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Toán học là một môn tưởng không dễ mà lại dễ không tưởng. Điều quan trọng là bạn cần biết cách học giỏi toán, bí quyết học môn Toán bởi Toán học không yêu cầu phải nhớ nhiều như các môn khác, điều quan trọng là phải hiểu được bản chất của vấn đề.
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 ) Bai 368 ( D A I H O C S U P H A M T P H C M - 2000) Cho h i n h chop tiif giac deu S.ABCD c6 day A B C D la h i n h vuong canh a va SA = SB = SC = SD = a 1/ Tinh dien t i c h toan phan va the t i c h h i n h chop S.ABCD theo a 2/ Tinh cosin ciia goc n h i d i e n (SAB, SAD) Giai 1/ Goi M ; N theo thiJ tif la t r u n g d i e m canh day BC vk canh ben SA va n h a n x6t t h a y cac mat ben h i n h ch6p tuf giAc deu S.ABCD la bon t a m giac deu canh a Ta CO t r u n g doan h i n h chop la : SM = Do dien t i c h toan phan S,p ciia h i n h chop S.ABCD la : Stp = S.p = a^ + - - ^ S,p = ( l + The tich V ciia h i n h chop S A B C D la : vai V=-.a o aV2 ^ = a^V2 , V = - Ti.h Sa^ h = SO = VSM^'-OM^ = Vaja^ (ycbt) aV2 a^ , - ^ ( y c b t ) 21 Cac mat ben (SAB), (SAC) la cac tam giac deu canh a JBN SA ^ DN SA =^ (p = B5?t) = [(SAB); (SAD)] Ap dung d i n h l y h a m cosin t a diToc : coscp = = - 2NB.ND y (ycbt) J DE Tl/dNG T i ; Bai 369 ( D A I H O C Y DlJCiC T P H C M - 1997) Cho dadng t r o n (O, R) dUcrng k i n h A B thuoc mp(P) Dudng t h a n g (d) (P) t a i A L a y diem S d va SA = h M la mot d i e m chay t r e n dadng t r o n (O) m a t phAng qua A , vuong goc SB tai H, cat SM t a i K a/ Chilng m i n h : A K m p ( S B M ) va K chay t r e n mot difdng t r o n co d i n h k h i M chay t r e n dudng t r o n (O) b/ Tim the t i c h h i n h chop S A H K k h i M la t r u n g d i e m cung A B 265 HvCdng a/ MB din (SAM) SBl(KHA) b/ Bai V = (gt): -3(2R^ + h ^ K h ^ ( D A I 370 H O C MB ± AK S B J_ A K (1) => dpcm (2) 4R^) T H A N G L O N G - K H O I D - 1998) C h o h i n h c h o p til g i a c d e u S A B C D , c a n h d a y a , t a r n I , dUcrng cao h i n h c h o p S I = a t a n X § l = -^2, tan^CsC = -2-j2, t a n A s B = a/ T i n h tanKSl, tan;CsC;, tanASfe a/ b/ T i n h t h e t i c h vk Stp h i n h c h p Hirdng dan Chuyen de 17: 2^ h/ W = - 4a^ HINH TRU TR6N XOAY L GIAN L V p c KI£:N TW(C • T I THI^U DNi : Cho hai hinh tron b^ng n h a u C ( ; R ) ; C'(0; R) hai mat phdng s o n g C O t r u e O O ' ( c u n g v u o n g g o c vdri h a i d a y ) U n g vdri m o i d i e m M M' E ( C ; R), doan MM' // O O ' \vt\x d p n g t a o t h a n h m p t h i n h g p i song e ( O ; R ) va l a h i n h t r u tron x o a y , d i t ^ c g p i tfit l a h i n h t r u K h i M c h a y k h S p d u d n g t r o n ( O ; R) va M ' c h a y k h d p > D o d a i O O ' : c h i u cao h c i i a h i n h t r u > O O ' t r u e h i n h t r u ( v u o n g goc v d i d a y ) > R : b d n k i n h day V C ( ; R); C'(0'; R) : h a i day > O' M' dUofng t r o n ( O ' ; R ) , t h i M M ' t a o t h a n h m a t