Tuyển tập 500 bài toán hình không gian chọn lọc phần 1 nguyễn đức đồng

516.23076 BAN T527T GI AO VIEN N AN G N J GU Y EN K H I EU TRUCiNG T H I eC/ C e O N G ( C hu b ie n) PHAN LOAI VA PHl/ OfNG PHAP GIAITHEO CHUYEN DE • BOI Dl/ dNG HQC SINH GIOI • CHUAN B! THI TU TAI, DAI HOC VA CAO D AN G BOG Ha NOI NHA XUATBAN OAI HO C Q UO C G IA HA NOI BAN GI AO V I E N NANG K H I E U TRl/CfNG T H I NGUY EN DLfC D N G {Ch u hien) TUYEN TAP 500 BAITOAN • HDIH i mm GI AN C H O N LOG • • • PHAN LOAI VA PHU dNG PHAP G IAI THEO CHU YEN • B oi difdng hoc s inh gioi • C h u a n b i t h i Tii ta i, D a i hoc va Cao da ng (Tdi ban idn thvt ba, c6 svCa chUa bo sung) THir ViEN TiiVH BiKH liik^m NHA X UA T BA N DA I H O C QUOC GI A 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 t r d ch ** n hiem xu a t ban: d oc - Tong bien tap: T S P H A M T H I T R A M tap: TH UY HOA bdi: TH AI Che ban : Trinh VAN N h a s a ch H O N G A N bay bia: TH AI V A N SACH LI E N 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 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 • [ ( A B C ) ; ( E F G ) ] : goc tao bori mp ( A B C ) va ( E F G ) (il • => : (i) keo theo • : k h o n g tUdng dilcfng • d> : k h o n g keo theo • = : dong n h a t -> • C > : Phep t i n h tien vectcf v V • D A : Phep doi xOmg true A • Do : Phep doi xiiTng true : 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 • Q( ; cp) : Phep quay t a m O, goc quay (p • V T ( ; k ) : Phep v i t u t a m , t i so k • V s AHc = V ( S A B C ) : the t i c h h i n h chop • D N : dinh nghla S.ABC • H Q : he qua • Sxq : D i e n t i c h xung quanh • D L : dinh ly • Stp : D i e n t i c h t o a n p h a n • B i : budc i • A ' = ''7(ai A : A ' la h i n h chieu ciia A • C M R : chiJng m i n h r i n g • V : The t i c h xuong m a t p h i n g (a) • A ' = ''Vfd) A : A ' l a h i n h chieu cua • T H i : t r u d n g hop i A • V T : ve t r a i xuong dtfcfng thftng (d) • 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 • 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 den mat phang ( A B C ) • d p c m : dieu p h a i chuCng m i n h • (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 • (3r 3^ : goc tao bdi h a i dUomg t h i n g d • 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 t r vC6c 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 • Mu on xac d in h n h ^ n h mot ma t p h ^ n g tr on g kh on g gian ta chon thu thu at thUc h a n h : M p t h i n h t a m g i a c , t ii" g i a c h o a c d a g i a c p h &n g ( 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 g p i c a c m& t p h ^ n g l a m ^i t 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 t hd'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 d o k h o n g l a b i e n c u a m a t p h d n g bi k h u a t d o , t h i d i ^d n g t h &n g d o c u n g tii'oTng