem ciia he: £=>Lzl.z + 3 3x + 4y + z - = B 33 35 67 26 26 V8987 17 93 AB = (—; )=> AB = —V7' +17' +93' = 26 26 ^26 26 26' 129 a) Pt mp (KHI) di qua ba diem: 2x + y - z - l PT giao tuye'n cua hai mp: X + z= X Dat z = t, ta CO PT tham so: 2) Ta c6: Duofng thang (d) Dudng thing ( A ) Di q u a d i e m M , ( ; - l l ; 16) Di qua diem M2(5; 2; 6) = (1;2;-1) M,M, Bie'u thiic [M, , = -t y = - +—t 2 z= t = (2;1;3) = (5;13;-10) ] M,M^ f PTmpla: = => (d) va A cung thu6c mot mp -1 -1 1 3 2 1J = (7;-5;-3) x - ( y + 1 ) - (z-16) = 7x-5y-3z-7 =0 3) Mp (P) chiia (d) va chie'u (d) theo phuong ( A ) nen no la PT mp of cau2 Vay PT tdng quat ctia hinh chieu la: r7x-5y-3z-7 152 ^ Mat phang c6 vecto phap tuye'n: =0 2x + 2y - 9z - = ^ - * u = ( l , 2,-1) [y = - l l + 2t = fl4x - lOy - 6z - 14 = 3x-2y-2z-l = j+Sx - lOy - lOz - = 4) X - 4z +9 = => X = -9 + 4z Duomg thing A phai tim song song v6i (?) nen song song v6i (d,), nghia x = - + 4t la CO vecto chi phuong u^ PT ciia ( A ) la: y = -14 + 5t Dat z = t ta CO PT tham s6': x = + 43t z=t PT chinh tac: x+9 y + l4 y = - + 30t z z = - + 23t 132 Goi hai mp da cho theo thii tu I la (P) va (Q) V i (P), (Q) cung • vu6ng g6c vdi mp (R) ntn giao : tuye'n ( A ) ciia chung phai vu6ng goc vdi (R) Do do, 131 Trudc het, viet PT mp (Q) chijra ducmg thSng (d) da cho va di qua A Dudng thang (d) c6: X = PT: I + 3t ^ y = - + 2t z = + 2t Hinh 104 No di qua B(2; -4; 1) va vecW chi phuong u = (3; 2; 2) va AB = (nen c6 PT: ( x - ) ^ 2 _^ + ( y + 4) _^ c -1 3 ^ + (z - 1) ^ ^ = 1 -1 -12 3 133 ) ^ = (2;0;2) = 2(1;0;1) Dudng thang AB c6 vecto chi phuong M = (1; 0; 1) di qua diem A(0; 0; - ) , nfen AB c6 phuong trinh: 12x - 8y - 12z - 28 = [x=0 + t 248 43 -30x + 43y + 248 = => x = — + — v 30 30- vay (d,) CO vector chi phucmg w, = (43; 30; 23) 2x + 3y + z = G '-42x + Sly + 12z + 276 = Dat y = t, ta CO PT (d,): -12 No cung la vecto phap tuyen ciia mp (R) nen PT mp (R) la: f-14x + 17y + 4z + 92 = Dudng thang (d,) c6 PT: BC thi mat CDD'C la mat ben Idn Theo tmh chat hinh hop thi BC mp CDD'C nen CD' la hinh chieu cua BD' tren mp Vay goc tao bai BD' va mat ben 1dm Hinh 106 CDD'C la goc BD'C = /? Tam giac vu6ng BCD' cho: BC = BD' sin /? = d sin p, CD' = BD'cosy5 =dcosy9 |Tam giac vuong BDD' cho: D'D = BD'sin« = d s i n a Tam giac vu6ng CDD' cho: CD = Vc'i)^ -DD^ De tim SH, ta dira vao tarn giac vuong SHK va phai ti'nh H K De tim H K , ta phai dua vao cac tarn giac vuong K H A va C H I : = d.