tuyển tập 500 bài toán hình không gian chọn lọc nguyễn đức đồng

516.23076 BAN GIAO VIEN NANG T527T N J G U Y E N KHIEU TRUCiNG THI eC/C e O N G ( C h u bien) PHAN LOAI VA PHl/OfNG PHAP GIAITHEO CHUYEN DE • BOI Dl/dNG HQC SINH GIOI • CHUAN B! THI TU TAI, DAI HOC VA CAO BOG Ha NOI DANG NHA XUAT BAN OAI HOC QUOC GIA HA NOI BAN GIAO V I E N NANG K H I E U TRl/CfNG THI NGUYEN DLfC D N G {Chu hien) TUYEN TAP 500 BAITOAN • HDIH imm GIAN C H O N LOG • • • PHAN LOAI VA PHUdNG PHAP G I A I THEO CHUYEN • B o i difdng hoc s i n h g i o i • C h u a n b i t h i T i i t a i , D a i hoc v a Cao d a n g (Tdi ban idn thvt ba, c6 svCa chUa bo sung) THir ViEN TiiVH BiKH liik^m NHA XUAT BAN DAI HOC QUOC GIA H A NOI NHA XUAT BAN DAI HOC QUOC GIA HA NQI 16 Hang Chuoi - Hai Ba Trcfng - Ha Npi Dien thoai: Bien tap - Che ban: (04) 39714896 Hanln chinli: (04) 39714899; Tong Bien tap: (04) 39715011 • Fax: (04) 39714899 * Chiu Gidm Bien Saa trdch ** nhiem xuat ban: doc - Tong bien tap: T S P H A M T H I T R A M tap: THUY bdi: THAI Che ban: Trinh HOA VAN N h a sach H O N G A N bay bia: THAI V A N SACH LIEN K E T TUYEN TAP 500 BAI TOAN HJNH KHONG GIAN CHON LOG Ma so: 1L - 195OH2014 In 1.000 cuon, kho 17 x 24cm tai Cong ti Co phan V3n hoa VSn Lang - TP Ho Chi IVlinh So xuat ban: 664 - 2014/CXB/01-127/OHQGHN 10/03/2014 Quyet dinh xuat ban so: 198LK - TN/QO - NXBOHQGHN 15/04/2014 in xong va nop IIAJ chieu quy il nSm 2014 LCilNOIDAU Chung t o i x i n g i d i t h i $ u den doc gia bp sdch: Tuyen t a p cdc b ^ i toan d k n h cho hoc sinh Idp 12, chuan b i t h i vao cac trucrng D a i hoc & Cao d i n g Bo sach gom quyen : T U Y E N T A P 546 B A I T O A N T I C H P H A N T U Y E N T A P 540 B A I T O A N K H A O S A T H A M SO T U Y E N T A P 500 B A I T O A N H I N H G I A I T I C H T U Y E N T A P 500 B A I T O A N H I N H K H O N G G I A N T U Y E N T A P 696 B A I T O A N D A I SO • T U Y E N T A P 599 B A I T O A N L U O N G G I A C T U Y E N T A P B A I T O A N RCJI R A C V A C l / C TRI NhSm phuc vu cho viec r e n luyen va on t h i vao D a i hoc b k n g phucrng phdp t i m hieu cac de t h i dai hoc da ra, de tiT n a n g cao va chuan b i k i e n thiJc can t h i e t De phuc vu cho cac do'i tUcfng t\i hoc : Cac bai g i a i luon chi t i e t va ddy d u , p h a n nho tCrng loai toan va dua vao cac phucfng phap hop l i Mac du chiing t o i da co g^ng het siic t r o n g qud t r i n h bien soan, song vSn k h o n g t r a n h k h o i nhiJng t h i e u sot Chiing t o i m o n g don n h a n m o i gop y, phe b i n h tii quy dong nghiep ciing doc gia de Ian xuat ban sau sach ducfc hoan t h i e n hcfn Cuoi Cling, chiing toi x i n cam cm N I l A X U A T B A N D A I H O C Q U O C G I A H A N O I da giiip da chiing t o i m o i m a t d l bo sach dUdc r a dcfi NGUYEN DtfC DONG • (i) B A N G K E CAC K I H I E U V A CHLf V I E T T A T T R O N