vlng k h u a t c u e bp h a y t o a n bp No i h a i d i e m m a i t n h a t c m p t d i e m k h u a t t h i dUpc m pt dUcfng k h u a t c ue bp h a y • M p t d i e m nhm t r o n g m p t m $ t ph&ng h i n h thuTc bi k h u a t t h i g o i l a d i e m k h ua t • t o a n bp : n e u h a i di i c t ag k h o n g l a b i e n c u a c a c m ^t phA ng 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 kh u a t cue bo, (d) c6 doan ve net dijft doan n k m du di (a) S • (d) b i ma t p h ^ n g (SAC) che kh u a t cue bo, (d) CO mpt doan ve duft doan n k m sau (SAC) (hien n h ien (d) cu ng d sau cac ma t (SAB ), (SBC)) • C a nh AC b i h a i ma t pha ng (SBC) v£l (SBC) che kh u a t toan bo, ca doan AC xem n h u hoan toa n d sau dong th d i h a i ma t p h ^ n g (SAB ), (SBC) -A A c./—1 —^VF J L ^ • • A ] H b i che toa n bo ca doan A ] H n k m sau ma t p h i n g ( AiAD 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 Ch o tiif gia c l o i A BC D c6 cac c a n h d o i k h o n g so n g so n g va d i e m S d n g o a i ( ABCD ) T i m giac t u ye n ciia : a/ ( SAC) va ( SBD ) hi ( SAB) va ( SD C) ; ( SAD ) va ( SBC) Giai a/ Xe t h a i m a t p h a n g ( SAC) va ( SBD ) , t a c6 : • • S l a d i e m c h u n g th uf n h a t (1) T r o n g tuT gia c l o i ABC D , h a i d u cm g ch eo AC n BD = O : d i e m c h u n g t h ijf n h i (2) ^ Ti/ (1) va (2) su y r a : ( SAC) o ( SBD ) = SO ( yc b t ) hi Xe t h a i m a t p h a n g ( SAB) va ( SD C) c u n g c6 : H a i c a n h b e n A B va CD cu a t i l gia c ABC D • S la m o t d i e m c h u n g • t h eo gia t h i e t k h o n g so n g so n g ^ AB ^ C D = E : l a d i e m c h u n g th ut h a i Do : ( SAB) n ( SD C) = SE ( yc b t ) Tu cfn g t i f : ( SAD ) n ( SBC) = SF ( yc b t ) ; vd i F = AD ^ BC; A D / / BC Bai Ch o t i l d i e n ABC D Go i G j , Ga l a t r p n g t a r n h a i t a m giac BCD va AC D La y t h e o thuT t i i I , J, K l a t r u n g d i e m ciia BD , A D , C D T i m cac gia c t u ye n : ( ABK) ^ ( CI J) = G,G2 d ( GiGa B) n ( ACD ) = GgK h oSc A K hi (G1G2C) n ( ABD ) = I J a/ (G1G2C) o ( AD B) aJ hi (G1G2B) n ( ACD ) c/ ( ABK) o (CIJ> Bai Ch o h i n h ch o p S ABCD c6 d a y ABC D l a h i n h b i n h h a n h t a m O T i m gia o t u ye n cu a h a i m St p h i n g ( SAB) va ( SCD ) hi T i m gia o t u ye n cu a h a i m a t p h Sn g ( SAD ) va ( SBC) aJ c/ T i m gia o t u ye n ciia h a i m a t p h ^ n g ( SAC) va ( SBD ) Giai aJ Xe t h a i m a t p h Sn g ( SAD ) va ( SBC) , t a c6 : De y A D c ( SAD ) ; BC c ( SBC) m a A D // BC • S l a d i e m c h u n g thur n h a t • Ta d u n g xSy // A D h oac BC [(SAD) = (xSy; AD) ^ | (SBC) = (xSy; BC) =^ ( SAD ) n ( SBC) = xSy ( yc b t ) hi Tifa n g t i r , d ifn g u Sv // A B h oft c C D => ( SAB) r ^ ( SCD ) = u Sv ( ycb t ) c/ Go i O = AC n B D , tiTcrng t a b a i => ( SAC) n ( SBD ) = SO ( ycb t ) Bai Ch o h i n h ch o p S ABCD c6 d a y la h i n h t h a n g AB C D v d i A B l a d a y Idtn G p i M la m o t d i e m b a t k y t r e n SD va E F l a d ifa n g t r u n g b i n h cu a h i n h t h a n g a/ T i m gia o t u ye n ciia h a i m St p h i n g ( SAB) va ( SCD) b/ T i m gia o t u ye n cu a h a i m a t p h S n g ( SAD ) va ( SBC) , c/ T i m gia o t u y e n cu a h a i m St p h a n g ( M E F ) va ( M AB ) Doc gia t u g i a i tUcfn g t u n h u cac b a i t r e n Bai Ch o h i n h ch o p S ABCD c6 AB C D l a h i n h b i n h h a n h Go i G, , G2 l a t r o n g t a m cac t a m gia c SAD ; SBC T i m gia o t u y e n cu a cac cSp m St p h a n g : a/ (SGiG^ ) va ( AB CD ) b/ ( CD Gi Gz) va ( S AB) Uv Cd n g 0/ ( AD G2 ) va ( SBC) d§Ln Go i I , J , E , F thur t a Ik t r u n g d i e m cac d o a n t h i n g AD , BC, SA, SB t h e o thur tvt d Th ifc h i e n cac l a p l u a n n h t f cac bai toan t r e n ; a/ (SG1G2) n ( ABCD ) = I J ( ycb t ) b/ ( CD GiGa ) n ( SAB) = E F ( ycb t ) c/ ( ADG2 ) ^ ( SBC) = xG2 y ( ycb t ) T r o n g xGay // A D h oSc 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 P H i r ON G PH ANG P H AP Ca sd cua p h a a n g p h a p t i m gia o d i e m O cu a d u d n g t h a n g (a ) va m a t p h Sn g ( a ) l a x e t h a i k h a n S n g xa y r a : n T r i r d n g h o p ( a ) ch iJ a d u d n g t h S n g ( b ) va (b ) l a i c&t d iicr n g t h d n g (a ) t a i O T i m O = (a ) n ( b ) => O la d i e m ca n t i m n T r t fd n g h a p ( a ) k h o n g chiifa dUcm g t h i n g n a o ca t (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 ca n t i m n CAC BAI TOAM G O B A N Bai Ch o tuf d i e n AB C D Go i M , N I a n l u g t la t r u n g d i e m cua AC va BC L a y d i e m K e B D ch o K B > K D T i m gia o d i e m ciia h a i d u d n g t h i n g CD va A D v d i ( M N K ) Giai 1/ Ha A H SB t a i H (1) De y t h a y : B C I A B (AABC vuong can t a i B) [ B C J_ S B ( d i n h l y b a d u d n g v u o n g goc) => B C K S A B ) Ma : A H c (SAB) => BC A H (2) TO (1) va (2) => A H _L (SBC) => A H = d [ A ; (SBC)] Ta c6: 1 • AB-^ • +SA^ (aV2)2 ,2 AH^ ^ 2a' « A H ^ = ^ c A H = ^ Vay : d[A; (SBC)] = 21 TO A H (SBC) (ycbt) => ( A H C ) (SBC) theo giao t u y e n H C Goi K l a t r u n g d i e m C H OK = A H ma O K (SBC) (ycbt) Vay : d [ ; (SBC)] = O K = DE Tl/dNG T i ; Bai 156 ( D A I H O C Q U O C G I A H A N O I - K H O I G - 9 ) Trong m a t phftng (P) cho d i e m O va m o t ducfng t h S n g d each O m o t k h o a n g O H = h L a y tren d h a i d i e m khac n h a u B , C cho g o H - COfi = 30° T r e n ducfng t h i n g vuong goc v d i P tai lay d i e m A cho OA = O B a/ T i n h the t i c h tuT dien O A B C b/ T i n h k h o a n g each tir O den m p ( A B C ) theo h HvCdng d&n b/ = 2^ a/ VoABc = Vv • Bai 157 ( C A O D A N G S i / P H A M H A N O I - K H O