Tuyển tập 500 bài toán hình không gian chọn lọc phần 2 nguyễn đức đồng

D i r n g B H I E => N H _L I E ( d i n h l y b a d t f d n g v u o n g go c) => g o c c i i a m a t p h S n g t h i e t d i e n v a m a t d a y l a : (p = N K B ( h o a c S F t ) D o A N B H v u o n g d B => ta n c p = NB HB ( 1) Ti T e a c h d i Tn g t a s u y r a N l a t r u n g d i e m c u a B B ' H B^ BI ' BE ^ c Va ^ ~ b — 2Va2 b^ ^ ^ ^ c ( a + b ) ab NB = ab Va : ( 1) => t a n t p = CO S a^b^ (p ab CO S ( p = V a V + b V + c V D i n h l y c h i e u h i n h c h o t a : S ' = S.co s ip ^ S = S' cosip (vi S l a d ie n tich th ie t d ie n ) = - VI v b V7 ^ (ycbt) U o a i S : M A T P f l A« G I A C C U A NflJ D i t N I P H U O N G P H AP Co so cu a ph ifan g phap di/ n g m a t phan giac th e o dinh n g h i a c a n p h a i t h i f c h i e n b a h xid c ca b a n : B : D u n g m a t y = (d; Mc ) vdi d = a • B2 : D irn g tia p h a n giac M c ciia • B i : D ifn g go c n h i d i e n • a ^ c u a n h i d i e n ( a ; d ; (5) aMfe r> (5 T h i y l a m at p h a n giac ciia ( a ; P ) O Gh i d en VL c h i i : Mot i C A C m at BAI d iem nhi T O A N t r en d ien m at bdng C O p n g ia c c6 k h o d n g ea ch nha u BAN B a i 178 Ch o h i n h ch o p S A B C D c6 l a t a m gia c S A B d e u v a A B C D l a h i n h vu o n g G i a s if ( S A B ) ( A B CD ) X a c d in h m a t p h a n giac ciia cac n h i die n ( B ; A D ; S ) Giai Go i G la tro n g t a m A A B C deu Ke A G n S B = M ( AMN ) o S C = N Th e o t i n h c h a t giac tu ye n s o n g s o n g t a c M N // C D De y AD ± (SAB) ! AD SA [ AD SB D o n g th c fi A M l a p h a n g i a c S AB D o m p ( A M N D ) l a m o t p h a n gia c go c n h i d i ^ n ( S ; A D ; B ) ( ycbt) 10 Biti 179 Cho hinh chop tuf giac deu S.ABCD ChiJng minh rkng SO la giao tuyen cua cac m at phan giac tucfng ufng vdi cac cap nhi dien (SA), (SB) va (SB), (SC) va (SC), (SD) va (SD), (SA) Giai Do tinh chat doi xtifng ta chi can xet mot cSp nhi dien (SA), (SB) De y den nhi dien (SA), dUng BM SA OM la i SA => O M la tia phan giac cua I5MB = => (SOA) la mat phan giac nhi dien ( S A ) Taong tif tinh doi xuTng : i(SOB) la mat phan giac nhi dien ( S B ) (SOC) la mat phan giac nhi dien (SC) (SOD) la mat phan giac nhi dien (SD) Do S O la giao tuyen tifong ufng ycbt (dpcm) tail G: XAC Dpjfl MAT CAU tiQl Tli? HlNfl CHOP DA GlAC Dt u LPHirONGPHAP Co so cua phuang phap thuomg sijf dung cho hinh chop tam giac deu qua ba budc : B i : Xac dinh true difdng tron d cija da gidc day (thong thifdng d qua dinh S) • B : Xac dinh mat phan giac (y) cua mot mat ben tiiy y va day • B : (d) n (y) = W la tam mat cau can t i m va khoang each tCr W den cac mat ben hay day cua hinh chop la ban kinh r cua mat cau Sau ta c6 the tinh ban kinh inft cdu ngi tiep hinh chop bdng the tich V va di&n tich todn phdn Stp Ghi chii : nhiC sau: r =• 3V n C A C B A I T O A N C O B A N Bai 180 Xac dinh tam va ban kinh mat cau noi tiep tiir dien deu canh a Giai Goi G la tam AABC deu => GB = GD = GC (1) Tuf dien deu ABCD canh a (2) => AB = AD = AC Tir ( ) va ( ) cho ta AG la true dudng tron ngoai tiep ABCD Diftig mat phan giac (y) cua goc nhi dien canh CD la ^ ynAG = W Taco: W e AG ^ d[W; (ABC)] = d[W; (ACD)] = d[W; (ADB)] We y ^ d[W; (ACD)] = d[W; (BCD)] i W la tam mat cau npi tiep tuf di§n deu A B C D (ycbt) [WH = W G - r la ban k i n h mat cau noi tiep tijf di?n 101 O D i n h l y difdng p h a n giac \k d i e n t i c h cho: I W = A I I G c o s ( \^) AI + IG = W G = V W ^ ~ G F = J ^^' - ^^' c o s ^( ^) ^G I ^ K ( A I ^^G I ) ^ Doc gia t i f g i a i => r = (ycbt) C a c h k h a c : T a c6 : V = - S , „ t p - r => r = 2 aV3 3V (1) = V3 a ; T r o n g d6 : V = i B h = l — a 3 ,2 (1) Do : r = 12 VSa^ ^ aV6 " (aV3 Y j K J f1 ,3 aV3 Y j -Jla^ 12 (ycbt) 12 Bai181 T r e n dudng v u o n g goc v d i t a m giac deu A B C canh a t a i triTc t a m l a y m o t d i e m S cho S H = 2a Xac d i n h t a m v a b a n k i n h m a t cau n o i t i e p tuf d i e n SABC Hi^dfng d i n Tifang t i f b a i t r e n : W l a t a m va r = W H l a b a n k i n h m a t cau n o i t i e p tuf d i e n S A B C V(Ji S,p = i B C S l V - B C A I U j Va V = - B h = 3 Do : r = | ^ = - a Stp Bai :.2a = (ycbt) ^ 182 xac dinh t a m va ban k i n h mat cau noi tiep tijf dien A B C c6 AABC deu canh a cho SA = SB = SC va goc n h i d i e n (BC) la a UiXdng din Doc gia t i f g i a i , t u o n g t u b a i t r e n Bai 183 T i m b a n k i n h m a t cau n p i t i e p h i n h ch6p tuf giac deu c6 canh ben b a n g c a n h day l a a Giai T a CO : S,p = a^ + aV3l fl = (V3+l)a^ 2 SO = I ,2 > Goi V l a t h e t i c h h i n h chop tuf giac deu 102 V= l Bh = l a ^ - ^ ^ = ^ a ^ 3 Do : r = 3V S,p (V3 - l ) a ^ 2(V3 (ycbt) 4-1) nLGlAlTOANTHI Bai 184 ( D A I H O C C O N G A N T R U N G LfONG - 1970) 1/ Ngifcfi ta xem mat trung triJc ciia canh cua mot tuf dien C h i i n g to rkng mSt ay giao tai mot diem 2/ Goi r va R Ian lifot la ban k i n h ciia dUdng tron npi tiep va cua du&ng tron ngoai t i e p ciia mot mat tam giac y ciia tijf dien tren ChiJng m i n h r k n g : ~ ^ — R Giai 1/ Cho til dien A B C D G o i I la giao d i e m cua ba di/dng t r u n g true ciia t a m giac B C D => I la t a m difdng t r o n ngoai t i e p (BCD) Hai mat t r u n g trUc cua h a i canh BC va C D c6 diem chung I , nen p h a i cftt theo giao tuyen d ^ d (BCD) Hai mat t r u n g trUc ciia canh B D va C D cung CO diem chung I , nen p h a i cftt theo giao tuyen (d') qua I va d' (BCD) ^ (d') ^ (d) ^ Vay ba m a t phSng t r u n g trUc cua ba canh cua t a m giac (BCD) c&t n h a u theo dUcrng t h i n g d vuong goc v6i m a t p h d n g (BCD) t a i I Canh A B x i e n goc so vdi m a t p h a n g (BCD), nen m a t p h a n g t r u n g triTc ciia canh A B cSt difcfng t h i n g d t a i m o t d i e m O => la diem chung ciia bon m a t t r u n g trUc cua bon canh BC, C D , D B va A B OB = OC = O D = OA Cac d a n g thiic OC = OA va O D = O A c h i i n g to O cung dong thcfi thuoc cac mat t r u n g trifc A C va A D Vay O la giao d i e m cua sau mat t r u n g triic cua sau canh cua t i i dien da cho (dpcm) • C a c h k h a c ta c6 the l|Lp l u ^n dcfn g i a n nhii s a u : Ro rang mot d i e m y thuoc dong t h d i ca sau m a t p h a n g t r u n g trifc ciia sau canh ciia t i i dien k h i va chi k h i d i e m each deu b o n d i n h ciia t i i d i e n Chi CO m o t d i e m nha't c6 t i n h c h a t n a y , la t a m cua h i n h cau n g o a i t i e p t i i d i e n =5 (dpcm) 2/ Gia sii goi AABC la t a m giac can xet t r o n g gia t h i e t va goi : Diicfng t r o n noi t i e p AABC c6 tam Oi va ban k i n h r ; d u d n g t r o n ngoai tiep AABC c6 t a m O2 va ban k i n h R • Diing t a m giac A i B i C i c6 cac canh d i qua cac d i n h ciia AABC va song song v d i cac canh cua \ABC • Diing cac tiep tuyen cua dadng t r o n t a m O2 (ban k i n h R) t a i cac t i e p d i e m T j , T2, T va song song v d i cac canh ciia t a m giac A i B j C i cho tiep tuyen A2 B2 song song vdfi A i B j va tiep diem T2 nhm t r e n cung A B (chiia d i e m C), ta difcfc t a m giac A2B2C2 103 Theo each dUng t r e n day, m i e n t r o n g t a m giac A i B j C i nkm han AAI B, CI a t r o n g cua mien t r o n g t a m giac A2 B C2 va 0 A A B C2 Goi t h e m R, la ban ki'nh dUcfng t r o n n o i t i e p t a m giac A,B,C, =j> R, < R M a t khac, t y so dong d a n g cua ban k i n h cua diTdng t r o n noi tiep h a i t a m giac dong d a n g A i B j C i va A B C bSng t y so cac canh ciia h a i t a m giac do, tilc l a : Ri ^ r ã = ô R i = 2r 2r (hien nhien) R - 2r > — ^ — (dpcm) R B a i 185 ( D A I H O C K H O I A - M I E N B A C - 1972) Cho mot k h o i tuf d i e n deu SABC Goi S H l a dudng cao ciia k h o i tuf d i e n d6 va I la t r u n g d i e m cua S H a/ ChuTng m i n h r k n g d i e m I , t r p n g t a m T ciia t a m giac A B C va t a m h i n h cau ngoai tiep k h o i tuf dien l A B C t h a n g h a n g b/ T i n h ban k i n h ciia h i n h cau n o i t i e p k h o i tuf d i e n l A B C theo canh a cua tuf d i e n deu SABC c/ Chufng m i n h r k n g ba dudng t h a n g A I , B I va C I tiTng doi mot vuong goc v d i Giai a/ De y t i n h chat deu ciia tuf d i e n S A B C nen S H chufa I la true dadng t r o n ngoai tiep AABC va T = H Goi r l a t a m m a t cau ngoai t i e p h i n h chop => 1 I H = IS r, I , I I t h a n g h a n g (dpcm) b/ Goi O l a t a m h i n h cau