G • [ ( A B C ) ; ( E F G ) ] : goc tao bori mp ( A B C ) va ( E F G ) -> • C > : Phep t i n h tien vectcf v V • D A : Phep doi xOmg true A • Do : Phep doi xiiTng true • Q(0; cp) : Phep quay t a m O, goc quay (p • V T ( ; k ) : Phep v i t u t a m 0, t i so k • D N : dinh nghla • D L : dinh ly • Stp : D i e n t i c h t o a n p h a n : The t i c h • C M R : chiJng m i n h r i n g A : goc • B i : budc i • T H i : t r u d n g hop i • V T : ve t r a i xuong dtfcfng thftng (d) (3r3^ SACH CAC K I H I E U T O A N HOC v A CAC T l / V I E T T A T : (i) tUcfng dUcfng (il • => : (i) keo theo • : k h o n g tUdng dilcfng • d> : k h o n g keo theo • = : dong n h a t : k h o n g dong n h a t • i • Sv\nc = S ( A B C ) = d t ( A B C ) : d i e n t i c h AABC • V s A H c = V ( S A B C ) : the t i c h h i n h chop S.ABC • H Q : he qua • Sxq : D i e n t i c h xung quanh • V • A ' = ''7(ai A : A ' la h i n h chieu ciia A xuong m a t p h i n g (a) • A ' = ''Vfd) A : A ' l a h i n h chieu cua • d [ M ; (D)l : k h o a n g each tiT d i e m M d e n ducfng t h i n g (D) • d [ M ; ( A B C ) I : k h o a n g each tii diem M den mat phang ( A B C ) • (a; P ) : goc n h i d i e n tao bcfi mfa m a t phang (a) va ( P ) • ( S ; A B ; D) = ( A B ) : n h i dien c a n h A B • tao bdi h a i dUomg t h i n g d • V P : ve p h a i • B D T : bat d i n g thijfc • y c b t : yeu cau b a i toan • d p c m : dieu p h a i chuCng m i n h • gt : gia thiet • K L : ket luan • D K : dieu k i e n • P B : phan ban va d' • [ H T C A B C T I : goc tao bdi du&ng t h i n g d va • C P B : chiTa p h a n ban mp(ABC) Chuyen de : TONG QUAN V E C A C KHAI NIEM T R O N G HINH H O C K H O N G G I A N • H i n h hoc k h o n g gian la m o t mon hoc ve cac v $ t t h e t r o n g k h o n g g i a n ( h i n h h i n h hoc t r o n g k h o n g gian) ma cac d i e m h i n h t h a n h nen v a t the t h u d n g thiTcrng k h o n g ciing n f t m t r o n g mot m a t phang • N h i f vay ngoai d i e m v a d i i d n g t h d n g k h o n g drfoTc d i n h n g h i a nhiT t r o n g h i n h hoc phAng; mon h i n h hoc k h o n g g i a n xay di/ng t h e m mot doi tuong can n g h i e n ciifu nCfa la k h a i n i # m m g t p h a n g c u n g k h o n g difoTc d i n h n g h i a K h i noi tori k h a i n i e m t a lien tuang den m o t m a t ban b a n g phang, m o t m a t ho nildc yen l a n g , m o t tb giay dat d i n h sat t r e n mot m a t da di/gc l a m phang No duoc k y hieu b d i cac chCf i n L a T i n h n h a : (P), (Q), (R), hoac cac chCf t h u d n g H y L a p nhU (a), ((5), (y), • M a t phang k h o n g ducfc d i n h n g h i a qua mot k h a i n i e m k h a c ; n h i f n g thifc te cho thfi'y mSt ph&ng CO nhutng t i n h chat cu t h e sau, goi la cac t i e n de : O T I E N D E 1: C o i t n h a t b o n d i e m t r o n g k h o n g g i a n k h o n g t h ^ n g h a n g (nghia la luon luon c6 i t n