Tuyển tập 500 bài toán hình không gian chọn lọc phần 4 nguyễn đức đồng

V A, B & cung ben so v& i (D) D img A' d o i x i J ng vdi A q ua (D) Luc : A' v a B d kha c b e n so v6i M A + M B > AB (D), n e n t r d v e t rUc f ng hop tren : MA' + M A ^ AB min(MA + M B ) = min(MA' + M B ) = A B ttfong Ofng : M a M Q = ( A ' B ) o O Ket • Xa c (D) l u a n : v a y t r o n g m o i t r i f d n g h g p t a x d c d i n h di f ac M t h o a y c b t d i n h d i e m m t r e n d Uo T n g t h S n g ( d ) de | MA - MB | T i f a n g tiT c a n p h a n b i e t h a i t rUc f ng h a p : V A, B & cung ben so v& i |MA - MB | < (D) AB M ma x| M A ~ M B | = A B Mo (d) tu a ng ling M = M Q = (AB ) r^ (D) V A,B d khdc ben so v& i (D) | M A - M B | < |MA' - M B | ^ AB yiy^ Vdi A' la h i n h doi xijfng ciia diem A qua (d), t h i A' va B I a cu ng p h ia (D) ma x I M A - M B | = ma x J M A ' - M B | = A B \ (D ) /Mo A' ta ong tjfng M s M Q = (A'B ) n (D) O Ket lu a n : vay tr on g moi tri/dng hap ta da xac d in h diem M thoa ycb t n Gl A l T OAN T HI Bai 393 ( D A I H O C K H O I A M I E N B A C - 1972) Cho mot kh oi t i l dien AB C D a/ Mot ma t phang song song vdri canh B C cat cac canh AB , AC, D C, D B d c^c diem M , N , P, Q Chufng m i n h rS ng tuT giac M N P Q la mot h i n h th a n g pha i thoa ma n dieu ki§n nao de tuT gidc la mot h i n h b in h h a n h ? la mot h in h c h a n h a t ? b/ Cho b iet cac goc B AC , C AD , D A B la vu ong, B C D la mot ta m g iic deu canh a Tin h th^ tich cua kh oi tijf dien theo a c/ Cho b iet B C D la mot ta m giac deu canh a va c6 ta m la diem O Ti n h doan OA theo a cho m a t cau ngoai tiep ti l dien AB C D n h a n dudng tr on (B CD ) l a m mot ducfng tr on Idn Tinh dien tich m a t cau tr on g trU dng hap ay Xac d in h vi t r i ciia d in h A tr en m a t cau ay de the tich h in h t i J dien A B C D I6n n h a t G ia i aJ Ta c6 : mp(P) // B C = (AB C) n (B CD ) => M N // PQ Vay th iet d i6n M N P Q la mot h in h th a n g (ycbt) 300 Muon cho M N P Q la h i n h b i n h h a n h ; tifang t u tren ta phai c6 t h e m dieu k i e n N P // M Q , (P) // A D Vay dieu k i e n de M N P Q la h i n h b i n h h a n h la mat phang (P) p h a i song song d o n g thcfi v d i : BC AD (ycbt) Hcfn nOa de M N P Q la h i n h chfl n h a t t h i ta p h a i c6 MN NP Vi BC // M N va A D // N P =i> BC AD Vay dieu k i e n de M N P Q la h i n h chff n h a t la BC A D (ycbt) b/ Tuf dien A B C D la tuT d i e n v u o n g d A fBC = CD = DB = a AB = A C A D = ^ = ^ Vay the t i c h k h o i tuT dien A B C D la : V= - A B - A C A D = i ( A B ) ^=i 24 (ycbt) c) De y dUcfng t r o n ( B C D ) la m o t dudng t r o n I d n cua m a t cau ngoai t i e p tiir d i e n A B C D va c6 la tam ciia t a m giac B C D canh a, nen t a m O cua t a m giac deu B C D cung c h i n h la t a m cua mat eau ngoai t i e p tuf d i e n A B C D ^0 A = O B = ^ Tir dien t i c h Sc ciia m a t cau ngoai t i e p tuf d i ? n A B C D l a : Se = 471 OA^ = 471 aV3 = — 7ca Goi A H la dUcfng cao ciia tuf d i e n A B C D t i f d i n h A suong mat day (BCD) AHOC ( f i = " ) =i> A H < O A Va t i n h duoc the t i c h cua k h o i tuf d i e n A B C D b a n g : V = ^ S , „c D A H = V3.a2 12 AH < I aV3 2 Vs.a 12 a AH OA (1) Dau dang thufc t r o n g (1) xay l i = A ( h i n h ch6p A B C D deu) 3maxV= 12 OA (ycbt) 301 Bai 394 (D A I H O C K H O I A - M I E N B A C - 1974) Trong mat p hlng (P), cho hinh vuong ABCD c6 canh b ing a Tren dUdng thi ng Ax di qua A va vuong goc vdi mftt p hing P, ngucri ta lay mpt diem S tiiy y, ro i difng mat p hing Q qua A c i t SB, SC, SD Ian lirat tai B', C, D' Biet (Q) i SC a/ ChuTng minh rSng SB vuong g6c A B' va SD vuong goc vdi A D' b/ Tim c^c quy tich ciia B', C, D' S chay tren Ax c/ Xac dinh v i tri cua S tren Ax cho hinh ch6p C'ABCD c6 the tich \dn nha't va tinh thi tich ay Gi&i a/ Ta c6 : CB A B C B± S A CB ± (SAB) A B' => CB A B' Mat khac ta c6 : SC _L A B' (vi A B' n^m mp(Q) ma SC Q) Do d6 A B' (SBC) SB => A B' ± SB (dpcm) ChiJng minh hoan to^n tUcfng tif ta