I d - 1997) Cho m a t phSng (P) va h a i d i e m A , B doi x i i n g n h a u qua (P) I l a giao d i e m cua A B v d i (P), la mot d i e m ngoai (P), c6 h i n h ehieu vuong goc xuong (P) l a H M l a m o t d i e m t u y y t r e n di/dng t r o n dUdng k i n h H I t r o n g (P) Ch\Jng m i n h : a/ M I la k h o a n g each vuong goc ehung cua B A va M O b/ Khoang each tiT O va H den mSt p h l n g ( B A M ) bSng T i n h t h e t i c h tuT d i e n B O M A biet BA = 2a; H M = b; M I = e Htfdng dan a/ [HM M I M i l I la trung diem A B b/ V = -HM.-AB.MI MO AB MI = M I l a dudng vuong goe ehung A B va M O -abc 85 L o al Z : DOAN WdNG G6 C CHUNG CCA HAI DUCJNG THANG CfltO NHAU L PHUONG PHAP, Ca sa cua phuang phap de timdoan vuong goc chung cua hai dudng thing cheo (a), (b) la : sijf dung dinh nghia doan vuong g6c chung qua budc ca ban nhif sau : • B i : Chon A e (a), B e (b); (a) va (b) cheo Sau d6 ta chufng minh : AB1 (a) va AB1 (b) • B2 : Ket luan AB la difcing vuong goc chung cua dUcfng cheo (a) v^ (b) d o an ch u n g O Ghi chii : Do ddi doan (duang) vuong goc chung cua hai duang cheo la khodng each ngdn nhdt giOa hai dUang thdng n PHiroNG PHAP, Co sd ciia phiJang phap la silf dung each diftig thuf I Do dai dtfcfng vuong goc chung la khoang each tif mot diem tren dacfng thang thil den mot mat phing chufa dUcfng thing thil II va song song w6\g thing thuf I, qua bifofc cOban sau: • B | : Trong mat (a) da chufa b difng a' // a va B = a' b • B2: Do (a; a') xdc dinh mp(p), nen dUng mp((3) doan BA (a) va A (a) • B : Ket luan doan vuong goc chung la AB=d[(a); (b)] (ycbt) O Ghi chii : AB = AB' Id cdc ddi doan vuong goc chung m PHtfONG PHAP3 Co so cua phuang phap la suf dung each diTng thuf II (a), (b) cheo (trUdng hop tong quat ehi suf dung (a), (b) cheo va (a) J (b) chifa tim difoc phifong an xuf ly cac trudng hop (trifdng hop dac biet) khae da biet tii trUdrc) d o an c h u n g d o an -1 c h u n g Chung ta can thuc hien c^c bifdc : • Dirng (a) (a) tai Ava (a) (b) Dung AB (b) (B e (b)) => AB la doan vuong goc chung cua (a) va (b) O Ghi chill : Khi dang ndy khong khd thi thi ta mai sii dung cdc dang khdc Chung ta can thifc hien cac bUcJc: • Difng (a) J (a) tai A' va chieu (b) xuong (a) la (b') • Dung A'B' (b') (B' e (b')) • Dung B'B// (a) (B e (b)) • Dung BA// B'A' (A G (a)) => AB la doan vuong g6c chung cua dUcfng thing cheo (a), (b) 86 Ghi chu2 : U day hinh tricang hap dinh chieu goc la doan vuong goc O ly duang vuong G h i chii : Do AB = A'B' (nen thUc hanh hay tinh dai A'B' cho dan gidn phep ta tinh) chung IV c A c B A I T O A M C O B A M Bai 158 Cho tijf dien deu A B C D canh a Xac d i n h va t i n h dai doan vuong goc chung cua A B va CD Giai Goi I , J l a t r u n g d i e m A B v a C D theo thuf t u do, va theo t i n h chat cua tiJ dien deu t h i cac m a t ben ciia no se