^cos^ p - sin^ a V = AB.BC D D ' = d^sina sin p Vcos^y^-sin'a (*) HA = Bie'n doi: l + cos2y9 cos p - s m a = l-cos2a _ HC = Thay vao (*) ducfc dpcm » a = HK 135 Cho A A B C c B = l v v a A = a , A C = a.Theod£ubM:(SAQ ( A B Q chop thi SH e(SAC) SI BC thi H I la hinh chie'u HI cos a cos a + sin a a sin 2a sin a cos a 2V2sin(45°+a)' Tarn giac vuong SHK cho :SH = H K t g Ke cua SI tren day ma SI Thay vao (*) duac dpcm Chii y: V i d^u bai kh6ng noi ro BC Chan ducmg cao thu6c mien day nen H I BC (theo dinh l i ba ducmg vu6ng goc) VSy sin a BC = H K ( ^ ^ + - ^ • ) v i ( H K = H I ) sma cos a = -(cos2/? + c o s a ) = c o s ( a + > ) c o s ( > - a ) nen ke ducmg cao SH ciia hinh HK Hinh chop hay khong, nen H c6 SIH the la giao diem cua A C keo dai la goc phang ciia nhi didn BC, theo d^u bai SIH = /? Xac va tia phan giac ngoai cua goc Hinh 107 B (hinh: 108) NeuAB>BC dinh tuong t u ta co S K H = /3 De tha'y A SHI = A SHK => H I = H K , nghia la H each diu hai canh goc vuong => H thuoc dudng phan giac cua B : HA = HC = H B I = H B K = 45? => B I H K la hinh vu6ng C V = ^dt(AABC).SH Dt A A B C = - A B A C s i n a sin a HI cos a HK = A B = AC.cos a = acos a = - a ^ sin a cos a = -a^sin2a va V = < 45") thi: HK a = HK(—!^ sin a De tim dt ( A ABC) ta t i m A B tuf tarn giac vu6ng ABC: (a ~ ) cos or a sin 2a.tgP 2V2sin(45°-a) a\s,\n^ laigp 24.V2sin(45°-a) = ^^^"^"'^^^ 2V2sin(45°+a) - Neu AB < AC ( « >45") (hinh 109) thi: 12 Khi th^ tich lom nha't la • - HI HC = cos a HK HA = sin a a = HK ( V= •'' 1 a.smla.tgp >HK = ^ a' • 12 ~ 48 A B / / C D => AB//(SCD)ma 2V2sin(a-45'') (P) chiia AB, M N = (P) n (SCD) =>AB// M N , de dang chiing minh A N = BM, ndn ABMN la hinh thang can 24.V2sin(a-45°) 136 1) Ta CO aS 137.1) Goi O la tam ciia day hinh chop, P, Q theo thii tu la trung diem canh day AB, CD, R = M N n SQ •) sin a cos a 'y BH ± AC (vi H la true tarn cua A ABC d^u) /' ' \\ /'' ' \\ A/M[ \ Ta CO SP AB, OP AB nfin BH M A (vi A M (ABC)) Suy BH (MAC) => MC MC , * MC 1 OS? la goc phang cua nhi ' dien tao bod mat ben va day BH BK (K la true tarn A BMC), suy MC ± (BHK) (BHK) => HK MC cung chiing minh tucfng tu ta c6: HK >HK (BMC) a Dodo M N = ^;PR = MB 2) Goi D la trung diem ciia BC thi K e M D Da dang chiing minh BC (MAD) Trong mp (MAD) ke LO AD (O e AD), de dang chiing minh dugfc: ^KABC = ~ => OPS = 60°, de dang chiing minh tam giac SPQ la tam giac d^u Ta CO PR la ducmg cao, M N la dudng trung binh cua A SCD H i n h 1 vay dien tich thie't dien ABMN, la: a _ AB + MN '^ABMN ^ t'K - a + 2- aS 2) Theo c6ng thiic tmh M tich khoi chop: ABC- OK Vay VKABC lo'n nha't va chi OK Idn nha't Do A H K D vudng K (HK (BMC) => K nam tr6n du5ng tron ducmg kfnh HD ndn O K \6n nha't ^ V=i.Bh B = dt hinh vu6ng ABCD Hinh no B = a^ H = SO cung la chieu cao cua tam giac d^u SPQ nen: _^S.a^ ~ Dieu kien du: Co he thiic be + 2a' = a( b + c) thi A ABC vuong a A h= so = aS That vay, ta c6: AB' + AC' = 2a' + b' + c' - a(b +e) = 2a' + b' + e' - (be + 2a') vay: = b' + c'-tK: = B C Theo dinh l i dao Pitago thi A ABC vuong A VAy3CD= ^ - Ta lai c6: ~ V SCO = > _ 1 SD ' _ i V A S B M = — Vs.ABCD' V A S M N = ' S.ABCD 3) Ke ducmg cao A H Ke A I ± OB thi H I la hinh chie'u eiia A I tren day ntn H I ± OB theo dinh l i ba du5ng vu6ng goe Tuorng tu: ke A K I O C eung CO HK ± OC Cac tarn giac vu6ng OAI va KAO bang (OA Chung, mot goe 60") =>AI = A K => H I = HK nghia la H each deu cac canh A c Hinh 112 ciia goe yOz = 60" n6n OH la tia 138 1) Da tha'y OABC la tii dien deu vi cac tarn giac OAB, OBC, OCA la nhirng tarn giac deu bang Khoang each tCr A den mp Oyz la ducmg phan giac goe HOI = HOK = 30°, OA = a => = - a va A I = cao tii dien, va tinh dugc bang V = - d t (AOBC).AH = tarn giac vu6ng HOI cho HI = O I t g " = A ^ = 12 2) Ap dung dinh if ham so cosin vao tarn giac OAB, OBC, OCA ta c6: ^ Ap dung dinh l i Pitago vao tarn giac vu6ng H A I cho: A B ' = OA^ + OB' - 2.OA.OBcos60" = a' + b' - ab BC' = OB' + OC' - OB.OC.COS60" = b' + c' - be AH' = A I ' - H I ' = 2.a' C A ' = OC + O A ' - OC.OA.COS60" = c' + a' - ac Dieu kien can: Cho BAC = Iv thi be + 2a' = a(b+c) That vay, ap dung dinh If Pitago vao tarn giac vuong ABC, c6: BC' = A B ' + AC' « b' + c' - be = 2a' + b ' + c' - a(b + c) =;>dpem V = dt (A BOC).AH =1.1 Theo cau 2, ta c6 be + a' = a(b +c) QK OC.sin60".AH = b,c la nghiem cua phuong trinh: - dx + ad - 2a'= (*) Co A = d' - 4(ad - 2a') = (d - 2a)' + 4a' >0 Nen phuong trinh (*) chic chin c6 hai nghiem khac Vi b, c > nen a(d - 2a) > => d > 2a Day la di^u kien d^ tinh duoc b, c Goi f(x) = x' - dx + ad - 2a' Co f(a) = - a' < 0, f(2a) = a (2a - d) < (do d > 2a ) z:>a va 2a d khoang hai nghidm ciia (*) => dpcm 139 a) Ke OK d thi OK = a va SK i d theo dinh If ba dudng vuong goc => d mp(KOS) ii>mp(S,d) ± mp (KOS) Tam giac vuong KAO cho: OA = KO sina a 3sina sma Tam giac vuong SAO cho: SA' = SO' + OA' o 25a' 64a' Ila tam mat cau ngoai tiep tir dien SOAB 10 cat SM tai G Ta phai chiing minh G la tam A SAB That vay Do A //SO (vi chiing cung vuong goc v6i (P) nen theo Talet, ta c6: ^ = ^SO = l2 ^ G S = 2GM(dpcm) ' y , 140 Bai toan khong sai neu ta cho hinh cau va hinh non long vao cho chan ducmg cao hinh non triing \6'\p diem cua hinh cau va (P), liJC tam hinh cau thuoc ducmg cao hinh non Hinh 114 la thiet dien qua true hinh non, cat hinh non theo mot thiet dien la tam giac can, cat hinh cau theo mot thiet dien la hinh tron Idn (qua tam hinh cau), c6 ban kinh bang R Do (Q) // (P) => A'l //AB nen theo Talet ta c6: SI DC SH HA R (h -x) Tam giac vuong HA'J cho: h - x IKi IK = — h h R A'l' = IJ.IH = (2R -x)x S = ;r (IK' + lA") = n: R' (h-xf +(2R-x)x (1) Khi h < X < 2R (hinh 114), thurc hien phep tinh nhu tren, ta van dugrc bieuthirc(l)vi (x - h)^ = (h - x ) ' b Biend6i.