c6 A D' SD (dpcm) b/ De y : B' e (SAB) A B ^ = 90° ; A, Bco dinh => Vay quy tich ciia nhiJng diem B' la dudng tro n (trong mat p hlng (SAB) dtfdng kinh A B (Doc gia tif lam phan dao) • Taang ti i ta c6 quy tich ciia nhufng diem D' mat p hing (SAD) difdng kinh A D (ycbt) • Tucfng tu ta cung c6 quy tich cua nhCifng diem C la dudng tro n tro ng mat p hing (SAC) dadng kinh AC (ycbt) 0/ Ha CO AC va thay OC // SA; SA P n6n : V = VC'ABCI) = — SA BCD - O C ' Vi hinh chU nhat A BCD la co d inh, nen the tich V se I6n nhat OC la Idrn nhat C luon luon nam tren diidng tro n dtfdng kinh AC Vi vay OC se Idrn nhat no la ban kinh (NhK vay CO hai v i tri Co' va Co" doi xufng vdi qua AC cung thoa man tinh chat d6 : C'oC'o ± AC tai O; C'oC'o c (SAC) Khi OC = OA = OC = hay tam giac vuong SAC co OC la difdng trung binh, taang ling do : AS = 20C = a-s^ Vay S (nam tren A x) each A mOt doan a ^/2 (co hai v i tri SQ va SQ' doi xufng vdi qua A) thi hinh chop C'ABCD c6 the tich Idn nhat, va the tich d6 la : 1V V = i a ^ ^ 3 a"-^V2 (ycbt) 302 Bai 395 ( D A I H O C Y - N H A - D L f d C - 1976) Cho h i n h vuong A B C D canh a Goi SA la doan thSng goc vdri mSt p h ^ n g ( A B C D ) vk M la m o t d i e m d i dong t r e n doan SD DSt S M = x SA = a a/ Mat phang ( A B M ) cat doan SC t a i N Chitog m i n h tiif gidc M A B N l a mot h i n h t h a n g vuong b/ Dat y = A M ^ T i n h y theo a va x d Khao s a t sif b i e n t h i e n va ve dudng bieu d i e n cua y = A M ^ k h i M ve t r e n doan SD Gi&i a/ Ta CO : A B // C D => A B // (SCD); A B c ( A B M N ) ^ Lai ( A B M N ) n (SDC) = M N // A B // C D CO : A B (SAD) AB AD AB I S A => A B A M => M N (SAD) => M N I A M Vay A M N B l a m o t h i n h t h a n g v u o n g h a i ddy l a A B va MN (ycbt) hi Goi H l a h i n h chieu cua M xuong canh CD ADMH oo ADAS MH MD SA DS aV2 - X M H = aV2 AAHM ^ AM^ = A H ' + HM^ aV2-X HMD vuong can => H D = H M = AH = AD - HD = a - a - n ^- A » , (aV2-x)2 Do do: A M ' = — + 2 ^ x^ + + - 2aV2.x = A M ^ = x ' - a V x + a^ Vay y = x ' - a V x + a ^ Vx e [0; a V ] (ycbt) d Mien xac d i n h cua y : Df = [0; a ^/2 ] ^ y' = x - a V ( aV2 = Bang bien t h i e n : Do t h i : tV2 X a^f2 y y' a' a aV2 aV2 x Dudng bieu d i e n la cung Parabola A S B 303 Ba i 396 ( D A I H O C B A C H K H O A - T O N G H O P - 1980) Trong khong gian cho ba tia Ox, Oy, Oz tifng doi mot tao wdi mot goc a (0 < a < 90°) tren Ox, Oy, Oz lay Ian lucft cac diem A, B, C cho : OA = a, OB = b, OC = c 1/ a, b, c phai thoa man h? thilc gi de tam giac ABC c6 goc A vuong ? Hay tim dieu ki§n can va du rang buoc b, c, a de tim dUcfc a thoa man he thufc ay 21 Gia sijf a co dinh (0 < a < 90°) v^ b = c co dinh Xac dinh a de tam giac ABC c6 g6c A Wn nhat Gia tr i Idn nhat ay cua goc A bang bao nhieu 3/ Vdi cac gia thiet cua Hay tinh the tich cua tiif dien OABC ufng vdi gia tr i \6n nhat cua goc A Gi a i 1) AABC vuong ta i A » BC^ = AB^ + AC^ (1) Dinh ly ham cosin cac tam giac : AOAB, AOBC, AOAC cho : (2) BC^ = b^ + c^ - 2bccosa (1) AB'^ = a^ + b^ - 2abcosa AC^ = + c^ - 2accosa (3) Thay (2), (3) va (4) vao (1) (1) b^ + c^ - 2bccosa = 2a^ + b^ + c^ - 2a(b + c)cosa a^ - a(b + c)cosa + bccosa = g(a) = a^ - [(b + c)cosa]a + bccosa = (5) De tim dtfcfc a thoa man (5) A = (b + c)^cos^a - 4bccosa > A = cosaKb + c)^cosu - 4bc] ^ (0 < a < 90° => cosa > 0) (b + c)^cosa - 4bc ^ (6) (ycbt) (6) dieu kien can va du rang buoc b, c va a de tim difac a thoa man (5) 21 Xet gia thiet : b = c Goi HB OA « AOAB = AOAC CH OA ^ B H = CH Xet hai tam giac can ABC va HBC; chung co canh chung BC JAB ^ H B ^ gAC < B Ht; => maxBAC = B Ht; tircJngilngA^H J A O H C AOBH jO H = bcosa ' H B = HC = bsina BC'^ = OB'' + OC' - 20B.OCcosa AOBC BC (SAB) BC MB Thanh thijf thiet dien MBCN \k mgt hinh thang vuong (6 = 1^ = ° ) B 306 ASMN MN _ SM ASAD M N = AD AD ~ SA SM = b SA 2a-X = 2a b "[^ 2a, Vi vay h i n h t h a n g vuong M B C N c6 dien t i c h : b + b X 2a = b ^[^77 a^ (ycbt) 4a j b/ Vi S > 0, S dat gia t r i lorn n h a t k h i Taco: = dat gia t r i Idm