l a c i c tarn gi&c deu ^ { => C D ( A B J ) IJ ; CDJ_AJ => CD I J b a n nOa A B I J ( A A J B can t a i J ) ^ I J A B va CD Vay I J la doan vuong goc c h u n g ciia A B va C D (dpcm) Taco : i I J = -/ ' ra> 2a^ > aV2 (yebt) Bai 159 Cho h i n h chop S.ABCD c6 SA ( A B C D ) l a h i n h c h i i n h a t D a n g doan vuong goc chung cua SA va CD Giai 'CD c (ABCD) De y : < SAl(ABCD) A D la doan v u o n g goc chung cua SA va C D (ycbt) Bai 160 Cho h i n h chop S.ABCD day A B C D l a h i n h t h a n g vuong t a i A va B A B = a; B C = a; A D = 3a; CD = aVv va SA = aV2 K h i SA ( A B C D ) , hay difng va t i n h d a i dacfng v u o n g goc chung ciia cac cap dacfng t h i n g : a/ SA va CD b/ A B va SD c/ A D va SC Gi&i a/ De y t h a y t r o n g A A C D : iAC^ CD^ t V^ f ^V7f = 9a^ ^ ^ ^C^ + CD^ « A A C D (C = 90«) ; A D ^ = (3a)''^ - 9a^ Luc A C = aV2 l a dadng v u o n g goc chung cua SA va C D (ycbt) b/ Ta CO Dong : A B (SAD) SD => A B SD A H SD t a i D ( t h i A H l a h i n h chieu) A H la doan v u o n g goc chung cua A B va SD 87 c/ A H^ SA " 18a^ 11 AD' 2a ^ AH = ^ 9a' 18a' (ycbt) 11 De y thay A D (SBC) SB ma SB la h i n h chieu cua SC xuo ng (SA B) ( v i CB (SA B)) iter A doan A K J SB ; K e SB Dirng : \tiiK doan KE // BC ;E e SC tii E doan EF // KA ; F e A D Theo each d ifng th i l I I => EF la d o an v uo ng goc chung ciia A D v a SC, ve dp d EF = A K, ta c6: 1 A K' SA^ A B'' Vay EF = A K = aj | - « A K^ = 2a a 2a' ^ (y cbt) O C ach : De y thay : f va (SCB) (SA B) theo giao en SB [(SBC) // A D Difng A K SB => A K la dp d d o an v uo ng goc chung cua A D va SC 'KE/ / BC ' ^ EF/ / A K : E c SC ; F e A D EF la d o an v uo ng goc chung ciia A D va SC v a EF = A K O -4 G h i chu: Neu gid thiet chi yeu cdu tinh ddi doan vuong goc chung cua AD vd SC thi Cdch to tien Igi han Cdch Bai 161 Cho h i n h iSng tr u A BC.A ' B' C cac m at ben deu la cac h i n h v uo ng canh a Xac d i nh v a t i n h d d o an v uo ng goc chung ciia A ' B v a B' C 2/ H i n h Iftng tr u ay c6 dAc d iem gi ? 1/ G i 1/ Do tat ca cac mat ben cua h i n h ISng tru deu la cac h i n h v uo ng bSng nen hai day cua h i n h lang tru la hai tam giac deu lAA'^ AB M at khac A A ' ( A BC) [A A '^ A C V ay lang tr u d g ia th i e t la m o t Iftng tr u tam giac deu, canh d ay bang a v a chieu cao cung bang a 2/ Gp i I v a r Ian luot la tru n g d iem ciia BC v a B' C De y thay B' C ( A TI) chijfa A ' l la h i n h chieu ciia A ' B xuo ng mat phang Dung EF / / I H ' (F e • Dung H E / / BC (E G A ' B) • Dang I' H ± l A ' ( H c l A ) • B' C) EF c hi nh la d o an v uo ng goc chung cua A ' B v a B' C • • De thay la : B C l ( A A TI) => B' C ( A A TI) => B' C ± I H B C EF ( tai F) Goc A E F CO EF / / ( A A T I) , c6 h i n h chieu xuo ng mat phang ( A A T I) la goc v uo ng A ' H I' nen c hinh goc A ' EF la goc v uo ng 