h a t d i e m d ngoai m o t m a t p h ^ n g tiiy y) O T I E N D E 2: N e u m p t dtfdng th&ng v a m p t m a t p h ^ n g c h a i d i e m c h u n g t h i dUcTng th&ng a y se n S m t r p n v ^ n t r o n g m a t p h a n g n e u t r e n O T I E N D E 3: N e u h a i m a t p h & n g c d i e m c h u n g t h i c h t i n g c v so' d i e m c h u n g : n e n h a i m a t p h S n g c S t n h a u t h e o m p t d U d n g t h ^ n g d i q u a v so' d i e m c h u n g a y Di/cfng t h a n g ay goi la giao tuyen cua h a i m a t ph^ng O T I E N D E 4: C o m p t v a c h i m p t m $ t p h a n g d u y n h a t d i q u a b a d i e m p h a n b i # t khong th^ng hang O T I E N D E 5: T r e n m p t m § t p h a n g t u y y t r o n g k h o n g g i a n c a c d i n h l y h i n h h o c ph&ng scf c a p (da hoc tCr Idp den Idp 10 va cac d i n h l y n a n g cao) d e u d i i n g O T I E N D E 6: M o i d o a n th&ng t r o n g k h o n g g i a n d e u c dp d a i x a c d i n h : t i e n de neu len sU bao toan ve dp dai, goc va cac t i n h chat lien thuoc da biet t r o n g h i n h hoc p h i n g • TiT chung t a c6 m o t so each xac d i n h m a t p h n g n h i / sau : O H E Q U A 1: C o m p t v a c h i m p t mfit p h S n g d u y n h a t d i q u a m p t d U d n g t h S n g v a m p t d i e m n S m n g o a i dt^dng t h a n g O O H E Q U A 2: C o mpt v a c h i mpt m^t p h d n g n h a t d i q u a h a i di^cAig t h ^ n g cSt n h a u H E Q U A 3: C o m p t v a c h i m p t m ^ t p h a n g d u y n h a t d i q u a h a i di^c/ng t h d n g song song • Dong t h d i t a phai hieu t h e m r k n g mot m a t phang se r o n g k h o n g bien gidi va dUcmg t h ^ n g c6 dai v6 t a n mac du t a se bieu dien no mpt each h i n h thiifc hflu h a n va k h i e m t o n nhU sau: • De thuc h i e n dirge phep ve c h i n h xdc m t h i n h h i n h hoc t r o n g k h o n g g i a n ngoai cac dudng t h a y ve l i e n n e t , t a can p h a i n a m chac di/pc k h a i n i e m di/dng k h u a t ve b k n g net dijft doan: Mpt dtfdng b i k h u a t t o a n bp h a y c h i k h u a t m p t d o a n c u e bp n a o k h i v a c h i k h i t o n t a i i t n h a t m p t m a t p h S n g du'ng p h i a trvC6c h o ^ c p h i a t r e n c h e n o m p t e a c h t o a n bp h o a c c u e bp ti^cAig uTng • Muon xac d i n h n h ^ n h m o t m a t p h ^ n g t r o n g k h o n g gian t a chon t h u thuat thUc h a n h : M p t h i n h t a m g i a c , tii" g i a c h o a c d a g i a c ph&ng ( k h o n g g e n h ) , dUcfng i r o n , l u d n x a c d i n h m p t m ^ t p h S n g t r o n g k h o n g g i a n T a gpi c a c m&t p h ^ n g l a m^it p h S n g h i n h thvCc v d i c a c k y h i p u ( A B C ) , ( A B C D ) , ( C ) , txictng vtng M p t dvictng t h d n g n ^ m t r o n g m ^ t p h & n g h i n h thd'c m a m a t h i k h u a t c u e bp • M a t p h d n g h i n h thu^c h i k h u a t n e u c m p t h a y n h i e u