n h a t r(4a-x)2(x^+a^) = - f i x ) ; Vx e [0; 2a] 16a' => fix) ^ = 2(x - 4a)(x'^ + a^) + 2x(4a - x)^ f (x) = 2(x - 4a)(2x^ - 4ax pv, , n Cho f (x) = +a') X = 4a a ( ±V ) V x = Lap difgtcbang bien t h i e n cua h a m so fix) t r e n doan [0; 2a] a(2-V2) a(2 + V2) 9 f(x) 16a' fix) a ' ( + 8V2) a'(71-8V2) DiTa vac b a n g b i e n t h i e n n^y t a t h a y : Vay amaxS^ o khi: X = AM = 3maxS = — ^^^iJ^ ^ + 8^2 k h i va chi (yebt) d Hien n h i e n h i n h chop S.ABCD c6 the t i c h : V= iAS.AB.AD= — De t i n h the t i c h V cua h i n h iSng t r u t a m giac cut M A B N C D , t a difng m a t phSng ((3) qua N vuong goc vdi BC; t h i m a t p h i n g (P) cat A D va BC I a n lifat t a i K va L , (P) chia l a n g t r u I cut h a i p h a n : l a n g t r u t a m gidc dtifng M A B N K L c6 the t i c h : V , = M N d t ( M A B ) = - abx 2a j va h i n h chop d i n h N , day K L C D , c6 the t i c h : Va = - N K d t ( K L C D ) = -bx^ 307 t h a n h t h i l ISng t r u cut M A B N C D c6 t h e t i c h : V = V i + Va = - a b x 2a J Yeu cau b a i t o a n can xac d i n h x ( h i e n n h i e n < x < 2a) cho V = 2V' 2a^b = — abx 3 2a ) X Phiicfng t r i n h c6 n g h i e m : - 6ax + 4a^ = = a(3 + V S ) x = a(3-V5) V i ^ X < 2a n e n c h i c6 t h e c h o n : x = a ( + V S ) ( y c b t ) B a i 399 ( D A I H O C K T - T H - SP - N N - 1983) Cho tuf d i e n S A B C , day A B C la tarn giac vuong t a i A , A B = 2a, A C = 3a, canh SB vuong g6c v d i day SB = a/ C h i ro tarn v a b a n k i n h m a t cau ngoai t i e p tijf d i e n SABC b/ M l a m o t d i e m d i d o n g t r e n canh SC, dat M C = x Goi H va K I a n liftft l a cdc h i n h chieu vuong goc ciia M l e n cac m a t phSng (ABC) va (SAB) Mat p h l n g K M H , cat A B t a i L ChiJng m i n h r k n g : K M H L l a m o t h i n h chi? n h a t V d i gia t r i nao cua x t h i K M H L la mot h i n h vuong c/ T i n h theo a va x d a i dudng cheo M L ciia h i n h c h a n h a t K M H L V d i gia t r i nao cua x t h i M L C O d a i nho n h a t ? l n g v d i gia t r i da t i m difdc cua x, hay neu l e n dac t i n h h i n h hoc cua dean M L d/ H a y t i n h theo a va x t h e t i c h V cua h i n h chop d i n h A , day PCMHL K h a o sat sif bien thien va ve t h i ciia h a m V k h i M d i dong t r e n canh SC e/ Xac d i n h x cho : V = 4V3 27 Giai a/ Goi O va I I a n lifat la t r u n g d i e m ciia SC va BC Cac t a m gidc SAC, SBC theo thii ti( vuong t a i A , B nen t a c6 : l a t r u e dUcJng t r o n ( A B C ) |ma Ola trung diem S C OA = OB = OS = OC Vay O l a t a m cua m a t cau ngoai t i e p tuf d i e n SABC va ban k i n h m a t cau l a : R = OA = — = 2a That vay : SC^ = S B ^ + B C ^ = S B ^ + A B ^ + A C ^ ^ S C ' = 33^ + a ^ + Qa^ = I G a ^ => SC = 4a => R = 2a (ycbt) 308 b/ KMHL la hinh chuf nhat MK ± (SA B) A C (SA B) ' => M K // AC ^ MK // HL // AC M H // SB M H // KL // SB M H (A BC)' SB (A BC) J =c Vay tuT giac KM H L la mot hinh binh hanh De y den SB AC => H L LK : nhvt vay tit giac KM H L la mot hinh chff nhat (dpcm) Dinh X de KM H L la mot hinh vuong Ta c6 : M H _ CM SB ~ MH = SC MK SM AC SC MK = SB.MC _ aVs.x _ V x SC ~ 4a " A C SM _ a ( a - x) _ ( a - x) SC ~ 4a ~ Vay : KM H L la mot hinh vuong « d Ta V3 x ^ ( a ^ 4 CO < ^ V x = 3( 4a- x ) ^ : ML^ = M H ^ + HL^ = 3x2 9(4g _ 16 16 x = a V ( V - 1) (ycbt) V3( x - a x + 12 a ) V3(x - 3af ML = : = -'^ M L> 9a' 3a 2 ; Vx G 3(x^ - a x + 12 a ) + 9a^ ^ ;Vx [0; 4a] ^^ [0; 4a] ( 1) Dau dang thilc tro ng (1) xay x = 3a 3a => minML = — , xay va chi x = 3a Ta CO : AB (MKLH) minML = 3a A B± M L = d[(AB); SC] Vay M L nho nhat thi doan M L la doan vuong goc chung cua hai difdng th i n g A B va SC (ycbt) d/ The tich V hinh chop A MHKL V = i d t(MHKL).A L = - M K.M H.A L 3 AB Ta CO (2) BS : AB ML o BS, M L, AC ciing vuong goc \6i A B AB AC BS, ML, AC cung nkm mat p hing song song Ap dung dinh ly Thales AL CM AB CS — 4a AT AL = — 309 c a t h i n h Idng tr u theo mSt p h i n g T i m giao tu yen cua ( M N P) Gidi (MN P) va ( AA, D D , ) TiT xac d in h diem Q