n o i t i e p tuf dien l A B C , t h i O n a m t r e n I H , dong thcfi O n k m t r e n m a t p h a n g p h a n giac ciia goc n h i dien canh A B , tufc l a O la giao d i e m cua I H va dirdng p h a n giac goc I M H , va r = O H la ban k i n h h i n h cau noi t i e p k h o i tuf d i e n l A B C Ta CO : S H = A/ SA^~MF = a^- (2 aS_ 3" ^ I H = ^ = ^ Xet t a m giac vuong I M H ta eo : tan ^ I H (1) fi aV3 M H O r A/ M H 104 Mat khac : IM= VMH^ + I H ^ = Theo cong thilc nhan : (i> a i 4" , V / 2tanM r ^ = V^ - tan^^Mj o V2 tan^ M j + t a n M i - tanMi = J-LJE tanMj = -1-V3 = — < (loai) > =0 « t a n M j = t a n " V2 Xet tam gi ac vu o n g OMH, ta c6 : r = OH = M H t a n O M H = ^ ^zA = V2 ^^^^^^ 6V2 , V pO Ghi chu : Doc gld c6 the tinli r = ——, bcti loan se dan gian hem nhieu! 3 £L Nhif ta da tinh difcfc cau b) : I M = — tuang t u ta c6 : I N = —, IP = — 2 ^ IM = ^ A B , I N = - B C , I P = - A C 2 => Cac tam giac AIB, BIC va CID deu vuong tai I , trung tuyen c6 dp dai bang nCfei canh => A I , BI va CI doi mot vuong goc (dpcm) iBai 186 (DAI HOC KHOI B - M I E N B A C - 1972) |_ I Cho hinh thang ABCD, vuong o A va D, AB = AD = a, DC = 2a Tren du&ng t h i n g vuong goc pp6i mat phing ABCD tai D lay mot diem S cho SD = a a/ Cac mat ben cua hinh chop SABD la nhufng tam giac nhu the nao ? b/ Xac dinh tam va ban kinh hinh cau di qua cac diem S, B, C, D [d Goi M l a diem giiira SA Mat phang DMC cat hinh chop SABCD theo thiet dien gi ? Hay tinh dien tich cua thiet dien Giai 1/ Mot each dan gian chung ta se chuTng minh duac cdc mat ben cua hinh chop la bon tam jp^c vuong Doc gia tu giai Chang ban ta t i n h duac : SC = VSD^TCD^ = Va^ + (2a)2 = aVs • SB = V S A ^ T A B ^ = V(aV2)2 + a^ = aVs ^ SC^ = SB^ + BC^ = Sa^ BC = aV2 =i> ASBC vuong B (dpcm) 105 hi G oi O la tr u n g d iem cua SC De y th a y h a i ta m giac SDC vu ong va SBC vu ong theo thuf t u ta i D va B nen ta c6 : OS = OD = OB = OC = SC 2 Vay S, B, C, D na m tren mat cau ta m O, ban kinh: „ SC K = aVs = c/ Hai mat p h i n g (D MC) va (SAB) c6 diem M chung nen chung cfit theo mot giao tuyen M N (N n km tren SB) va vi AB // CD nen M N // CD (va M N // AB) => C D M N la h in h tha ng, la th iet dien cua h in h chop vdi mSt phAng D MC M a CD J CM => Th iet dien C D M N la h in h th a n g vu ong ta i D va ta i M (ycbt) Do M la tr u n g d iem SA nen ta c6 : D M = i SA = ; M N = - AB = 2 - Vay dien tich th iet dien C D M N bSng : S = CD + M N DM = Bai 187 (DAI HO C 2a + /r aV2 2 KH O I A - M I E N BAC - (ycbt) 1973) Cho h in h chop tuT giac deu S.AB CD d in h S, day la mot h i n h vu ong AB C D canh b ang a, cac ma t ben l a m vd i ma t pha ng day vd i mot goc (p Ta dUng ma t phSng pha n giac cua goc nhi dien canh BC tr on g h in h chop (tilc la goc n h i dien cua h in h chop xac d in h b a i ma t BCS va ma t B C AD ); ma t pha ng pha n giac cMt SD d M va SA d N Ti n h the tich cua h in h chop S B C MN theo a va ip Gi G oi I , L theo thuf tiT la tr u n g diem cua B C, AD Va h in h chop S.AB CD la h in h chop tiif giac deu => ASIL la ta m giac can ma goc a day S i t = SL)I = cp va S H la du dng cao cua hinh chop H = AC o B D M a t pha ng p h a n giac (a) cua n h i dien canh BC ch in h la ma t pha ng d i qua BC va difdng pha n Silt giac IK ciia va K la tr u n g diem cua M N Tif de dang th a y r a n g th iet dien B C M N la mot h in h th a n g can D ien tich h i n h th a n g B C M N duoc ti n h theo cong thufc : SBCMN = - ( B C + MN).IK IL Theo d in h ly h a m s in tr on g A L KI ta c6 : sin 106 Theo dinh l y h a m sin t r o n g A S K I , ta c6 : SK SI SK - ^^^^2 singlk singKl SI g.^3(p Nhifng : MN AD => SBCMN = = SK = SL SK ^^^^ SK M N = AD SI SI • ^ '"2 = a (0 sin— • sin— sincp —a + a 3(p Sep sin— sin- s i n — + sin— 2; sin (pcos— - a sm sincp • 3q) sin 2 Do (SKI) ( B C M N ) nen difcfng vuong goc SJ t i f S xuong I K cung c h i n h \k dUdng cao cua hinh chop S B C M N T r o n g t a m giac vuong S J I ta c6 : • asm— SJ = S l s i n ^ = ^ c o s (p (p (I) a s i n (pcos— a s m — V s BCMN= - S ( B C M N ) S J = o ^ ^