(l): ,R S= n: i^-\)x^ voi < X < < X +2R0 R O a) )x + R Sn,in X 2R Do thi la parabol c6 dinh cue tieu I o Rh 141 a V s A B c - IsA.dt (AABC) ma SA = a, A ABC la tarn giac deu canh a n^n 2:i.R'.h R +h' R +h R > h thi parabol c6 be 16m quay Ifin tren nSn c6 la tung dp dinh: 2nR'h S^i^ = R +h dt(AABC)=^ (hinh 116) Vay V s A B c = - « - - — R - Neu —- - < R < h thi Parabol c6 be 16m quay xu6ng dudi ntn Hinh 118 ^V3 12 CO cue dai S^ax la tung dinh: Sn^^^ = R +h (hinh 117) Goi I la trung diem eiia BC - Nfi^u R = h thi S = ;r R^, thi la doan thing song song \6i true hoanh (hinh 117c) A SBC can dinh S (SB = SC) ^ SI I B C , nen dt (A SBC) = - a SI ma SI = -JSA'+Al' ^dt(ASBC)=ia.^ = 2 V Neu • T I I I O D ^ V s A B c = - h d t ( A S B C ) = Ih ^ H ^ ^ goi h la khoang each tCr A den = mat phang (SBC) thi = 12 h = Hinh 116 167 b) Ghon he toa d6 nhu hinh ve 118 ta c6 A(0; 0; 0) la g6'c toa d6, S(0; 0; a), C(0; a; 0) Vi SIC = SAC = 90" suy tii dien S AIC noi tie'p mat c§LU c6 ducmg kinh la SC = Neu goi K la trung di^m cua AC, thi K = K R= 2'2) la tam mat cdu 142 a) AB = (2; 3; 4) Ducmg thang d di qua A va B nhan vecta AB lam vecto chi phuofng va di qua A(0; -2; 0) ndn c6 phuong trinh tham s6' la: = 2t y = - + 3t z = 4t b)M ed => M (2t;-2 + 3t; 4t) 2t-2 + 3t-4t + d(M,(a))= — X t+3 V5 d(M,(«)) = 2.V3 o - ^ = 2.V3 t+3 = V29 Ta c6: d(I, ( a ) ) = V3 7V3 V29 (S) CO tam la trung diem I ciia AB „ ——- < —— = R nen mat phang ( a ) cat mat c^u (S) 143 a) OA = (4;0;0) => OA =4 Tuong tu ta tinh duoc OA = OC = AB = BC = => ABCO la hinh thoi Co OC = (0;4;0) =^OA.OC = ^ OA IOC => ABCO la hinh vu6ng Mat khac ta lai c6 SO = SC = SA = SB = VlT => SABCO la hinh chop d^u b) Dien tich hinh vu6ng ABCO la : OA^ = 16 Mat phang (ABCO) c6 hai vecta chi phuofng OA = (4; 0; 0), OC = (0; 4; 0) nen no chinh la mat phang Oxy, nen phuong trinh mat phang (ABCO) la: z = 0; d(S, (ABCO)) = I Vay VsABco=^-6.16 = 32(dvtt) t=3 t = -9 Vdi t = ^ M, (6; 7; 12) V6i t = -9 M2 (-18; -29; -36) c) Mat ciu (S) CO ducmg kinh AB va CO ban kinh R = AB Mat c^u (S) CO phuofng trinh: (x -1)^ + (y + )2 + (z _2)2 = ^9 ngoai tiep tii dien SAIC Vay phuong trinh mat c^u ngoai tiep tii dien SAIC la: x^ + ( y - f ) ^ + ( z - 2^ ) ^ = ^ 2) 1(1; c) Goi M la trung diem ctia SO => M = (l; 1;3) Mat phang (P) la mat phang trung true ciia SO di qua M c6 vecta phap tuyeh Ik OM = (1; 1; 3) Vay phuong trinh ciia (P) la: ( x - l ) + ( y - l ) + 3(z-3) = / iW i\\ /' // /' \ //'' \\ / > \ ^.1 \ \ (0:4.i)) U ' ' ' / -'X B (4:4:(n Hinh 119 ^ =>x + y + z - l l = T De dang c6 toa 1(2; 2; 0) 7S = (0; 0; 6) Vay phuong trinh tham so cua ducmg thang X=2 IS: ] y = z = 6t Tarn I cua mat c^u la giao diem cua mat phang (P) va ducmg thing IS, thay gia tri x = 2; y = 2; z = 6t vao phuomg trinh mat phang (P) ta c6: + + t - l l = = > t = — J ( ; 2; - ) 18 ,, ,^ =i>R = OJ = 121 _ n ~ Do phuong trinh mat ciu ngoai tiep hinh chop S OABC la: AH ±b ^ (^t - y ) + l(t + 16) + ( t - ) = - r = - y «t = - s u y r a H ( - ; - l l ; - ) Dia'm A' la dilm doi xirng ciia diem A qua A2 nen H la trung diem ciia AA' suy AA' = 2AH mkAH= (-10; -13; -8) ^ A'= (-17;-24;-11) b) Goi (Q) la mat phing di qua A, va song song A2 phuong trinh mp (Q) CO dang: a ( x - y + 3z-5)+ /?