88 = 90" => A'B E F (ta i E) Qua each diTng ta th a y E H I T la h i n h chO n h a t => E F = I' H Tarn giac Il' A' vu ong d I ' : Trong : 1 I'H'' IT I ' A' IT = a va I'A' = aj3 I'H ' 1_ ^ 3a2 ^ 3a2 E F = d[(A'B ); (B 'C )] = ( y c bt ) Bai 162 Goi d, va d la i dUdng thS ng cheo nha u va vu ong goc nha u n h a n A B = a l a m doan vuong goc chu ng (A e d,; B s d ) G oi O la tr u n g d iem cua AB , (r) la difdng tr on ta m O b an kinh R n&m tr on g ma t tr u n g trUc (P) cua AB , M la mot d iem thuoc (r) 1/ Chufng h ton ta i n h a t mot doan t h i n g d qua M c&t ca h a i dU dng t h i n g d i va d 21 Goi E; F Ian lugt la giao diem cua d vdi d i va d Dat AE = x; B F = y C hiing m i n h r i n g : + la mot dai li/cfng kh on g tha y doi k h i M di dong tr en (r) Gi a i 1/ (P) la mat p h i n g tr u n g trUc cua A B cho nen ; d] // (P) va d // (P) Chieu vuong goc d] va d xu ong (P) th a n h d'l va d'2 Ta c6 : d'l // d i va d'2 // d vi dj d n e n d'l d'2 Khi M cho s i n , ta t i m difcrc n h a t diem I e d'l va J e d'2 cho M la tr u n g d iem cua I J Luc ta du ng h a i diem I , J nhif sau : -! D a n g O ' d o i x i i f n g v6i O qua M D g n g O ' l X d 'l v a O ' J d'2 O' d ': O Do tiir g i a c O IO 'J la h i n h chif n h a t nen IJ d a difcfc x a c d i n h v a n h a n M l a m t r u n g diem Tii I dung di f dng t h i n g s o n g s o n g vdi AB , no c i t dj or E H ien n h i e n O A E I la h i n h chOT n h a t nen I E = OA TO J d i i n g dac rng t h i n g s o n g s o n g \6i AB D u dng t h i n g n a y c I t d d F Ta c un g c6 E F = OB V i t h e n e n : IE = J F TO g i a c E I F J la h i n h b i n h h a n h n e n E, M , F t h i n g h a n g va M la t r u n g d i e m c u a E F Tom la i : q ua M t o n ta i d uy n h a t m o t d u d n g t h i n g d c I t dj d E va c I t d d F ( y c bt ) 21 Ta CO ; x^ + y"* = 01^ + OJ^ = IJ^ = M ^ = = c o nst (dpc m ) 89 V.GIAITOANTHI B a i 163 ( D A I H O C Y - N H A - DLfOC - 1976) Cho m o t tuT d i $ n deu S A B C c6 canh l a a 1/ H a y t i n h chi4u cao SO p h a t x u a t tCf d i n h S va t h e t i c h V cua tiir d i e n SABC 2/ Goi M v ^ N I a n l u a t la t r u n g d i e m cua SA va BC C h i l n g m i n h M N l a doan vuong g6c chung cua SA vk B C T i n h d a i cua M N theo a Giai 1/ O l a t r o n g t a r n cua t a m gidc deu A B C Doc gia t u g i a i V = dt( AABC) X SO = i X ^ 3 X^ = 2ay De d a n g t h a y cac t a m giac S A N va B M C I a n lifat can a d i n h N , M => M N ± S A va M N B C => M N l a doan vuong goc c h u n g ciia S A va B C (dpcm) 2b/ T a c6: M N " ' ' = S N ' ' ' - S M ^ MN^ = 3a' a' y2j 2a' 4 a^f2 MN = (ycbt) B a i 164 ( D A I H O C T O N G H O P T P I I C M - 1991) Cho h i n h v u o n g A B C D c6 canh b a n g 2a ve Bx va D y l a h a i nijfa dUcfng t h a n g vuong g6c vdi m a t p h a n g ( A B C D ) va a cung m