G ia sijf H la giao diem cua PQ vdi ma t pha ng ( A B i C i O SD , // B i N PB, SP PN AS M N CO PQ // M N , SP ^ = => SN ~ MN SN PQ SP PQ = i M N = V a yPQ = | B a i 16 Cho h i n h la n g tr u tarn giac AB C AjB iC , Goi M la diem tr en dUcfng cheo A B j ciia ma t A B B iA j cho ~ ' MB ^ AM D ifng th iet dien ciia la n g tr u vdi ma t phftng u qua M va song song vdi cdc diTdng cheo AiC, B C i Xac d in h t i so ma ma t pha ng a chia canh CC, Gidi Goi P la ma t pha ng qua A^C va song song vdi B C] (5 va (B B ,C iC ) la difdng th a n g qua C va // BC, dUcmg cat B B , ta i Sj A, S i o A D = S2 u // p M a t pha ng ( AB B , A, ) cat ma t pha ng theo giao tu yen S3S1 P qua M va // vdri AB^ M a t pha ng (AB C) cat P theo giao tu yen CS a II B C] nen cat ma t pha ng (AiB ,C ,C ) theo giao tu yen Sr,Se song song B C, (AB C) // ( A, B iC , ) nen a cat ( A i B i d ) theo giao tu yen S3S7 song song S , S Vay th iet dien la ngij giac SyS^SsSgSv r^- Ti ' CS, S,.C, SO B a i 17 Cho h i n h hop A B C D A iB iC iD , Goi M , N , P Ia n lifcrt la tr u n g diem cua AD , B B , va ta m ciia A ] B , C iD i D ifng th iet dien ciia la n g tr y b i cat b di ma t pha ng a qua M , N , P Xac d in h t i so ma t pha ng a chia canh AB Gidi Ti m giao diem N P vdi ma t p h l n g ( AA, D D , ) du dng th a n g n a m tr on g ma t phIng ( B B , D , D ) cat ma t pha ng ( AA, D , D ) ta i S, (S, = N P ^ D D ,) TiJong tU t i m giao diem S^ ciia N P vdi ma t pha ng (AB C D ) ( S2 = N P o BD) a cat (AB C D ) theo di/dng S2N Ta t i m d in h cua th iet dien S3 = S iN ^ D ,A,; S, = S2N " AB 402 Ss = S P r B , C i ; Sf, = BC a Ngu giac M S S N S t h i e t dien p h a i diTng (MS;, // N S , S S // MS,,) Tarn giac BS,;N = t a m giac BiSgN: cho BSg = BjSg B la t a m doi xiJng ciia h i n h b i n h h a n h A i B j C i D i => BiSs = D,S i TCr t a c6 BS„ = D,S,.i Tit gia t h i e t A M = D M t a c6 A S i / B S , = D N A ) , S , T Tam giac D S , N co t a m giac D,S,S:, cho t a DM/DjSa = D S , / D , S , Tam-giac S , D , P = t a m giac N B j P cho D , S i = B N B i N = 1/2 B , N = 1/2 D , D D,S = 1/2 D , D DS, = 3/2 D D , I)M/D,S;i = D S , / D , S i = 3/1 A S /BS., = D N / D , S = 3/1 Vay canh A B dtfqc chia theo t i so : t i n h tiT d i e m A Bai 18 Cho h i n h l a n g t r u luc giac dfiu, dirdng cheo \dn n h a t c6 dp d a i d va vdri m a t ben d i qua mot dau di/dng cheo ay goc u T i n h dien l i c h xung quanh cua h i n h ISng t r u M a t ph^ng (P) d i qua canh I a n lucft nkm t r o n g day song song v6i n h a u va k h o n g njm t r o n g ciing m o t m a t ben a) Xac d i n h m&t cit tao b o i (P) va l a n g t r u b) T i n h goc n h i d i e n tao b d i (P) va mSt day theo d k h i a = 30" c) T i n h d i e n t i c h t h i e t dien Gidi Dien t i c h x u n g quanh: N h a n x e t rSng a d i e m cua h i n h l a n g t r u , chftng b a n B c6 difdng cheo l a B D i , B E , va B E ] cac du&ng cheo c6 h i n h chieu d t r e n day A i B i C , D i E , E i l a B j D i , B , E , , B , E i t r o n g B i E , la lorn n h a t nen B E j l a dadng cheo \dn n h a t Ta CO B E , = d v i B j E , l a difcfng k i n h cua dUcfng t r o n ngoai t i e p day A i B j C i D i E i F i n e n A,E, A,B, Suy r a A , E , ( A B B , A , ) vay B A , l a h i n h chieu cua B E , t r e n m a t ben ( A B B , A , ) nen goc AjBE, = a B C 403 D i e n t i c h x u n g q u a n h cua h i n h l&ng t r u 1^ S,, = p h = A B B B , T i n h A j B i v a B B , T a m giac B A , E i vuong t a i A i CO goc B = a cho t a : A i E i = B E i s i n B = dsina va AjB = BEjCosB =dcosa T a m giac B i A j E i vuong t a i A , , c6 goc B i = 60° cho ta A j B i = A i E j C o t g B i = dsinacotg60° = T a m giac A i B i B v u o n g t a i B i cho B B i = J An' => = ^ - A R ' = , d ' cos' a " • " = 4- ^3 cos' a - s i n ' a VScos' a - s i n ' a = 2d''*sina x/Scos' a - s i n ' a a) Xac d i n h m a t cSt giOTa m a t p h a n g (P) v a l a n g t r u : X e t m a t p h a n g (P) d i qua canh song song BC va E i F i I a n l u g t n a m t r o n g day va k h o