m ^ t ph&ng n a o c h e n o • h a y t o a n bp v a k h i dUcTng t h ^ n g k h o n g l a b i e n c u a m a t p h d n g b i k h u a t do, t h i di^dng th&ng c u n g tii'oTng vlng k h u a t c u e bp h a y t o a n bp Noi h a i d i e m m a it n h a t c mpt d i e m k h u a t t h i dUpc mpt dUcfng k h u a t cue bp h a y • Mpt d i e m nhm t r o n g m p t m $ t ph&ng h i n h thuTc b i k h u a t t h i goi l a d i e m k h u a t • t o a n bp : n e u h a i diictag k h o n g l a b i e n c u a c a c m^t phAng h i n h thufc c h e no • C A C H I N H A N H M I N H HQA \(d) • (d) b i (a) che k h u a t cue bo, (d) c6 doan ve net dijft doan n k m dudi (a) S • (d) b i m a t p h ^ n g (SAC) che k h u a t cue bo, (d) CO m p t doan ve duft doan n k m sau (SAC) (hien n h i e n (d) cung d sau cac m a t (SAB), (SBC)) • C a n h AC b i h a i m a t p h a n g (SBC) v£l (SBC) che k h u a t toan bo, ca doan AC x e m n h u hoan t o a n d sau dong t h d i h a i m a t p h ^ n g (SAB), (SBC) -AA c./—1—^VFJL^ • • A ] H b i che t o a n bo ca doan A ] H n k m sau m a t p h i n g ( A i A D D i ) , mSc dij no d trU H a i m a t p h l n g (a), (P) thuf tif chiJa h a i difdng t h i n g ( d i ) , (da) ma (dj) n (da) = I => S I la giao tuyen can t i m > H a i m a t p h l n g (a), (P) thuf t i f chtifa h a i difdng t h i n g ( d i ) , (da) ma ( d i ) // (da) S_ D i f n g xSy song song v d i (dj) h a y (da) => xSy la giao t u y e n can t i m m C A C B A I T O A N C O B A M Bai Cho tiif giac l o i A B C D c6 cac canh doi k h o n g song song va d i e m S d ngoai (ABCD) T i m giac tuyen ciia : a/ (SAC) va (SBD) hi (SAB) va (SDC); (SAD) va (SBC) Giai a/ Xet h a i m a t p h a n g (SAC) va (SBD), t a c6 : T r o n g tuT giac l o i A B C D , h a i ducmg cheo A C • S la d i e m c h u n g thuf n h a t • (1) n B D = O : d i e m c h u n g thijf n h i (2) ^ Ti/(1) va (2) suy r a : (SAC) o (SBD) = SO (ycbt) hi Xet hai m a t p h a n g (SAB) va (SDC) cung c6 : H a i canh ben A B va C D cua t i l giac A B C D • S la m o t d i e m chung • theo gia t h i e t k h o n g song song ^ A B ^ C D = E : la d i e m c h u n g thut h a i Do : (SAB) n (SDC) = SE (ycbt) Tucfng t i f : (SAD) n (SBC) = SF (ycbt); v d i F = A D ^ BC; A D / / BC Bai Cho t i l d i e n A B C D Goi G j , Ga la t r p n g t a r n h a i t a m giac B C D va A C D L a y theo thuT t i i I , J , K la t r u n g d i e m ciia B D , A D , C D T i m cac giac tuyen : aJ (G1G2C) o ( A D B ) hi (G1G2B) n ( A C D ) c/ ( A B K ) o (CIJ> a/ (G1G2C) n ( A B D ) = I J (ABK) ^ (CIJ) = d (GiGaB) n ( A C D ) = GgK hoSc A K hi G,G2 Bai Cho h i n h chop S A B C D c6 day A B C D la h i n h b i n h h a n h t a m O T i m giao t u y e n cua h a i mSt p h i n g (SAB) va (SCD) hi T i m giao t u y e n cua h a i m a t phSng (SAD) va (SBC) aJ c/ T i m giao t u y e n ciia h a i m a t p h ^ n g (SAC) va (SBD) Giai aJ Xet h a