o t p h i a v d i ( A B C D ) Gia suf B' chay t r e n B x va D ' chay t r e n D y cho n h i d i e n ( B ' , A C , D ) luon luon vuong D a t B B ' = u va D D ' = v aJ T i m m o t he thiifc l i e n lac giiJa a, u va v b/ Goi O la t a m h i n h v u o n g A B C D H a vuong g6c v d i B'D' C h i l n g m i n h l a doan vuong goc c h u n g cua A C va B'D' c/ Chufng to O I = aV2 Suy r a n h i d i e n (A, B'D', C) cung vuong Giai a/ De y B'B ± (ABCD) BD >AC ( B B ' D ) = ( B B ' ; D D ' ) AC => B'Otr = (B'.AC, D ) = 90° T r o n g m a t p h a n g ( B B ' ; D D ) dUng D'E // D B t h i B D D ' E l a h i n h chiJ n h a t , n e n : D'E = D B = aV2 ; B' E = I u - v I 90 Ta CO : fAED'B'=>B'D'2 =(2aV2)2+(lu-vl)2 = 8a^ + ABOB' OB'^ + - 2uv = u'^ + ( a V = 2a^ + ADOD' => OD'^ = AODB' ^ B ' = OB'' + OD'' o + ( a V f = 2a^ + a ' + u ' + v ' - 2uv = a ' + u ' + v ' o uv = a ' (*) (ycbt) b/ Theo each d i m g : O I B ' D ' (1) Theo cau a: A C (OB'D') => AC ± (2) (1) va (2) c h i l n g to l a doan vuong goc chung cua A C vk B ' D ' d Da c6: AC ( O B'D ') AC B 'D ' Theo gia t h i e t 1 B 'D ' B'D' ( l A C ) Xlt! Xet A O B D ' = (A, B'D', C) O I B ' D ' = OD'.OB' 01 = V2a^ V2a^ + Vsa^ + -2uv 01'= + - u v + 8a^ 2a 2( u 2+v 2) + 8a^ a ' ' (u-^ + v ^ ) + 4a^ u 2+v2+4 a + = 2a^ + 4a^ I = aV2 Ta CO : => OI = OA = OC = Alt: = 90° => N h i dien : ( A , B ' D ' , C ) l a n h i d i e n vuong (dpcm) Bai ( D A I H O C Q U O C G I A H A N O I - K H O I A - 1997) Cho A B la dUcrng vuong goc chung cua h a i dUcrng thSng cheo x; y L a y A e x; B e y, A B CO dinh va A B = d M e x; N G y; M ; N t h a y d6i va A M = m ; B N = n ( m ; n ^ 0), dg t a luon luon c6 m ' + n ' = k > 0, k k h o n g d o i a/ Xac d i n h m ; n de d a i doan t h i n g M N d a t gia t r i nho n h a t , I d n n h a t hi Trong t r i f d n g hop x y va m n * h a y xac d i n h m ; n theo k va d de t h e t i c h tuT d i e n ABMN dat gia t r i \dn n h a t v a t i n h g i a t r i Gi a i a/ Goi a = (^cfy^ v d i < a < — • • Neu m = : M s A v a k = n' => MN' = AN' = AB'+ BN' => M N ' = d' + n ' = d' + k (1) Neu n = : N = B v a k = m ' ^ M N ' = d' + k (2) 91 => n* = MN V MA + AB +BN - A M + AB + BN > + M N = + d^- AM.AB - 2AM.BN + 2AB.BN M N ^ = k + d^ - 2mncos( A M ; B N ) (3) T L / (1) (2) va (3) t a CO t h e v i e t t o n g quat : fd^ + k : neu m n = +k-2nmcosa d^+k + 2mncosa : neu m n ^ va : neu m n va \AM; BN = a \AM; B N = i - a (4) (5) Gia t r i l(Jn n h a t , nho nha't cua M N : • N e u a = — t h i cosu = va M N ^ = d^ + k ; V m ; n thoa m a n + n^ = k max M N = m i n M N = V d ^ + k • N e u < a < — t h i cosa > va m n < + n^ = k (dang thijfc xay r a k h i : m = n = => d^ + k - c o s a < ) (6) d^ + k^ - 2mncosa m in (v(Si d k ( ) , ( ) ) < d^ + k^ + k + 2mncosa < d^ + k + c o s a max (vdidk(5),(6)) m i n M N = d ^ + k - k cos a m e