d t ( A B C D ) = aV2 2a^ _ 2^f2a^ ^/3 • V3 " V = a>/2 2y[2a^ V3 • 4a^ ^ , = — ^ (ycbt) 9V3 Trade h e t t a n h ^ n x e t r k n g cde m a t ben (SAB), (SBC), (SDC), (SAD) l a bon t a m giac bkng T h a t v a y , v d i SA = SC, SB = SD v a tCr gia t h i e t : A B = BC = C D = A D t h i t a m gi^c d6 c6 c^c canh t a o n g ilng bSng H a B H SA, ABSA la t a m giac can ( B A = BS) n e n B H cung l a t r u n g t u y e n cua t a m gidc AS BSA, turc l a H A = H S = T r o n g t a m giac v u o n g O S A t a eo : SA'-* = OS^ + OA'^ = AH = SA BH 4a' SA = 2a a - HA^ AB' = Ja a V3 a aV2 dt(SAB) = - SA.BH = H A B H = ^ ^ V3 V3 a^yf2 Cuoi cung t a ducte : Sip = d t ( A B C D ) + d t ( S A B ) = 4a^ y f2 2a^ 42 3 => S,p = 2a^V2 (ycbt) e/ N o i D v(Ji H => D H l a dadng cao ciia t a m giac can D S A Do vay i J H f i = ( ^ Ta CO = [(SABiTisAD)] : D H ^ = BH^* = 2a^ 108 4a' DII^ + B H ' = = BD'^ 3 => ABHD vuong tai dinh H => Goc nhi dien canh (SA) la nhi dien vuong (ycbt) Bai 189 (DAI HOC K I I O I B, N - 1975) Cho mot tarn giac vuong can ABC, AB = AC = a BB' va CC cung vuong goc vcJi (ABC), d cung mot phi'a doi vdi mat phang va BB' = CC = a a/ ChiTng minh rSng tarn giac AB'C la tarn giac deu hi Tinh the tich cua hinh chop c6 dinh la A va day la tuf giac BCC'B' d Chufng minh rang nam diem A, B, C, C , B' ciing nSm tren mot mat cau T i m the tich ciia hinh cau tifang ijfng Giai a/ Cac tam giac ziABC, ABAB', ACAC vuong can => BC = AB' = A C = a>/2 => B'C = A C = AB' = a => AAB'C la tam giac deu (dpcm) hi Ha AH BC Va thay BB' (ABC) => BB' ± A H Lai CO : A H _L BC A H (BCC'B) A H BB'J Dieu chufng to A H la difcfng cao ciia hinh chop A.BCC'B' aV2 Trong A vuong can ABC : => A H = Vay VAHCCIV = ^ A H dt (BCCB') = \ o (a.aV2) = ^ (ycbt) c/ Goi O la tam hinh chuT nhat BCC'B' Ta c6 : OB = OB' = OC = OC = aVs Trong tam giac vuong HOA => A H = (1) 1V2 Do OH la dtfdng trung binh cua tam giac BCC ^ OH = - C C = - ^ OA = V A H + O H ^ Tif (1) va (2) 1V3 = OA = OB = OB' = OC = OC = (2) iV3 Do O la tam hinh cau di qua nam diem A', B, B', C, C va the tich Vc ciia hinh cau bkng: V , = - : R = -n 3 S1 (ycbt) 109 Gi a i a/ The tich V cua h in h chop S AM B N la : V= i d t ( A M B N ) X SA (1) Trong : d t( A M B N ) = d t( AB N ) + d t( AB M ) d t( AM B N ) = 2d t( AB M ) (vi M , N doi xufng qua AB ) d t( AM B N ) = X - A B X M H => d t( AM B N ) = 2R X A M s in a => d t( AM B N ) = 2R X A B cosa.sina => d t( AM B N ) = 2R^ (2s ina cosa) = 2R^sin2a 2R'^ V = - X 2R^sin2u x R = s in2a (ycbt) 3 hi Xet : V(S B MN ) = - d t( B M N ) x SA = - x i M N x B H x SA => V(S AMN ) = - 3 dt( A M N ) x SA = - x - M N x A H x S A V(S B MN ) = 3V ( S AM N ) » B H = 3AH BH AB AH ~ BH + AH ~ 3+1 ~ ~ ~ « cos^ a = AH AH Ma A M = A B cosa = o AH = ^ 3R BH = R = — cos a AB „ n ^ n , cosa = — ( v i O < a < — ) = > a = — (ycbt) 2 Bai 273 ( D AI HOC B A C H KH O A - TO N G HCfP - K H O I A - 1977) Cho mot h in h chop tiif giac deu S.ABCD (S la dinh) Cat h i n h chop ay b a ng mot mSt phSng khong song song vdi ma t day M a t p h l n g cat cac canh ben SA, SB, SC, C D Ian liXcrt tai cac diem M , N , P, Q; M P va N Q cat ta i L D a t S M = a, S N = b, SP = c, SQ = d, SL = I, ^ = a (SH la difdng cao ciia h in h chop S.ABCD) 1/Tinh dien tich ta m giac S MP theo a, c, a 21 Chufng h he thijfc : — + — = a c / 3/ Chufng h he thiJc : — + — = — + — a c b d Gi 1/ Do hinh chop SAB C deu, nen day AB C D la mot h in h vu ong va dacfng cao S H la true cua dudng tron ngoai tiep h i n h vu ong AB C D Do H la ta m ciia h i n h vu ong AB C D Mat khac: S H = (SB D ) ^ (SAC) 185 ^> Q N r^ M P = L e S H dt(ASMP) = i SM.SP.sin KiSt' = — SM.SP.sin2a = — acsin2a = acsina.cosa (ycbt) 2 21 X e t h e t h ^ c : l + a c a + c o cos a = i = ^ ^ I a + c ac (1) 2cos a s in a = / (a ac 0) /si na a/sina + d s i n a = 2acsinacosa = acsin2a a/si na « a cs in 2a c /si na dt(AS ML) + dt(ASLP) = dt(ASMP) (2) : (luon dung) 1 2cos a , => — H — = (dpcm) a c / O G h i c h i i : Doc gid c6 the dung phuang phdp dien tich 3/ Ly lu an nhif cau 2/ vori cac ta m giac SQL, S LN va SQN, ta difcfc : 1 cos a b ^ d"^ (3) } Tif (2) va (3) suy : — + - = — + — (dpcm) a c b d Bai (DAI 274 HOC BACH KHOA - 1987) Cho mot h in h sau canh loi A B C D E F vdi cac d in h n a m tr en difdng tr on co