i m a t phSng (SAD) va (SBC), t a c6 : De y A D c ( S A D ) ; BC c (SBC) m a A D // BC • S la d i e m c h u n g thur n h a t • Ta d u n g xSy // A D hoac BC [(SAD) = (xSy; AD) ^ |(SBC) = (xSy; BC) =^ (SAD) n (SBC) = xSy (ycbt) hi Tifang t i r , difng uSv // A B hoftc C D => (SAB) r^ (SCD) = uSv (ycbt) c/ Goi O = A C n B D , tiTcrng t a b a i => (SAC) n (SBD) = SO (ycbt) Bai Cho h i n h chop S A B C D c6 day la h i n h t h a n g A B C D v d i A B l a day Idtn Gpi M la m o t d i e m bat ky t r e n SD va E F l a difang t r u n g b i n h cua h i n h t h a n g a/ T i m giao t u y e n ciia h a i mSt p h i n g (SAB) va (SCD) b/ T i m giao t u y e n cua h a i m a t phSng (SAD) va (SBC), c/ T i m giao t u y e n cua h a i mSt p h a n g ( M E F ) va ( M A B ) Doc gia t u g i a i tUcfng t u n h u cac b a i t r e n Bai Cho h i n h chop S A B C D c6 A B C D l a h i n h b i n h h a n h Goi G,, G2 l a t r o n g t a m cac t a m giac SAD; SBC T i m giao t u y e n cua cac cSp mSt p h a n g : a/ (SGiG^) va ( A B C D ) b/ (CDGiGz) va (SAB) UvCdng 0/ (ADG2) va (SBC) d§Ln Goi I , J , E, F thur t a Ik t r u n g d i e m cac doan t h i n g A D , BC, SA, SB theo thur tvt d6 Thifc h i e n cac l a p l u a n nhtf cac bai toan t r e n ; a/ (SG1G2) n ( A B C D ) = I J (ycbt) b/ (CDGiGa) n (SAB) = E F (ycbt) c/ (ADG2) ^ (SBC) = xG2y (ycbt) T r o n g xGay // A D hoSc BC L o a i : T l M G I A O D I £ M C U A D U d N G T H A N G 1fA M A T L PHirONG PHANG PHAP Ca sd cua phaang phap t i m giao d i e m O cua dudng t h a n g (a) va m a t phSng (a) l a xet h a i k h a nSng xay r a : n T r i r d n g hop (a) chiJa dudng t h S n g (b) va (b) l a i c&t diicrng t h d n g (a) t a i O T i m O = (a) n (b) => O la d i e m can t i m n Trtfdng hap (a) k h o n g chiifa dUcmg t h i n g nao cat (a) T i m ( P ) ^ ( a ) v a ( a ) n ( P ) = (d) > T i m O = (a) o (d) => O la d i e m can t i m n CAC BAI TOAM G O B A N Bai Cho tuf d i e n A B C D Goi M , N I a n lugt la t r u n g d i e m cua A C va BC L a y d i e m K e B D cho K B > K D T i m giao d i e m ciia h a i dudng t h i n g CD va A D v d i ( M N K ) c a t h i n h I d n g t r u theo mSt p h i n g T i m giao t u y e n cua ( M N P ) Gidi (MNP) va ( A A , D D , ) TiT xac d i n h d i e m Q Gia sijf H l a giao d i e m cua PQ vdi mat phang ( A B i C i O SD, // B i N PB, SP PN A S M N CO PQ // M N , SP ^ = => SN ~ MN SN PQ SP PQ = i M N = VayPQ= | B a i 16 Cho h i n h l a n g t r u tarn giac A B C A j B i C , Goi M l a d i e m t r e n dUcfng cheo A B j ciia m a t A B B i A j cho ~ ' MB ^ AM Difng t h i e t d i e n ciia l a n g t r u vdi m a t phftng u qua M va song song v d i cdc diTdng cheo AiC, B C i Xac d i n h t i so m a m a t p h a n g a chia canh CC, Gidi Goi P l a m a t p h a n g qua A^C va song song v d i BC] (5 va ( B B , C i C ) la difdng t h a n g qua C va // BC, dUcmg cat B B , t a i Sj