t r u v a M M ' di/gc goi l a d u d n g s i n h / cua m a t t r u ( h a y h i n h t r u ) > • Ducfng t r o n ( O ; R ) v a ( O ' ; R) : Ducfng c h u i n cua d a y h i n h t r u DN2 : Thie't di§n q u a true e u a h i n h t r u l a thiet d i $ n tao bdi mpt m a t p h ^ n g d i q u a true eiia h i n h t r u • T i n h c h a t : Cac t h i e t d i e n q u a t r u e l a nhCfng h i n h chO' n h a t bang • : T h i e ' t d i ^ n v u o n g g o c vdri t r u e l a t h i e t d i ^ n AvLffc t a o boFi m p t m a t DN3 phang v u o n g g o e vdri t r u e e u a h i n h t r u d o • T i n h c h a t : T h i e ' t d i e n v u o n g goc v d i true la m o t h i n h t r o n (thie't d i e n bat k y khong s o n g s o n g \6\e l a m o t h i n h E l l i p e • T i n h e h a t : (Sif t u a n g giao ciia m a t t r u va m a t p h i n g s o n g s o n g v d i t r u e h i n h t r u ) * M o t m a t p h a n g s o n g s o n g \di t r u e cua h i n h t r u l a t i e p d i e n c i i a mat t r u k h i v a c h i m a t p h I n g d o chijfa t i e p t u y e n c u a d u d n g t r o n d a y * M o t mat phang s o n g s o n g v d i t r u e c u a h i n h t r u se c6 tUcfng g i a o h o a e k h o n g c4t mat t r u , h o a e c a t m a t t r u t h e o h a i diTdng s i n h , h o a c c6 c h u n g v d i mat t r u m o t dUdng sinh n h a t (trifcfng h a p n a y goi l a t i e p dien) D N : M p t I d n g t r u duTng g p i l a n g o a i t i e p m p t h i n h t r u n e u h a i d a g i a c d a y c u a • n o n g o a i t i e p h a i d a y c u a h i n h t r u L u c d o t a c o n c h i n h t r u n p i t i e p t r o n g lang t r u 266 L a n g t r u duTng ngoai t i e p mot h i n h t r u t h i c6 c^c m a t ben n S m t r o n g nhufng tie'p d i ^ n cua mat t r u • Tinh c h a t : Dien t i c h x u n g quanh h i n h t r u : | S,Q = 2nBl Tinh c h a t : T h e t i c h h i n h t r u : V = TtR^h n GiAi T O A N T m Bai 371 ( D A I H O C B A C H K H O A T P H C M - 1977) Tir mot t a m t o n h i n h vuong canh a (cm) ngUcfi t a muon cat r a m o t h i n h chC n h a t v a Mnh tron cung difcrng k i n h de l a m t h a n va cac day cua m o t h i n h t r u T i n h t h e t i c h Idn n h a t cua hop t r u dugc l a m r a , bie't rSng cac canh ciia h i n h chC n h a t p h a i song song hoftc t r u n g v(Ji ciic canh ban dau cua t a m t o n Giai Goi h i n h cha n h a t da dUcfc cSt tiT m i e n g t o n h i n h vuong de l a m t h a n h i n h t r u l a A B C D : xla ban k i n h day cua h i n h t r u Co h a i each uon h i n h chff n h a t A B C D t h a n h t h a n h i n h t r u D, , ,C D, , a - 2x iC a - 2x B B T H j ; Chon A D l a m chieu cao h i n h t r u , canh A B se dugc cuon theo chu v i ddy Ta CO 0 B ' D l a dadng k i n h dUcfng t r o n day v a = 45" D e n day t a c6 : ADCB' la t a m giac v u o n g t a i C=> B D ^ = a^ + • ABCB' l a tarn giac v u o n g can t a i B' => C B ' = B B ' = • aV2 3a2 a^ B D = ^ ^ D O = ^ D i e n t i c h x u n g q u a n h cua h i n h t r u : S,q= 27t.DO.BB' ^ ^ (ycbt) The t i c h h i n h t r u : V=,t(DO')='.BB'=.?