n g d t r e n cijng mSt ben Goi M v a N l a giao d i e m cua (P) vBC//MN AD c (ADDjA) M a t khac, h a i mat ben A B B j A , v a D E E j D j song song v d i n h a u n e n B M // N E j va mat cheo A E E j A i , B D D i B , song song vdi n h a u n e n giao tuyen ciia c h i i n g v6i m a t phang (P) la M E i v a B N cung song song v d i Vay B N E i M l a h i n h b i n h h a n h , do B M = N E j T u a n g t u : M F j = C N De t h a y M N = C N v a y B M = M F j = O N = N E j Suy r a M , N l a d i e m giOra cua A A i , D D ] Mat cat la h i n h luc giac B C N E i F i M gom h a i h i n h t h a n g can b ^ n g n h a u B C N M va E,F,MN b) Goc cua (P) v a m a t day ( A B O D E ) (P) v a m a t p h l n g ( A B C D E ) c6 giao t u y e n l a BC Ta c6: EC v u o n g goc BC suy r a E j C vuong goc BC Vay goc p h ^ n g ciia n h i d i e n tao b d i c h i i n g l a goc E j C E T a m giac E j E C vuong t a i R cho t g ^ ^ = EE, EC Wdi E E i = B B , = 4- v ' c o s ' a - s i n ' a V3 = 4-N/3COS' " - s i n ' 30° V3 = ^ x/3 va EC = A j E , = dsina = dx/2 o , ^r7>, Vs 2V2 Suyratgf;cl: = ^ 2^ =- ^ = — 404 c) Dien t ich m at cSt: Luc giac deu ABCDEF ch fn h la h in h chieu cua m at cat t r e n day (ABCDEF) n en t a c6 cos = — VtJi ZE]CE la goc p h i n g cua n h i d ien t ao bd i m at cat va dAy S,„c: d ien t ich m at cdt S = d ien t ich day = 6.dien t iich t a m giac deu can h AB AB'Va 3v/3 „2 ' ' cos2£c5: = — - i ' 3S d sin a =- tg^E,CE + l 3_ (^^^ 11 + Suy cos Suy _ _ _ _ d ^ S„,e = cosE,CE + l B a i 19 Cho kh oi lap ph uan g ABCD A,BiC,D i Goi M va N la d iem d t r e n cac dudng cheo AB, va BCi ciia cac m at ABBAj va BBiC,C cho M N song song vd i m at ABCD Ti m cac , AM , BN MN t i so va vcfi BC, AB, AB Gidi S4 A, c, S3 S, D B Gia sQ M N // (ABCD) Ke dadng t h an g SiSa qua M a t r o n g m at AAjBiB song song vcJi AB, m a t p h i n g xac d in h bdi MN va S1S2 song song vd i m at ph an g (ABCD) Th iet dien vd i kh o i lap ph uan g la h in h vuon g S1S2S3S., — Dat = X, AB = a • AB, Tam giac MB1S2 don g dan g t a m giac ABjB suy r a AB, = AB = 5& B,B Tir M B, = ( - x)AB,, t a n h an difgc MS2 = ( - x)AB = ( - x)a 405 B,S2 = (1 - x ) A B i va BS^ = B B , - BiS^ = x B B , Tarn giac BS2N dong d a n g tarn giac B B , C i B N BS, S,N Suy r a —-— = = ~ =x B,C, BCj B B , „ _ , BN AM Suy SoN = 2a v a = =x BC, A B , T a m giac MS2N vuong, t a c6: M N = — a MS2 = (1 - x)a; S2M = xa Theo Pytago t a c6: - a ^ = (1 - x ) V + x V « 9x2 - 9x + = « ^- = X,, = — ' Tif t a CO v i t r i cua M N thoa dieu k i e n ciia bai toan BN , AN BN , A N Fra l o i : = = — hay = = AB, BC, A B , BC, B a i 20 Cho h i n h h o p A B C D A i B , C , D i G o i — M va N l a d i e m I a n luat t r e n canh A A , va AM CN CCi cho = m; = n M a t p h a n g (u) qua M va N song song v d i difcrng cheo BD AA, CCj cua day Xac d i n h t i so m a m a t p h a n g (a) chia canh B B ] Gidi B D c ( B B i D i D ) suy r a ( a ) v a ( B B , D i D ) cat theo di/dng t h a n g // vdi B D D u n g giao d i e m cua M N v CE = a ED AD / / L ~ ~1|1 L D E 0' - - / A =* C L = - 0 ' = - 3 JN D S , „, = / / C, P f AEOO' CO CL // OO' Dinh l i Thales CL EC OO' EO' VxAnK= D, K B 2a' a— \v — — CL.S„„„ - — 36 2a= _ a;^ Z5_ 36 36 ^ 29a^ 7a' V, = a^' 36 36 V Do -yL, = 29 — Bai 22 Cho hinh lang tru tuf giac deu Qua trung diem cua hai canh lien tiep a &ky , difng mat phang cat canh ben va tao vdi day a Tim dien tich thiet dien biet rkng canh day bang b Gidi Goi K, L la trung diem cua AD va AB, qua giao diem E giffa K L va AC ve EN tao vdi BC mot goc a (EN nftm mat phang cheo AA'CC) Qua di&m O" (giao diem cua true CO' vdi EN) difng PM // BD (trong mat phang cheo BB'D'D) Ngu giac KLMNP la thiet dien can t i m Ap dung cong thijfc S' = S.cosa 7b^ 7b^ S' la dien tich cua KLBCD = — nen S = 8 cos a Bai 23 Cho hinh lap phuong ABCD.A'B'C'D' vdri canh a a) Xac dinh hinh chieu ciia hinh lap phUong len mot mat p h i n g vuong goc vdri mot dudng cheo Tinh dien tich cua hinh chieu b) Dung thiet dien di qua tam va vuong goc vdi