dinh ta m O ban k in h R, ngoai A B = C D = E F; BC = D E = FA D at AB = a, BC = b, i C O B = 2a, ^OC = 2p 1/ Ti m h§ thufc lien he giCa a, b, a, p, tif suy ra ng : ta n a = aV3 ^ „ bV3 — ; tanp = a + 2b b + 2a 2/ Tin h dien tich S ciia h in h tr en theo R va a Cho b iet giA t r i Idn n h a t co the co cua S va y nghia h in h hoc Vdi gia t r i nao ciia a th i S = 3/ Tin h chu vi P cua h in h tr en theo R va a Cho b iet gia tr j \dn nha t co the co ciia P va y nghia h in h hoc 4/ Ti m he thufc giOfa a, b va dUa tr en he thufc t i m la i ket qua d cau 3/ Gi 1/ De y : 3(2a + 2p) = Ta CO — = : => a + P = 2:1 A F (1) Rsina • => b s ina = asinp - = Rs inp 186 n bsi na = a s i n do(l) a u V3 bsi na = a si n a cos a o {2b + a) si n a = a Vs c o sa (y ebt ) t an a = (dpcm) 2b + a (dpcm) Hoan t oan t uang tif t a t i n h difac : t anp = 2a + b f 2/ Ta CO ; S = 3(S,VVOB + SABOC) = - si n 2a + - v2 s i n 2(3 3R^ => S = ( s i n a + s i n 2(3) = R ^si n(a + (i)cos(a - p) S=3R ^COS 3V3 ^ 2a - (2) 3j Uau dang thufc t ro ng (2) xay r a c o s ^ a - ^ ^ 3j Vay : S„ «, = R ^ , t UOng ufng a = (? = - ( y c bt ) 2, = a = — Vay t r o n g c a c h i n h c a n h n o i t i e p t r o n g di /dng t r o n t r e n t h i h i n h c6 d i e n t i c h Idn n h a t l a hinh luc g i ac d e u - , o sR^Ve Tong quat xet : S = «> cos a 3j 2a = ^ ^ 3V3 „ ^2a-^^ R'^ c o s = c o s a 3j 2P = ^ sR^Ve 71 = COS — • a = p = — (y cbt ) a = — => 2p = — 12 12 3/ P = 3(a + b) = ( R si n a + R si nP ) = R ( si n a + si np) 71 P= R s i n ^ ^ c o s ^ ^ = 6Rcos a 2 ^ Dau dang thufc t ro ng (3) xay r a COS a Vay ?,„ ,, = R « a = p = - ;6R (3) 7:^ = ! < = > « = — b Vay t r o n g c a c h i n h l u c g i ^ c n g i t i e p t r o n g d Ud n g t r6 n t r e n t h i h i n h l u c g i d c d e u la h i n h c6 chu vi l(Jn n h a t ( y c bt ) 187 T a CO : a'^ + + ab = (a + bf - ab = (a + b)^ [{a + b)^ - (a -b)^] a^* + b^ + ab « -(a + hf = -ia + hf +- ( a - h f =3 R - i ( a - b ) = SR^* B A C A (1) M a CO ± m p ( B A O ) => CO B A Tir (1) va ( ) (2) B A m p (CAO) ( ) Ma BO ± mp(CAO) (4) Tii ( ) va (4) cho t h a y : qua B co h a i difdng t h i n g phan b i e t cung v u o n g goc vdi m p ( C A O ) t a i O (v6 l i ) Vay dieu gia suf A A B C vuong t a i A la sai (dpcm) V d i cac goc k h a c cung chiifng m i n h t u o n g t i i N g h i a la A A B C k h o n g p h a i la t a m giac vuong (dpcm) b/ VoAHC = - A O d t (ABOC) = - A O - BO.CO = ' A1 « » AO.BO.CO = V.M.BAO + V M.CAO + V M.Uf)€ - O A O B O C = - c.dtA(BAO) + - b.dtA(CAO) + - a.dtA(BOC) 3 188 » - O A.B O C O = i c AO B O + - b AO C O + - a.BO.CO 6 6 c = b + CO a BO + (5) AO Dg y r a n g : VQABC = ^ A O B O C O = ^ abc 6 ^ a b OA c OB OC Ap du ng B D T Cauchy cho ba so du ang va sOf du ng (5); ta c6 : a a a c b D c c OA ^ O B ^QC OB O C b OA => V o A B c ^ T ^ a b c D a OA b — = —abc.- — = —abc J _ (6) ^3 c 27 • +O B +O C Dau b at d i n g thuTc tr on g (6) xay r a k h i vsk chi k h i : f A = 3a AO (6) Do BO B = 3b C O C O = 3c V = — abc ; taong iifng : A = 3a O B = b (ycbt) O C = 3c d Theo B D T B u nhiacovsky ta c6 : (Va + Vb + Vc j < a b c (OA + O B + OC) = O A + O B + OC , A • + O B +O-C J Do : min(O A + O B + OC) = ( Va + Vb + Vc )^ xay r a k h i v^ chi k h i : rz OA ( A rr VOA \ B O V C O VoA VBO VCO -y/b Vc Va+Vb+Vc Va+Vb+Vc OB OC A O+BO + CO (V^ + V b + V c f A O = A/a(Va + -y/b + Vc) • B O = Vb (Va + Vb + Vc) (ycbt) C O = Vc(Va + Vb + Vc) Va+Vb+Vc Bai 276 (DA I HOC Y DLfOC TP.HCM - PB - 1996) Cho hinh chop S.ABCD day ABCD la hinh chCr nhat vdi A B = a, A D = b Canh AS = 2a cua hinh chop vuong goc vtJi day Goi M la diem tren canh AS, vdi A M = x; (0 < x < 2a) 1/ Mat phang (MBC) cat hinh ch6p theo thiet dien gi ? Tinh dien tich thiet dien Hy 21 Xac dinh x de mat phang (MBC) chia hinh chop hai phan vdi the tich b ing G iai 1/ Goi : N = (MBC) r> (SD) JM N = ( SA D ) n ( M BC ) MN / / A D/ / BC ^ [A D/ / EC ^ => MNCB la hinh thang Mat khac, ta c6 : A D (SAB) => A D ± BM r:> M N BM; BC BM; BM ? T = C B ^ = - Vay mat ph^ng (MBC) cat hinh ch6p theo thiet dien la hinh thang MNCB vuong tai M va B (ycbt) B M = V A M ^ + A B^ Ta CO : { SA AD SM M N M N = = Va^ + x^ 2a SA - SM A D (2a