^^^ 16 (ycbt) 268 Chuyen de 18 : HINH NON - HINH NON C U T LGlANLPrpCKIEN THUTC T I T H I ^ U n DNi : C h o h i n h i r o n C ( ; R ) v a m p t d i e m S c o d i n h t r e n t r u e h i n h t r o n G p i M la mpt d i e m b a t k y t h u p c C ( ; R ) I Khi diem M chay khap h i n h t r o n C ( ; R) t h i doan S M tao t h a n h mot h i n h goi la h i n h non t r o n xoay (goi tSt la h i n h non) y S : d i n h h i n h non r H i n h t r o n C { ; R) : day h i n h non dadng t r o n C ( ; R) la dUcrng chufin r SO : true h i n h non r Dp dai SO : chieu cao h cua h i n h non r K h i M chay k h i i p difcing t r o n (O; R) t h i S M tao t h a n h mot h i n h goi la m a t non va doan S M dUcfc goi la dudng sinh / cua m a t non hay h i n h non r Dadng t r o n (O; R) : la di/crng t r o n chuan day h i n h non n DN2 : T h i e t d i § n q u a t r u e c i i a h i n h n o n l a t h i e t d i ^ n t a o n e n b d i m p t m a t di q u a t r u e e u a h i n h n o n phftng • T i n h c h a t : Cac t h i e t dien qua true eua mot h i n h non la nhCfng tarn giac can b k n g n DN3 : T h i e t d i ^ n v u o n g g o c v Ngoai sU tUcfng giao mSt non v d i mot mSt p h i n g cho t a eac Conic DN4 : M p t h i n h chop goi la npi tiep mpt h i n h chop k h i h i n h c h o p co d i n h t r i i n g vdri d i n h c u a h i n h n o n v a c o d a g i a c d a y n p i t i e p t r o n g d a y c u a h i n h non due ta noi h i n h non ngoai t i e p h i n h chop do) • T i n h c h a t : H i n h chop n p i t i e p t r o n g mpt h i n h n o n t h i co dudng cao b a n g diidng cao cua h i n h n6n va co cac canh ben la nhflng dadng s i n h ciia h i n h non Q DN5 : M p t h i n h c h o p g o i l a n g o a i t i e p m p t h i n h n o n k h i h i n h c h o p c o d i n h t r u n g vdfi d i n h c i i a h i n h n o n v a c o d a g i a c n g o a i t i e p d a y c u a h i n h n o n (luc ta noi h i n h non n p i t i e p t r o n g h i n h chop do) • T i n h c h a t : H i n h chop ngoai t i e p m p t h i n h non t h i co m a t ben n a m t r o n g nhOfng t i e p dien ciia m a t non • T i n h c h a t : D i e n t i e h xung quanh cua h i n h non : « T i n h c h a t : The t i e h ciia h i n h n6n : V = ^ jtR^h 269 L o a l I: Htefl NON CVT L G I A N L i r p C K i t N THli'C T & I T H i f u • D N i : H i n h n o n c u t l a m g t p h a n h i n h n o n dvlffc g i d i h a n boTi m S t d a y v a mOt thiet d i ^ n v u o n g g o c v d i t r u e (hay song song vori ddy) • Cho h i n h n o n c6 day l a h i n h t r o n C ( ; R) va true SO Goi C'(0'; R) l a m o t t h i e t d i $ n song song v d i ddy • Xet h i n h n o n cut la p h a n h i n h n n g i d i h a n b d i C ( ; R) va C'(0'; R ) , t a c6 : ^ R' R SO' SO L a y m p t d i e m M t r e n dtfcfng t r o n (O; R) va S M cSt ducrng t r o n (O'; R') t a i M ' t a c6 MM' * Chieu cao OO' : => h = d[(a); (P)] * OO' : true * H i n h t r o n C ( ' ; R ) : day nho * H i n h t r o n C ( ; R) : ddy I d n * la dudng s i n h / K h i M chay t r e n dUdng t r n (O; R) t h i M M ' tao t h a n h m a t non cut • DN2 : Txidng txi h i n h n o n T h i e t d i # n q u a t r u e e u a h i n h n o n e u t l a nhutng hinh thang c a n bling • T i n h c h a t : D i e n t i c h x u n g q u a n h h i n h non cut : [ S,g = 7i(R + R')/ • T i n h c h a t : T h e t i c h h i n h ch6p cut : V - -^TthCR^ + R ' ^ 4^ R R ' ) n GiAi T O A N T H I B a i 373 ( D A I H O C M I E N B A G - 1970) M o t m a t p h i n g d i qua h a i difdng s i n h M P , N Q ciia mot h i n h non cut cat cac difdng tron day t h a n h nhQng cung 6O", cac d i e m M , N n k m t r e n dUdng t r o n day nho T i n h d i e n t i c h xung quanh va the t i c h cua h i n h n o n cut b i e t chieu dai du&ng s i n h l a Giai M a t p h a n g d i qua h a i dUcmg s i n h M P , N Q cat h i n h non cut theo t i e t d i e n l a h i n h t h a n g can M N P Q M P ^ = PfQP = P , P M = N Q = / Goi OO', R, r' l a t a r n va ban k i n h t u o n g iifng cua day Idn va day nho cua h i n h n o n cut Theo gia t h i e t t a c6 : MO^ = " va P O ^ = ° jMM = NO' ^ |PQ = Q = R Goi r, I la t r u n g d i e m theo thtif t i f ciia M N va PQ, t a c6 : ro' = 10 = 270 TO r dudng I H 01, the thi : h = 0 = I H va I H = I O - H O = ^ - I ^ = ^ ( R - r ) 2 Vay dien ti'ch xung quanh S^, va the tich V cua hinh n6n cut 1^: 'S,q = (R + 7t V = — (R2 (1) T)l + + Rr) (2) Dinh ly h^m so sin APNQ, ta c6 : PQ PN ^ sin a sinp NQ sinrrt - ( a + P)l /sina = PQ R PN = s i n ( a + P) /sinp s i n ( a + P) Ha difdng cao N K ta c6 : I ' l = N K = NQsinP = /sinP => K Q = /cosp = - (PQ - MN) = - (R - r) 2 => PK = PNcos[7t - ( a + P)] = ^^'"^ cos(a + p) s i n ( a + P) = -Zsinpcot(a + p) = - i ( P Q + M N ) = - (R + r) 2 J R - r = 2/cosP ^ [ R + r = - Z s i n p c o t g ( a + p) (3) => R r = Z^sin^p.cot''*(a + P) - Z ^ s ^ P Ma + r'^ + R r = (R + r)^ - R r => R^ + r^ = 4/^sin^Pcot^(a + p) - Z^sin^Pcot^(a + P) + Z^cos^p => R^ + r^ = Z^[3sin^Pcot^(a + P) + cos^p] (4) Xet tarn giac vuong I ' I H , ta c6 : Z^sin^p- h = zVsin^P-3cos^P V3 —2Zcosp = Z -Jcsin p + Vs cos P X s i n p - V3 cos p) h = / Jl sinp + tan—cosp sinp - tan—cosP h = I 71 ' sm sin = 21 sm sin (5) cos— The (3) vao (1) ta c6 : S,,,, = -27i/^sinpcot(a + P) Vi > nen cot(a + p) < 0, tiJcIa — < x + p < 7t 271 The' (4) va (5) vao (2) va v6i dieu k i e n : — < x + p < 271/ V = Jsin sin B a i 374 ( D A I H O C K H O I A M I E N B A G ~ thi : [3sin'^Pcot^(a + P) + cos^p] (ycbt) 1974) M o t h i n h non t r o n xoay c6 ban k i n h day R va dUcfng cao h T r o n g t a t ca cac m a t phSng di qua d i n h h i n h n o n , hay xac d i n h m a t phSng cSt h i n h non theo t i e t d i e n c6 dien t i c h Idn nhat va hay t i n h d i e n t i c h ay Giai Gia siJf m a t phfing d i qua d i n h G cua h i n h non cat h i n h non theo t h i e t dien GAB The t h i G A B la m o t t a m giac can vcri CA = CB Goi