difdng cheo n6i tren Tinh dien tich thiet dien t i m t i so dien tich thiet dien vdi dien tich hinh chieu 407 Gidi a) K h o n g l a m m a t t i n h chat t o n g quat cua b a i t o a n , t a t i m h i n h chieu cua h i n h lap phiXcfng t r e n m&t p h a n g d i qua d i e m A dong t h d i vuong goc v d i diicfng cheo A C De t h a y r S n g A C cSt (CB'D') t a i d i e m I va I C = i A C = 3 B' A' — \ M » "« / ^ Qua A dirng A D ' i // va b a n g I D ' , A B j ' // va bSng I B ' , song song va b a n g I C De t h a y r a n g cac d i e m doi xuTng cua D', , B[, C i doi v d i A l a B ] , D j , A'^ va dong t h d i l a h i n h \ Suy r a d i e m B j , C i , D j , D ' , , A[, R D' ' ' ^ j \ Q / / Xi' -X x\ 11 D 1V '\'''"X B C N chieu cua B , D , A' B[ n&m t r e n m a t p h i n g vuong goc v6i A C t a i A T a CO t h e chiifng m i n h r a n g h i n h chieu B j C i D i B'jA'jD'j la luc giac deu canh DiCi = AD, = ID = ax/e Do dien t i c h B j C j D i B'^A'^D; = dien t i c h t a m giac A C , B i = aVe b) T h i e t d i e n d i qua O va vuong goc v d i diTdng cheo A C p h a i // vori mSt p h i n g (CB'D') do t r o n g h i n h chOr n h a t D C B ' A ' qua C diTng SP // A ' D T r o n g h i n h vuong A B C D , difng P N // B D ; t r o n g h i n h vuong A A ' B ' B dUng S M // A ' B va dang SR // D ' B ' ; RQ // D A ' Cac d i e m M , N , P, Q, R, S c h i n h l a t r u n g d i e m cua cac canh B B ' , B C , C D , D B ' , A ' D ' , A ' B ' , suy r a M N b a n g N P b a n g PQ b a n g QR b a n g RS bang SM bang va OS b a n g O M b k n g O N b a n g OP b a n g OQ b a n g OR b k n g Do t h i e t d i e n l a luc giac deu canh D i e n t i c h M N P Q R S = d i e n t i c h t a m giac = IV2J' 4 T i so d i e n t i c h cua t h i e t d i e n va dien t i c h h i n h chieu b k n g sVSa' B a i 24 Cho h i n h chop t a m giac deu c6 difdng cao h , canh ben I t i m dien t i c h ciia t h i e t dien song song v d i day va each day k h o a n g l a a ( h > a) Gidi • all ( A B C ) n e n : a n (SAB) = D E ( D E // A B ) a n (SBC) = E F ( E F // BC) a n (SAC) = D F ( D F // A C ) T h i e t d i e n l a t a m giac D E F dong d a n g v d i t a m gidc A B C • T i m dien tich thiet dien Goi h = S H b a n g chieu cao h i n h chop S.ABC h i = S H i b k n g chieu cao cua h i n h chop S.DEF S l a d i e n t i c h t a m giac A B C Si l a d i e n t i c h t a m giac D E F 408 Taco: = Suy r a Si = S = (h - af (h - a ) ' Goi SK la difdng cao ciia tarn giac SBC suy r a Tarn giac S A H cho: A H = V S A ' - S H ' = V / ' - h ' AK= - A H = -V/^-h^ 2 K B = AK.tan30° KB= = s = aVs s, = V3(f-h^) ( h - a ) Bai 25 Cho W dien deu S.ABC canh a Goi E, F Ian luat la t r u n g d i e m cua SB va BC M a t p h i n g a d i qua EF va vuong goc v d i A B chia the t i c h h i n h chop theo t i so' nao? Gidi Goi I la t r u n g d i e m ciia A B M a t phang a A B nen a S I va do a cat m a t phang (SAB) theo giao tuyen E K // S I Vay t h i e t dien la t a m giac E F K can t a i K • The t i c h h i n h chop aW| l a ; ^f3 ^ VsABC = — 12 EKBF = ^ h is ' 3• 32 •a ^hV3 192 VsAKKC = i'hVs a'hVi 15a'hx/3 12 192 192 T h i e t dien chia the t i c h h i n h chop theo t i so: a'hx/3 _ ISa'hVs _ 192 192 ~ 15 • Bai 26 Cho h i n h chop tuf giac deu SABCD T r e n CD keo dai lay diem M cho M D = 2DC Qua M , B va t r u n g d i e m E cua SC diing m a t p h l n g M a t p h l n g chia the t i c h h i n h chop theo t i so nao? Gidi M a t phang ( M B E ) cat ( A B C D ) theo B M , cat (SBC) theo B E M a t phang ( M B E ) cat (SDC) theo giao tuyen qua E va M , SD n E M = F =i> T h i e t dien la tijf giac B E F G — =— =i MD GD suy r a A G = icD suy r a D G = - A D 409 Ve E'E // C D : EF FM CD E E ' _ 2_ 2CD MD SuyraE'F= i - E ' D = -SD 5 Goi difdng cao ciia h i n h chop la h , canh la a Ta c6 VsABcn = ^ a^h Ke E E i ( A B C D ) : E , s A C E E j = Ke F F i ( A B C D ) : F, e B D va F F , = ^ h VlM ^EGDK = Ve h o p M B C K " isHCM.EEi= V E R r M = V K n M = i w F F , = ViicDGKK = -ah / VMGDF i i a a | = ^a^h i.i.|.a.2a.