x)b f ( 2a- x) b + b 2a Va^ + x^ ( 4a - x)b MNCB (ycbt) 4a 2/ Trong mat phSng (SAB), ta ditog : SO BM Suy : SO (MNCB) tai O Ta CO : ASOM &o A BA M ^_ SO A B SM = BM AB SM SO SO BM a(2a - x) = , /a „2 + X Khi do, the" tich hinh ch6p S.MNBC la : V = A SO S S.MNCB V 3' • - Va^ + x ^ ( a - x)b ^^^^ 4a MNCB b(2a - x)(4a - x) S.MNCB 12 The tich hinh chop S.ABCD la: V„ = - SA S S.ABCD = ABCD , 190 Xet • V = b(2a - x)(4a V - x) o a ^b = 12 ; (0 < X ^ 2a) o ,2 ( < x < a ) ( a - x ) ( a - x ) = 43^^; o x ^ - a x + a ^ = ; (0 < x ^ 2a) X] = a - aVs (c6 : A' = 5a^ > ) : thoa < x < 2a X2 = a + aVs : k h o n g t h o a < x < a Vay vori : x = a ( - VS) t h i y c b t d u g c t h o a Bai 7 ( D A I H O C Q U O C G I A T P H C M - K I I O I A - 9 ) Cho h i n h c h o p S A B C D c6 d a y A B C D l a h i n h v u o n g c a n h b a n g a, c a n h S A ( A B C D ) v a c6 dp dai S A = a M o t m a t p h a n g d i q u a C D c a t cac c a n h S A , S B I a n l i /a t a M , N D a t A M = x l/TijT giac M N C D l a h i n h g i ? T i n h d i $ n t i c h tuf g i a c M N C D t h e o a, x 2/ Xac d i n h gia t r i c u a x d e t h e t i c h c u a h i n h c h o p S M N C D bang - I a n t h e t i c h h i n h ch6p S.ABCD Hvtdng dan Tirang t i f , k h i x e m D e D A I H O C Y D l /O C - P B - 9 Doc g i a CO t h e t h a y B bang D ; A D = a ; S A = a se d e d a n g c6 iMc : 1/ M N C D l a h i n h t h a n g v u o n g t a i M ; D ( y c b t ) SMNCD = ^ ( a - x)Va^ + x^ 21 X e t : VgMNCD - X = — a o (ycbt) g VgABCD (ycbt) Bai 278 ( D A I H O C Q U O C G I A T P H C M - K H O I A - 9 ) Tren cac c a n h ciia goc t a m d i e n v u o n g O x y z l a y cac d i e m : A e O x v a O A = a > ; B e O y v a OB = b > ; C e O z v a O C = c > K e O H m p ( A B C ) , a/ ChiJng m i n h A A B C c6 cac g c d e u n h o n v a H l a t r U c t a m A A B C b/ Chijrng m i n h : ( d t A A B C ) ^ = ( d t ABAO)'"^ + ( d t A C A O ) ^ + ( d t A B O C ) ^ d Gpi M ; N ; P t h e o t h a tif l a t r u n g d i e m A B ; B C ; C A C h u f n g m i n h b o n m a t c u a til d i e n POMN la cac t a m g i a c b k n g n h a u T i n h t h e t i c h cua n o t h e o a; b ; c il Cho A ; B ; C c h a y t r e n c a c c a n h c i i a g o c t a m d i e n n h i m g v S n t h o a m a n d i e u k i § n : a^ + b ^ + = ( k > c h o t r a d e ) K h i n a o t h i A A B C c6 d i e n t i c h I d n n h S ' t ? C h i J n g m i n h r a n g k h i d o thi dean O H c u n g d a i n h a t Giai a/ Tuong t i f d e D H N O N G L A M - K H O I A - 9 (dpcm) 191 b/ => dt(ABOC) = —ab => dt(ACAO) = —ab => dt(ABAO) = —ab (2) (dtACAO)^ = -a^c^ (1) (dtABAO)^ = - a ^ b ^ (dtAB0C)2 = Il bl ^ c ^ (3) Cong (1) + (2) + (3) theo ve ta c6 : (dtABAO)^ + (dtACAO)' + (dtABOC)^ = - ( a V + a^c^ + b V ) (4) Mat khac : dt(AABC) = - BC.AI => (dtAABC)^ = - B C l A I ^ = - (b^ + c^XAO^ + OI^) 4 Trong tam giac vuong BOC cho : 2„2 b=^C 1 O I ^ " OB^ ^ OC^ b^ Do : (dtAABC)^ = -(b^+ c^) a So sanh (4) va (5) c2 + 2„2 b^c b^+c=^ = - ( a V + a V + bV) (dpcm) c/ Tam giac vuong BOA cho : MO = ^ AB = PN Ttfang tu : N O = - BC = P M (dpcm) PO = - A C = M N Ta CO : AMNP oo AABC => dtAMNP = - dtAABC Tur dien OMNP va tuf dien OABC c6 cung ducrng cao OH nen : 'OMNP = - OH.dt A M N P = - O H - dtAABC = - V,OABC VQABC OABC = -g OA.dt ABOC = -a.~ BO.CO = -6 abc => V o M N P = d/ — abc (yebt) Theo BDT Bunhiacovky : a^b^ + b^c^ + c^a^ < a^ + b " + c" Mat khac : ( a^ + b^ + c )2 = a" + b ^ c" + 2{a^h^ + ah^ « k^= (*) + b^c^) a^b"+cn2(a2b2+b2c2+cV) k^ ^ 3( a^b^ + b^c^ + c^a^) = 12 (dtAABC)^ [theo (5)] dt AABC < 192 Dau ddng thufc tr on g (6) xay a = b = c dtAABC Idrn n h a t => max (dtAAB C) = (ycbt) V3 Cac tarn giac vu ong O A I va BOC cho : 1 O H^ => OV + 4r a' 1 OA^ = ^ OH^ „ \ + -4- > / + OC^ Ol2 (B D T Cauchy) „2 Ta cung c6 : a^ + b^ + -(a' ^OHU > > / a V c ^ + b' + c ') = ~ 9 => O H ^ - k (7) Dau dang thiifc tr on g (7) xay k h i a = b = c => k h i O H dai n h a t tu ong litng vdi dt AABC Idn n h a t (dpcm) Bai 279 (CAO D A N G K I N H TE D O I N G O A I CP - 1999) Cho h in h vuong AB C D canh bang a I la tru ng diem AB Qua I dOng du&ng vu ong goc vdi mat phSng (AB C D ) va tr en lay diem S cho 2IS = a V s 1/ ChuTng h r a n g tarn giac S AD la ta m giac vuong 2/Tinh the" tich h in h chop S.ACD roi suy r a khoa ng each tiT C den ma t p h l n g (SAD ) Gi 1/ Xet: S I ( AB C D ) ^ S I A D AB AD (SAB ) ^ AD SA AD => A S A D vu ong ta i A (dpcm) 2/Ta CO : = \I = ^ ^ a ^ ^ = ^ (ycbt) Goi h la khoa ng each liT C den ma t phSng (SAD ) Kh i : (1) = - SA.AD Ma : Ss Ai) Vdi : SA = J — + QA Sa^ 1^^ T = a o AD = a Khid6:^ h = ^ ^ = ^ ( y e b t ) 193 B a i 280 (DAI HOC QUOC GIA TP.HCM - KHOI A - D dT - 1999) Cho hinh chop ta m giac S.AIBC c6 day AB C la ta m giac deu canh a, SA -L (ABC) va SA = a M la mot diem tha y doi tr en canh AB D a t = a, S H vu ong goc vdi dadng t h i n g C M 1/ Ti m quy tich d iem H Suy gia t r i Idn n h a t ciia the tich tuf d i^ n S AHC 2/ Ha A I i SC, A K S H Ti n h dai SK, A K va the tich tuf dien S AKI Gi 1/ Goi N la chan difcrng cao ke tif S tr on g ta m giac SBC, ta c6 : B C _L (S AN ) => B C A N Ta CO : SH i (1) M C => M C (SAH) => A H HC (1), (2) ==> H lu on n h in AC dudri mot goc vuong N h a n th a y : M = A ^ M HE B ^ H A H = N Doc gia tiT l a m pha n dao, t h i ta c6 : Quy tich d iem H la cu ng A N thuQc ducfng tr on dacrng k i n h AC (ycbt) The tich tuT dien S AH C la : 'SAHC = - S A A C H H ' = - a ^ H H ' 6 (vdi i r la chan dUcrng cao tir H tr on g AAHC) Suy : m ax CV g ^Hc) m axCHH') H la tr u n g d iem A C Kh i d o : (VsAHc)n 2/ Ta c6 : • s ina = (ycbt) 12 AH AC A H = a s in a SsAH = - S A A H = S H A K • S H - V SA ^ + A H ^ = aVl + s in^ a AK = SA.AH a sma a.sm a a V l + s in^ a S K = VSA^ - A K ^ (ycbt) V l + s in^ a = (ycbt) s in " a Tr on g A S H C , dUng IJ S H ASAC c ant A Ma IJ // C H SI = I C AI I SC I J = - C H = ia.cosa 2 M a t khac : C H ( S A H ) => I J ( SA K ) 194 Vay the tich hinh chop SAKI 1^ : a.(sin a)a V OA = OB = OC => DO la true ducrng tron ngoai tiep AABC DO hien nhien la dircfng cao ti3 dien DABC Goi I la trung diem DC Ta c6 : DH = HC => ADHC can => H I l DC Khi : H I = VHC^ - IC^ = ^ Tif; Si„ic = - DO.HC = - HI.DC 2 D0 = V- XT HI.DC HC ^nnc 3a^ DO = Vay : VABCD = - D O S A B C a^-x^ = a.Vsa^ - x^ 4a^ - x^ y i.^ ^ a T^ 4a^ - x^ 195 VABCD = -^axVsa^ - (ycbt); (0 < x < aVs ) A p dung B D T Cauchy, t a c6 : ^ABCD - 144 x^+Oa^-x^) l2 144 144 9a^ a^ a^ : max(VABci)) = — (3) (4) Dau d i n g thijfc t r o n g (3) va (4) x a y r a Vay x^ = 3a^ - x^ x = aVe tifcfng uTng x = — — - B a i 282 ( D A I H O C H U E - aVe (ycbt) 1999) Cho tuf d i e n A B C D c6 ba c a n h A B , A C , A D vuong goc v d i n h a u t i l n g doi m o t va A B = a, A C = 2a, A D = 3a H a y t i n h d i e n ti'ch t a m giac B C D theo a Htfdtng d a n G o i h l a dp d a i dirdng cao c u a t i l d i e n v u o n g of A v a k e tiJ A t h i d a c h i l n g m i n h difoc : 1 AB^ AC^ + The t i c h h i n h tuT d i e n : 4a^ 9a^ , => h = — a V = - A B A C A D = a^ Do d6: d i e n t i c h A B C D \k : S = B a i 283 h 7a^ 3V (ycbt) ( D A I H O C Q U O C G I A T P H C M - K H O I A - D O T - 2000) Cho h i n h chop t i l giac S.ABCD c6 day la h i n h t h a n g A B C D vuong t a i A va D; A B = A D = a; CD = 2a Canh ben SD vuong goc vdti mat phSng ( A B C D ) ; SD = a 1/ C h i l n g m i n h rkng t a m giac SBC v u o n g T i n h d i e n t i c h t a m giac SBC 2/ T i n h k h o a n g each tii d i e m A den m a t p h i n g (SBC) Giai 1/ Goi E l a t r u n g d i e m cua C D Do t i n h chat h i n h t h a n g vuong A B C D , suy r a : BD DC = a = B C => E B D C , E B = - ( t r u n g t u y e n b k n g nijfa canh huyen) => A D B C v u o n g t a i B Theo d i n h l y ba dUcfng vuong goc => S B J- B C (vi B C S D , B C ± B D ) => A S B C vuong t a i B (dpcm) Luc d i e n t i c h S ciia A S B C l a : S = - S B B C = - V S D ^ + DB^.