O la t a m h i n h t r o n day va H la t r u n g d i e m cua A B CO ± (ABC) |CH : d u d n g x i e n O H : la h i n h chieu M a A B O H =^ A B G H D a t : X = A H = - A B t h i d i e n t i c h S ciia ACAB la : S = - AB.CH = x.CH Nhung : G H ' = CO' + O H ' = CO' + OA' - A H ' = h ' + R ' - x' => S = S(x) = X Vh-^ + - D K xac d i n h A B : 2x < 2R x < R D a t : t = x ' ; Vx e [0; R] ^ < t < R ' -•^ S' = x ' ( h ' + R ' - x ' ) g(t) = t ( h ' + I I ' - t ) Ta v i e t : g(t) = - t ' + ( h ' + R ' ) t ; vdi < t < R ' N h a n t h a y t h i ciia g(x) c6 dang n h u sau, theo h > R hay h < R • T H i : Neu h > R, t h i S' = g(t) (0 t s: R ' ) dat gia t r i Idn n h a t k h i t = R ' , Vay S dat gia tri lorn n h a t k h i x = R ( A B la m o t dUdng k i n h day), va t a c6 ; maxS = Rh • T H : Neu h =S R, t h i S' = g(t) (0 < t < R ' ) dat gia t r i Idn n h a t k h i : t= -(h^ +R'^) R' (ycbt) maxS = — ( h ^ + R ) tiicrng Ofng x = ) B a i 375 ( D A I H O C K Y T H U A T P I I U T H O vA D A I H O C K I E N T R U C - 1977) Cho m o t h i n h non cut t r o n xoay c6 chieu cao h , cac ban k i n h day la r va R (r < R) Tim k i c h thiTdc ciia h i n h t r u t r o n xoay c6 cQng true doi xijfng, n p i t i e p t r o n g h i n h non cut va c6 the t i c h I d n n h a t Giai Goi X la ban k i n h , z la chieu cao cua h i n h t r u T a c6: r ^ X < R ^ z < h 272 Gia sijf r k n g h i n h t r u noi tiep t r o n g h i n h non cut n h u t h i e t dien qua true nhir h i n h ben T h i e t dien cSt h i n h non theo hinh thang can ABB'A, cftt h i n h t r u theo h i n h chOf nhat: H K N M SO' O;A; so " OA ~ R SO' r SO-SO' R - r rh SO' = R- r Ma ; "so SO' OO' SO = OiM ^ SOi OA s, T_ rh + h = R - r _x SO2 * * O A " SO R ( S O - z) SOi = SO - z => X = SO The tich V h i n h t r u la : V = V(x) = nx'z = ~ ( S O ~zfz SO^ = ^ so^ TTR V(x) = (SO^ + - SOz)z ^ ( z ^ - S z ^ + S O ^ z ) (0 < z < h) SO^ =^ V ( x ) = - ~ ( z - S z + S O ^ ) SO^ => V'(x) = « z = SO Bang bien t h i e n : X z = SO = so-^ z = z - SO^ Rh Rh (R-r) Rh 3(R - r ) 3(R - r ) V (z - S O ) z = Rh (R-r) h V(x) Bmaxy CD CD V(x) z = Rh X 3(R - r ) z = h => De y r k n g : < z < h , ta c6 ; z = X = = 2R r Rh 3(R - r ) < h — < — R Ketluln : r „ • • — < — : the t i c h ciia h i n h t r u Idn n h a t k h i h i n h t r u c6 k i c h thifdc ban k i n h ddy: x = R Rh chieu cao z = 3(R - r ) r • - ^ — < 1, h i n h t r u c6 the t i c h I d n n h a t k h i h i n h t r u c6 k i c h thudc ban k i n h day R ' va chieu cao z = h Chuyen de 19 : HINH L G I A M L i r p C K i t N TBdC • cAu - T&I CH6M CAU - QUAT cAu TBxto D N i : M a t c a u ( S ) l a tlip h S = IcD.BH < CD.BO' 2 (4) Dau dang thiJc (4) xay o BH = BO' H = O' maxS = - CD.BO' Tifong ufng : CD A C (ycbt) c/ Do AB (ACD) va BH CD nen CD A H (dinh ly ba dUdng vuong goc) => H nam tren difdng tr6n dirdng kinh AO' Dao lai, lay H' la mpt diem y tren difdng tron difdng kinh AO', noi H ' vdi A, B va O; keo dai H'O cat (L) tai C