| h 4a^h ah = ah f 45 180 29 o, 31 o, a^h = a^h V.SARFKG = — a h 180 180 Vay t i e t dien chia h i n h chop theo t i so: ' 29 2^ 31 2, 29 ah : ah = — 180 180 31 B a i 27 Cho h i n h chop 111 giac deu S.ABCD Goi F, K, N la t r u n g d i e m cua cac canh A D , AB, SC Dirng t h i e t d i e n d i qua F, K, N va h i n h chop noi tren.Chiifng to r i n g mat p h i n g thiet d i e n c h i a h i n h chop r a t h a n h h a i k h o i da dien tiTOng dUOng (Co the t i c h bSng nhau) Gidi N o i F K cat C D d E va CB d M N o i N E cat SD a P N o i M , N cat SB a L T h i e t d i e n can t i m la ngu giac N P F K L Goi V ] la the t i c h k h o i da d i e n chuTa d i n h C va V la phan l a i , t a c6: Vl = VNKMC - VpEDp - V^BKM T r o n g m a t p h a n g (SCB) ke N R // CB V i B M = A F = -BC = N R nen A N R L = A M B L suy r a B L = ^ B R = i B S Tifang t u : P D = - SD Goi dirdng cao ciia tiif giac deu la h va canh day la a t h i V = a'h i 1„ , VNCEM= - S „ , A 3 h = g-2«-2^-2 = — a h 48 410 Vay VxcKM = ^ a'h VsADc = ^ „ „ a^h VpKn,.-= V , K B M 1 h 91 2^ = 6-2-2"l = ^ " ^ Dodo taco: V , = — a ' h - — a ' h = ^ a ' h = - V 48 96 Bai 28 Trong hinh chop tOf giac deu, cac mat ben lam vdfi day goc a Qua canh cua day difng mat phang tao vdri day goc (3 Canh cua day bang a Hay tinh dien tich thiet dien Gidi Thiet dien BCB,C, la hinh thang can Goi M , N la trung diem cua canh AD va BC Mat phang (MNS) cat thiet dien theo N K (K la trung diem B,C,) Ta CO j5 J M ^ = K I N S = al^lNl? = P Ta tinh K N va B,C, tarn giac M K N c6 "• K N a • sin(cx + f!) Vay KN = a sin a sin(a + (3) A Tarn giac ADS dong dang v : " ' Tirang tir RD = i SD = 4 M \ // ^ " c) SH = h nen V = - a.h E Mat phdng (MNP) chia hinh chop hai phan Goi Vi la phan the tich chila dinh S va V j la phan the tich ke vdi day ABCD V2 = VpcEF - [VQHEM - VKDNF] 3a 3a h i 2J • ~ h _ 3a^h 16 h Sa (Difdng cao tiT P la - va CE = CF = — ) 2 KDNF V,= = ^ - ;^ BM ~ = (Difdng cao tii Q va tif P deu bang — ) 6.16 a^^ ^ - 16 6.16 SABCD alh Do V i = VsARci) - V2 = - a ' h - ^ a ' h = = Vo V a y V i = V2 414 MUC L U C Ldi noi dau Bang ke cac k i h i e u va chuT v i e t t i t t r o n g sach Chuyen de : T o n g quan ve cac k h a i n i e m t r o n g h i n h hoc k h o n g gian Chuyen de : Quan he song song 14 Chuyen de : Phuong phap t i e n de 23 Chuyen di : Quan he vuong goc 30 Chuyen de : FhiTOng phap t r a i co the t r e n m o t m a t p h i n g 45 Chuyen de : Xac d i n h va t i n h cac loai goc t r o n g k h o n g gian 49 Chuyen de : Cac loai t h i e t dien tao t h a n h v d i v a t t h e h i n h hoc 61 Chuyen di : Cac dang k h o a n g each va dudng vuong goc chung 76 Chuyen de : M a t cau ngoai t i e p - m a t ciu n o i t i e p 94 Chuyen de 10 : D i m g h i n h t r o n g k h o n g gian 144 Chuyen de 11 : Quy t i c h mot d i e m t r o n g k h o n g gian 147 Chuyen di 12 : PhUcfng phap t h e t i c h 170 Chuyen de 13 : K h o i da d i e n : 200 Chuyen de 14 : Tijf dien - cac loai tuT dien dac biet 204 Chuyen di 15 : H i n h l a n g t r u - h i n h l a n g t r u cut 235 Chuyen di 16 : H i n h chop - h i n h chop cut 250 Chuyen di 17 : H i n h t r u t r o n xoay 266 Chuyen di 18 : H i n h non - h i n h non cut 269 Chuyen de 19 : H i n h cau - chom cau - quat cau 274 Chuyen di 20 : Pho'i hop cac k h o i h i n h hoc k h o n g gian 279 Chuyen di 21 : Phep bien h i n h va phep d6i h i n h t r o n g k h o n g gian 291 Chuyen de 22 : B a i toan cue t r i t r o n g h i n h hoc k h o n g gian 298 Chuyen di 23 : PhifOng phap vecto va tpa dp t r o n g k h o n g gian 336 Phuluc 366 : 415 -w^w^^v n h a s a c h h o n g a n c o r - n v n E m a i l : n h a s a c h h o n g a n (ghotmail.com C N g u y e n Thi Minh Khai - Q I - T P H C M O T : ( ) - 7 - 9 • F a x: ^^^^ ca ij^e /A^^ [...]... 