(x + y - z ) = vdia^+y9'^0 /? = Vay phuong trinh mat phAng (Q): 4x + 3y + z-5 = c) Ta CO d (A,, A2) = d( A2 (Q)) = d(B.(Q)) Goi H la hinh chieu vuong goc ciia A len Aj 4.(-y) + 3(-14) + 1.0-5 H(-Y + ^ t ; - - t ; t ) Taco AH = (1- t - —17; - t - ; t - ) 2 ncnng lb rigL = ^(2a + J3)-l(-a+2/3)+1(3a-J3) (x-2)^ + (y-2)^ + ( z - ^ ) ^ = 1211 ^ 11 27 145 a) A,: 69 ^/4'+3'+l' x+y+z-4=0 Dat z = t ta c6: 2x - y + 5x - = X = A,: - 2t = (3; 5: 6) y =2+t Tim mot diem M e A z =0+t M (1; - ; 1) Vay phuofng trinh chinh tac ciia ducmg thang A la: => A, di qua M , (2; 2; 0) va c6 vectachi phucmg u^ = (-2; 1; 1) A2 di qua diem M2 (1; 0; 2) va c6 vecto chi phuong x-l _y + = (-2;3;1) l^2-1 ~ c) Goi H la hinh chieu vuong goc cua A tren mat phang (P), H la [u[, ].M^M-^ =6^0 phuong trinh d u ^ g thing fx - 9y + 5z + = hirffng I K H O l D A D I E N V A THE TfCH CLIA C H U N G [x - 2y - 5z + = hirang I I M A T N O N , M A T T R U , M A T C A U A , : •< M6t vector chi phuomg ciia A la M J phuofng v6i M , va = (55; 10; 7), M J kh6ng ciing => A cit A , va A J 148 a Ta da bie't neu hai mat phing vu6ng goc \6i thi hai vecta phap cua hai mat phang vu6ng goc vdi va ngugc lai ^ = (1;6;2) J^' ^ = 2.1 - 1.6 + 2.2 = o ^ o ( a ) (/?) b) Mat phang (P) chiia giao tuyen ciia hai mat phang {a)\kifi) phuong trinh dang: (P): A ( x - y + z - l ) + ;/(x + y + 2z + 5) = o ( A + / / ) x + (->l+6//)y + (2/l+2//)z + / / - A = Mat phang (P) di qua gd'c toa d6 O nfin D = => // - A = cho A = => fi =>{V): l l x + y + 12z = =1 ntn c6 19 I Mat non, hinh non, khdi n6n 19 I I Mat tru, hinh tru, khoi tru 23 I I I Mat c^u 27 On tap chuong I I 32 hUtfng I I I P H U O N G P H A P T O A D O T R O N G K H O N G Tac6;C=(2;-l;2) GIAN 32 I He toa d6 khdng gian 35 I I Mat phang 48 I I I Dudng thang 58 On tap chirang I I I 70 NTAPCUOINAM 74 ...£^ ndi ttdiL Cu6''n sach Kien thiJtc ca ban vcl ndng cao Hinh hoc 12 nham giiip cac em hoc sinh lop 12 nSm viing noi dung ca ban va nang cao kien thiJc Hinh hoc Ke''t ca''u cudn sach duac chia... hoc tap va on t h i H i n h hoc 12, ma la t a i '' lieu tham khao t i n cay cho cac thAy c6 giao giang day mon hoc Cuon sach Kien thAc ca ban vd ndng cao Hinh hoc 12 tai ban Vkn c6 chinh l i va... tich day nhan vdi chi^u cao V = B.h Thi tich kh6''i chop bang ^ dien tich day nhan vdi chi^u cao V = ^ Bh The tich khoi chop cut bang tong the'' tich ba hinh chop c6 cung chieu cao v6i chop cut, c6