va D' la dadng kinh ciia (L) nen tam giac C A D ' vuong tai A 275 => C A I T la mot vi t r i cua g6c xAy cSt (L) d C va D', v i AB vuong goc vdi (AC'D) va A H ' ± C D ' nen theo dinh ly ba difdng vuong goc ta c6 BH' CD', tuTc la H' l£i hinh chieu cua B len diicfng thang CD' Vay quy tich cua hinh chieu H cua B len dUdng t h i n g CD k h i goc xAy quay quanh diem A la dacfng tron difcrng kinh AO' co dinh (do A ya O' co dinh) (ycbt) B a i 377 (DAI HOC K I E N TRUC TP.HCM - 1991) Cho mot hinh cau ban kinh R, (L) la giao tuyen cua mot mat p h l n g (P) each tam mat cau mot khoang each b^ng h (0 < h < R) A la mot diem co dinh tren (L) Mot goc vuong xAy mat p h l n g (P) quay quanh A : Cac canh A X , A Y c i t (L) d C, D DUcrng t h i n g di qua A vuong goc v C D ^ = A D ^ + A C ^ => A B ' + C D ' = BC' + A D ' Tif (1) & (2) (2) (3) A C ' + B D ' = AB'+ C D ' Mat khac, goi la tam cua difcrng tron (L) t h i : CD = 2AI = V R ^ - h ^ va AB = 201 = 2h => C D ' + A B ' = ( R ' - h') + h ' = R ' (const) (3) => BC' + A D ' = B D ' + A C ' = R ' (const) k h i goc xAy quay quanh A (ycbt) 2/ H a B H C D Vi A H CD BAl(ACD) Do CD = - h ^ (const) 3maxdt(BCD) 3maxBH = B I AH = AI « A I DC Vay : CD ± A I t h i dt(BCD) Idn nhat Vi BI = V A B ^ + A I ^ = V h + R - h = V R ' + 276 Nen 3maxdt(BCD) = - C D B I = V c R ^ ^ - h ^ X R ^ ^ ^ S l T ) 3/ H n h i n d o a n A I co d i n h dudi m o t goc vuong n§n (ycbt) H t h u o c d i / d n g t r n difdng k i n h AI m S t p h i n g (P) co d i n h Dao l a i , l a y H ' t u y y t r e n dUcfng t r o n dtforng k i n h A I Difdng t h ^ n g H ' l gftp ( L ) t a i C D ' Lai CO : AB (AC'D') AH' CD' va CD' l a m o t dUdng k i n h c u a ( L ) => B H ' C D ' => H l a h i n h chi§'u cua B tr§n C D ' Vi goc x A y lay t a t ca cdc v i t r i t r o n g mp(P) k h i quay quanh A , nen H ve ca dUdng t r o n di/dng k i n h A I Vay quy t i c h cua H h i n h chieu cua B l e n C D , la dUdng t r o n difdng k i n h A I (vdi I la t a m cua (L) nam t r o n g mSt p h ^ n g co d i n h (P)) (ycbt) Bai 378 ( D A I H O C K I E N T R U C - 1993) Cho mot m a t cau (S) t a m O t i e p xuc vdi m a t p h ^ n g (P) t a i I Ggi M la m o t d i e m d i dong tren (S) H a i t i e p tuyen cua (S) t a i M dt (?) d A va B 1/ Chufng m i n h r a n g A M B = Xlfe 21 r la diem doi xiitng cua I qua A B Chufng m i n h r a n g bon d i e m I , I ' , M , O cung n a m t r e n mat phang va I ' M d i qua m o t d i e m co d i n h J thuoc (S) 3/ Cho M d i d p n g t r e n (S) cho A M B = — , cac d i e m A ; B l i n lucft chay t r e n h a i dUcfng thSng d, d' n a m t r o n g (P) va d, d' vuong g6c v d i t a i K H a y chufng to r a n g k h i m a t cau dUdng k i n h A B luon chiira mot dUcrng t r o n co d i n h , I ' chay t r e n m o t dUcfng t h i n g co d i n h , va M di dong t r e n m o t dtfdng t r o n co d i n h Giai J 1/ Ta CO : A I t i e p x i i c (S) t a i I => A M t i e p x u c (S) t a i M => O i l l A : AOIA vuong O M