4^ 3 27 27 2, , 4V3 (4a - x) =^ — X 32 Rx) = x^ - 4ax^ + De y thay : f 3 0 a 27 128 27 i = 0 a^ = 0 4a 3 32 (4) o 3 3 8a 4a 4a x (3) - 4a nen ta c6 : 4a X = (3) O 9 8 x'^ - - a x 3 / 128ay 727 / / 0 8a / 0 _ a"^ = 0 9 4a X = — x = i(l±V^^ 310 Vi X e [0; 4a] nen cdc n g h i ^m cua (3) la : x = 4 J3 Vay : V = ^ a ^ ^ t « 4a 4a 4 x = ^ V X = - ( 1 + V3 )a r- v x = ^ ( 1 + Vs a) (ycbt) Bai 40 0 ( D A I H... _ ( 4 a - x ) - — = 3 4 4 2 Vs o —x^(4a-x) 32 (ycbt) Sau day ta khao sat sif b ien th ien va ve do t h i cua V tr on g he true (OxV) M ien xac d in h : D y = [0; 4] "x = 0 =i> V = 0 Dao h a m: V = ^ (- 3x^ + S a x) = 0 32 27 3 V" — ( - 6 x + 8a ) = 0 « => V = X = — 4a^V3 Vay (C) la do t h i ciia V tr on g ycbt 8V3a^ 4^ 3 , e/ D in h X de ; V = ~ ~ a^ 27 Xet : V = VS 27 ' (C ) 4V3a^ , 4^ 3 27... SAC " = -(BC^SI^ +SA^SB^ 4 +SA2.SC2) = - [ B C ^ S I ^ + S A ^ I S B ^ + SC^)] 4 = i(BC2.Sl2 4^ I S +SA^BC2) ' = - [ B C ^ C S I ^ + S A 2 ) ] = -(BC2.Al2) = SiBc 4 4 (1) Ap d u n g B D T S c h w a r t z : ( S S_l i C + S , S„B A +SsAc) 'SBC + ^I B A + S|AC ) 2/ Goi t h e t i c h tiir d i e n S A B C l a V : V = - AS.S„„,, = - S A S B S C = - a x ( k - x) ^ 3 - X + k - X (2) 24 Dau d a n g thijfc t... Ta OH = l ( a V + b V +a V ) 4 _ 02 , o2 , 02 SQAB + S Q BC + SQAC ON = - B C 2 PM = - B C 2 CO : Ttfongtu ^ OABC (PN = OM; OP = M N 325 Vfiy : A O M P = A O N P = A O M N = A M N P => (dpcm) T a CO : VQMNP = — - O H S M N P o => VQMNP = — O H ( — SABC) 3 4 24 4 abc 1 = > VQMNP = — -VQABC = " T — (ycbt) 4/ T a CO : S ^ c = - ( a ^ b ^ + b^c^ + a^c^) 4 A p d u n g B D T Cauchy,... N // QP // CD |(a)n(BCD) = QP//CD (4) Tifang tu : M Q // N P // A B (5) (3) Til: K h i do : (4) ; (5); (6) (6) A B ± CD ^ thie't dien M N P Q l a \\inh chO n h a t Dien t i c h t h i e t dien : S(x) = SMNPQ(X) = M N P Q MN AE d - X 2n(d - x) MN = CD AJ d Trong do : 2inx MQ EF X MQ = AB AB ~ d S(x) = 4 m n x ( d - x) 4mn => S(x) < 2 a 2 fx + d- x 4mn (Va^ - 2 - ) 4 - n'' - m => S(x) < m.n (7) Dau d... MC ^ —> ^ 2 f ^ => S = 2 + MG+GA 2 + MG+GB ( + MG+GC + J MD V / 2 _> MG+GD V > = 4 M G2 + G A ^ + G B ^ + G C ^ + G D ^ S + 2MG GA +GB + GC+GD = 4 MG2 + G A ^ + G B ^ + G C ^ + G D ^ I = 4 MG2 + G A ^ + G B ^ + G C ^ + G D ^ 3 24 S > G A ^+ GB^+ GC^+ GD^ (const) Vay : m i n i = GA^+ GB^+ GC^+ GD^ M = G (ycbt) Bai 41 2 (DAI HOC QUOC GIA TP.HCM - KHOI A - K I N H TE - 1997) Tren ba canh Ox, Oy, Oz... n g go c c h u n g c u a A B v a S I N h i fn g : EF = A N , n e n t a t i n h do d a i A N I A S A K vu o n g l AN ^ l AS^ 1 1 1 4 + • AN ^ AK^ b^_ +4h^ « bh AN = AN ^ Vb ^ + 4 h 2 D o d ^ i d o a n vu o n g go c c h u n g c u a A B v ^ S I l a : S EF = 2/ , Vb ^ + 4 h 2 ( yc b t) D e y A C ± ( S A B ) 3 M J , t r o n g m a t p h l n g ( S A B ) , d ito g A H 1 M J => A H l a d o a n v u o n g... i(aV+bV+aV) max(SABc) = khi a = b = c = 6 1 M a t khdc : 1 1 1 + — r+b^ c2 OH^ 2„2 (5) a^+b^+c^ Dau d i n g thuTc t r o n g (5) x a y r a a = b = c => m a x ( O H ) = - k h i a = b = c = 3 Bai 3 (ycbt) 41 3 ( D A... i i EF 4a' - b-^ - 2/ Theo tren CD 1 (ABF) EF = (0 < X < V4a^ - b ^ ) (ycbt) A FS = Goc nhi dien do vuong khi va chi khi 4a'-h' AB x = — X = Khi do : VABCD = VcAiif• + VDAHF = - CF.SA RF + 1 *ABCD = - SA I)F(CF + D F ) = - SA BK.CD = - D F S'ABF SABF 3 317 D o FA VABCI) I d n n h a t k h i SABF I d n n h a t = FB= i Vi l ^ n e n VAHCD Id n n h a t SAUK = i F A F B s i n X p ^ = - ( 4 a ^... (SBM) (dpcm) 2/ Ta CO : M B = V A B ^ - A M ^ = V 4 R 2 - x2 The tich tuT dien : "SABM = - S A S A B M = 4 h x V4 R 2 3 6 Nen : 3max(VsABM) max(SABM) -x^ (D Trong AABM ha M H 1 AB SABM = - A B M H 2 = R.MH (1) o 3 m a x ( M H ) ... thay : f a 27 128 27 i = a^ = 4a 32 (4) o 3 8a 4a 4a x (3) - 4a nen ta c6 : 4a X = (3) O x'^ - - a x / 128ay 727 / / 8a / _ a"^ = 4a X = — x = i(l±V^^ 310 Vi X e [0; 4a] nen cdc n g h i ^m cua (3)... V = X = — 4a^V3 Vay (C) la t h i ciia V tr on g ycbt 8V3a^ 4^ 3 , e/ D in h X de ; V = ~ ~ a^ 27 Xet : V = VS 27 ' (C ) 4V3a^ , 4^ 3 27 27 2, , 4V3 (4a - x) =^ — X 32 Rx) = x^ - 4ax^ + De y... _ aVs.x _ V x SC ~ 4a " A C SM _ a ( a - x) _ ( a - x) SC ~ 4a ~ Vay : KM H L la mot hinh vuong « d Ta V3 x ^ ( a ^ 4 CO < ^ V x = 3( 4a- x ) ^ : ML^ = M H ^ + HL^ = 3x2 9(4g _ 16 16 x = a V