(EB.V2 ) =-^la^ + (aSf (aV2) = ^ a ^ (ycbt) 196 2/ Ha A H ( S B C ) t a i H => h = A H = d[A; (SBC)] a^h Ve The t i c h V cua h i n h chop A S B C l a : V = - S h = —a'^.h = 3 (1) Mat khac t h e t i c h V cung l a t h e t i c h h i n h chop S A B C V = — S A B C - S D = — (SABCD - o o Sa' a Si)Ac)a a' a' a = — — a = — So sanh (1) va (2); t a c6 : a' a^h Vay : d[A; (SBC)l = ^ (2) h = (ycbt) Bai 284 (FDAI H O C S U P H A M H A N O ! - K H O I A - 2000) Trong k h o n g gian cho cac diem A ; B; C theo thijf t u thuoc cac t i a Ox; Oy; Oz vuong goc v d i tirng doi m o t , cho OA = a (a > 0); OB = a V2 ; OC = c ( c > 0) Goi D l a d i n h doi dien vdi cua h i n h chCf n h a t A O B D va M la t r u n g difim cua doan BC (P) l a m a t p h i n g d i qua A M va cat m a t p h ^ n g (OCD) theo m o t dUdng t h a n g vuong goc v d i difctng t h S n g A M 1/ Goi E la giao diem ciia (P) vdi ducmg t h ^ n g OC T i n h dai doan t h a n g OE 21 Tinh t i so thfe t i c h cua h a i k h o i da d i e n duac tao t h a n h k h i cat k h o i h i n h chop C.AOBD b d i mat phang (P) 3/ Tinh k h o a n g each tii d i e m C den m a t p h a n g (P) Giai 1/ Goi N la t r u n g d i e m O B ; P la t a m h i n h chijf nhat A B C D Xet he toa dp : • O A ; AN f a^ = AD = (ADO) ] Ta c6: D O = (-aV2; a) AN.DO = (aS {-ay/2) + a.a = AN DO tai R (1) Ma : M N D O (vi M N / / C O ( O B D A ) OD) (2) T i r ( l ) va (2) cho : D O J_ A M Lai t h a y : EQ A M v d i : EQ = (P) n (OCD) ^ EQ // OD 197 Tinh chat dudng trung binh cho : MP = — AC Goi : F = (P) n CP ta c6 : AFCA t/J AFPM FC FP MP FC m a ^ = ^ OE OE = - O C Vay:OE=|(ycbt) II Dat : Q = (P) ^^ CD Thi§'t di$n ma (P) cat hinh ch6p C.AOBD se chia hinh ch6p C.AOBD thsknh hai kho'i da dien Dey : C A C O C D Vc.AOD CA VcAEQ CB Vc.BOD CM Vc.MEQ _ C E CQ CE •CO CQ •CD 2 "as 2 •3 •3 Vi : Vc.OAD = Vc.BOD = 'C.AEQ => Vc.AEQ + V,C M E Q — Vc.OABD 9^9 + VcMEQ = -r A • ~ - Vc.OABD (goc) Vc.OABD o Vay ty so the tich cua hai khoi da dien mat phing ( P ) c^t hinh ch6p C.OABD tao la : - hoac (ycbt) O G h i chii : Bgc gid c6 the xem each gidi khdc d TUYEN TAP 500 BAI TOAN HINH HOC GIAI TICH eua eung nhom tdc gid 3/ Ha CK ± AM tai K => CK = d[C; (P)] D6c gid tif chon phuong dn t i n h CK => (ycbt) B a i 285 ( H O C V I E N N G A N H A N G - P H A N V I E N T P HCM - Khoi B - 2000) Cho hinh chop til giac deu S.ABCD c6 day la hinh vuong ABCD canh a, dudng cao SO = h 1/ Tinh theo a va h ban kinh mat cau ngoai tiep hinh ch6p 2/ Tinh theo a v a h dien tich toan phan cua hinh chop, tif d6 tinh ban kinh mat cau n6i tifi'p hinh chop Giai 1/ Hinh ch6p S.ABCD deu SA = SB = SC = SD S O la true dudng t r o n A B C D ^ |SOI(ABCD) Ttf trung diem M cua canh SB dong mat phIng trung trUc (a) cua canh SB (a) n SO = I Luc I la tam mat cau ngoai tiep hinh chop S.ABCD va c6 ban kinh R = SI ASOB SI _ SM SB " SI = SM.SB SO SO 198 SI = SI = SB^ f , SM = VI: 2.S0 SB^ J SO^+OB^ 2h aV2 2h Vay : R = 4h (ycbt) 4h 2/ Gpi J la t r u n g d i e m canh A D t h i t r u n g doan SJ la : -7 A SJ = p h a n Stp l a cu => S,,p = a ' + 4, — a - V2 I r : S,p = + 4( AD.SJ) = a|a + V a ^ i - h " j (ycbt) So sanh h a i the t i c h V ciia h i n h chop : V = -Bh ah r- — - V = A rS • ^^"^ V a " + h - ) (ycbt) a + Va^ + 4h^ DE Tl/dNG T I / Bai 286 ( D A I H O C Q U O C G I A T P H C M - K H O I A , B - 1997) Day h i n h chop S A B C D l a h i n h vuong, canh a, SA ( A B C D ) va SA = a M p qua C D cSt SA, SB t a i M , N D a t M A = x a/ T i m dien t i c h t h i e t d i e n b/ T i m X de the t i c h h i n h chop S M N C D b a n g | the t i c h h i n h chop S A B C D Htfofng d a n : a/ S = - ( M N + C D ) M D = - (2a - x) V x ^ T a ^ 2 Bai 287 ( D A I H O C D O N G D O H A N O I - K H O I A - hi x = - a 1998) Canh day h i n h chop t a m giac deu bSng a, goc tao b d i mSt ben Va day b a n g 60° a/ T i m the t i c h va S,p h i n h chop b/ T i m t i so the t i c h h a i p h a n cua h i n h chop m a t p h i n g p h a n giac ciia goc n h i dien 60° d t hinh chop Hrfdrng d i n : a/ V = 24 b/ ^ =i V, 199 ... B = R x = > A M = V2Rx B M ^ = B H X A B = 2R(2R - x) B M = V2R(2R - x) M H ^ = A H = x(2R - X B H x) M H = Vx(2R - x) M S = M H = Vx(2R - x) SA^ = MS^ + M A ^ = x(2R - x) + 2Rx = x(4R - => S... theo (1), thi : l2 + tan2- SH= HI t a n - (2) maHC = HI V2 Lay (3) SH (2) (3) tancp = HC V2 t a n — + J ;2 \ + tan2-^ (4) Vdi gia thiet cosa = ^f2 - 1, da c6 duoc : tan^— = V2 - (3) tancp = V... - O j + JC^ = R2 = + (c - 10 )2 = +OF + (IJ- (1) Olf + c % ^ - 2C.0I (2) + O r = b^ + c^ + O r - 2c OI TCr (1) va (2) , suy : 114 01 = 2c Thay 01 = ^ vao (1) 2c = a^ + b^+c^-a^^'' 2c Vay : R = —