A M : A O M A vuong Nhifng : = O M = R => A O I A = A O M A => A I = A M 277 2/ T a a n g t i f => B M = B I V a y : A M A B = A I A B => X l ^ = AMB X e t : ( S ) t i e p x u c ( P ) t a i I => O I ( P ) Khi : M B tiep xiic (S) M A tiep xiic (S) (dpcm) :=> 1 A B (1); 01 A B AB ir (1) (MAB) tiep x u c (S) => M ( M A B ) = > Mat k h a c : TU MO-LAB A B ( O I D =^ o r A B ( ) v a ( ) => B o n d i e m I ; I ' ; M ; O c u n g (2) (3) nam t r e n m o t m a t p h a n g q u a O v u o n g goc v d i A B ( y c b t ) Vdri J l a g i a o d i e m cua (S) va M I ' ; g o i E l a t r u n g d i e m ciia i r T a CO : ( M A B ) t i e p x u c ( S ) => M E t i e p x u c ( S ) => M E O M , h a y A O M E v u o n g t a i M ^ AOME = AOIE ^ Hay ME = ir M E = IE A I ' M I vuong t a i M V a y J d o i x u t n g vdi I q u a O , h a y M I ' q u a d i e m J co d i n h t h u o c ( S ) 3/ (dpcm) D e y d e n m a t c a u diTcfng k i n h A B l u o n chijfa dUcfng t r n co d i n h diTcmg k i n h K I nam t r o n g mat p h & n g v u o n g goc vdri ( P ) K h i r l i A i d o n g t r e n d i f d n g thang K x c h o d ' l a p h a n giac cua M d i d o n g t r e n difcfng t r o n co d i n h l a g i a o d i e m cua ( S ) va mat phang ( J K I ) (dpcm) 278 Chuyen de 20 : PHOI HOP C A C KHOI HINH H O C KH6NG GIAN LpmroNGPHAp Ca SOT cua phi/cJng phap Ik sit dung cac dinh nghia ve sir npi tiep, ngoai tiep giOa hai kh6'i hoc khac va siT ti/Ong giao giifa cac khoi vdi Ch^ng han hinh cau noi tiep h i n h non, hinh tru n6i tiep hinh cau, phan chung cua hai hinh chop, qua hai budrc ca ban : hinh • Bi : Phoi hop dinh nghia va quan ly gia thie't • Ba : Pho'i hop cac cong thuTc the tich, dien tich cua cac hinh khoi, sau ket hop vdi cAc dinh ly hinh hoc so cap de tim cac quan he lien thuoc : giila dien tich, the tich, dudng cao ban kinh, diTdng sinh, canh ben, canh day, cac loai goc, n cAc BAI TOAN C O BAN Bai 379 Tim the tich cua mot hinh iSng tru dufng c6 day la hinh thoi ma goc nhon la a, ngoai tiep mat cau CO the tich bang V Giai De y thay hinh la mot dudng tron Dudng tron c6 day trung vdi tam mat cau chieu cua mSt cau tren day hinh iSng tru noi tiep hinh thoi a day hinh iSng tru tarn la hinh chieu ciia tarn hinh cau tren hinh thoi va c6 ban kinh bSng bdn kinh BV- - - - ->C' • < Goi ban kinh mftt cau la r, ta c6 : A' ; ; • o ih ; \\^.1 J -\ i / J i V = — T t r ' => r = -M \1 Dung DI AB ^ DI = 20H = 2r c °' = sin a sin a Ducfng cao lang tru cung bSng dudng kinh hinh cau : h = 2r ' ADA! (i = 90 Vi, = — — sin u 6V , —• (ycbt) n sm a B Bai 380 Tim ban kinh hinh cau tiep xiic vdri tat ca cac canh cua tiJ dien deu canh a Giai De y thay hinh cau tiep xiic vdi cac canh ciia tuf dien deu n^y se noi tiep hinh lap phuong, tiJc la tiep xiic vdi cac mat cua hinh lap phuong tai tam ciia mat (tam hinh vuong) => Cac tiep diem la trung diem cac canh tuf dien deu => Duorng kinh hinh cau bang canh hinh lap phuang c6 dudng cheo bSng a (canh tuf dien deu) 279