A , S i o A D = S2 u // p M a t p h a n g ( A B B , A , ) cat m a t p h a n g theo giao tuyen S3S1 P qua M va // vdri AB^ M a t p h a n g (ABC) cat P theo giao tuyen C S a II BC] nen cat m a t p h a n g (AiB,C,C) theo giao t u y e n Sr,Se song song BC, (ABC) // ( A , B i C , ) nen a cat ( A i B i d ) theo giao tuyen S3S7 song song S , S Vay t h i e t d i e n l a 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 i B i C i D , Goi M , N , P I a n lifcrt l a t r u n g d i e m cua A D , B B , va t a m ciia A ] B , C i D i Difng t h i e t dien ciia l a n g t r y b i cat bdi m a t p h a n g a qua M , N , P Xac d i n h t i so m a t p h a n g a chia canh A B Gidi T i m giao d i e m N P vdi m a t p h l n g ( A A , D D , ) dudng t h a n g n a m t r o n g m a t phIng ( B B , D , D ) cat m a t p h a n g ( A A , D , D ) t a i S, (S, = N P ^ DD,) TiJong tU t i m giao d i e m S^ ciia N P vdi m a t p h a n g ( A B C D ) (S2 = N P o BD) a cat ( A B C D ) theo di/dng S N Ta t i m d i n h cua t h i e t d i e n S3 = S i N ^ 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 , S3S5 // 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 AS4 /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 o = ^ x/3 va EC = A j E , = dsina = dx/2 , ^r7>, Vs 2V2 Suyratgf;cl: = ^ 2^ =- ^ = — 404 c) Dien tich mat cSt: Luc giac deu ABCDEF chfnh la hinh chieu cua mat cat tren day (ABCDEF) nen ta c6 cos = — VtJi ZE]CE la goc p h i n g cua nhi dien tao bdi mat cat va dAy S,„c: dien tich mat cdt S = dien tich day = 6.dien tiich tam giac deu canh AB AB'Va 3v/3 „2 ' ' cos2£c5: = — - i ' 3S d sin a =- tg^E,CE + l 3_ (^^^ 11 + Suy cos Suy S„,e = _ _ _ _ d ^ cosE,CE + l B a i 19 Cho khoi lap phuang ABCD.A,BiC,Di Goi M va N la diem d tren cac dudng cheo AB, va BCi ciia cac mat ABBAj va BBiC,C cho M N song song vdi mat ABCD Tim cac , A M , 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 thang SiSa qua M a mat AAjBiB song song vcJi AB, mat p h i n g xac dinh bdi MN va S1S2 song song vdi mat phang (ABCD) Thiet dien vdi khoi lap phuang la hinh vuong S1S2S3S., — Dat = X, AB = a • AB, Tam giac MB1S2 dong dang tam giac ABjB suy AB, = AB = 5& B,B Tir MB, = ( - x)AB,, ta nhan 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 S,N B N BS, 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 , A N BN , AN BN 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 AEOO' CO CL // OO' Dinh l i Thales CL EC OO' EO' =*CL= - 0 ' = - 3 VxAnK= JND.S,„,= D, / Ta c6: V , = V^MW - V,.KCE A / / / L ~ ~|1 f 1 / P C, 1 L D E 0' - - / 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 1V D '\'''"X B N chieu cua B , D , A' C 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 = Vehop 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.|h4a^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 OT: (08) - 7 - 9 • F a x : ^^^^ ca ij^e /A^^