i x M N = d^ + k ^ + k c o s a b/ Do X y va A B l a dudng vuong goc chung cua x va y n e n A M ( B A N ) va A B A N vuong d B =j> V = - d t ( A B A N ) M A = - - B A B N M A 3 l m ^ + n ^ => V = — m n d $ — d 6 12 kd ( 7) Dau d a n g thuTc t r o n g (7) xay r a m = n = Vay ; m a x V = — k d (vdi m = n = — ) (ycbt) V 12 92 Bai 166 (D AI HO C Y D tfO C TP H C M - 1999) Trong mat phang (P) cho h i n h vuong AB C D canh a G oi O la giao di§m cua h a i difcfng cheo cua hinh vuong AB C D Tr en difdng thftng Ox vuong goc (P) la y diem S G oi a la goc nhon tao bdi mat ben va day ciia h i n h chop SAB CD 1/ Tinh the tich va dien tich to^ n pha n ciia h i n h chop S AB C D theo a va a 2/ Xac d in h du dng vu ong g6c chu ng cua SA va CD Ti n h dp d a i difdng vu ong g6c chu ng d6 theo a va a G ia i 1/ Goi M la tr u n g diem D C => S M D C => D C O M => gMO = a ; < a < ta n a = Taco: SO OM SO = — a t a n a SM = VSO^TOM^" = Vay: VSABCD = - o va Jia^tan^a+i V4 1 - SO.SAB CD = - a " " t a n a (ycb t) Stp = 4.S.SDC + SABCI) = 2.S M.D C + SABCD „2 / + a^ = a^ cos a 1+ (ycb t) cos a J 2/ Goi : N = M O o A B Taco: M H l S N ( ca ch ditog) AB KSMN)!^ A B IM H => M H l( S AB ) r ^ M H lS A Tir H difng H I // AB ; (I SA) => H I // D C Tif I dirng I J // M H Khi do, I J la dtfdng vu ong goc chu ng ciia SA va D C V d i: • SM = SN = a 2.cos a MN = a • SO = — a.tana Thi: SsMN = - SO.MN = - M H S N SO.MN M H = I J = —^^^^— = a.sina (ycbt) 93 Chuyen de 9: MATCAUNGOAI TIEP - MATCAUNQI TIEP : X A C DlSIfl M A T cAu NGOAI T T £ P HiNfl CHOP B A N G G ^ C W O N G L PHirOMG P H A P , C a sd cua phuang phap can thifc h i ^n btfdc ca ban: B i : Quan ly gia thiet de tim duac (n - 2) dinh c6 s i n nhin hai dinh co dinh l a i cua hinh chop n dinh dudi mot goc vuong (xin hieu n dinh gom dinh h i n h ch6p va (n - 1) dinh cua da giac day) B : Suy h i n h chop noi tiep mat cau c6 difcfng kinh la khoang each giuTa hai dinh co dinh Luc tarn mat cau la trung d i l m doan noi hai diem co dinh a tren ban kinh la ntta d6 dai doan noi hai diem co dinh (va l A = I B = I M j = I M = R la bdn kinh mat cau ngoai tiep h i n h chop) n c A c B A I T O A N C O B A N Bai 167 Cho hinh chop S A B C co S A ( A B C ) ; A C = a; X B C = ° va S A = 2a D i n h t&m va tinh ban kinh mat cau ngoai tiep h i n h chop d6 Giai T a c6: S A J_ ( A B C ) fSB : la dudng xien |A B : la hinh chieu Do A A B C (fi = ° ) => B C ± A B => B C S B (dinh ly di/dng vuong g6c) '^SAC = ° Trong khong gian n h a n xet : I [ S B C = 90° => A ; B thuoc mat c l u difdng k i n h S C Mat cau tarn I trung diem S C co Mn k i n h R = i S C ngoai tiep h i n h ch6p S A B C Trong : R = i s C = -^ISPJTAC^ 2 = -^[{2^^ =: ? ^a (ycbt) Bai 168 Cho h i n h ch6p S A B C D co A B C D la hinh cha nhat va S A J ( A B C D ) D i n h tfim va tinh ban kinh mat cau ngoai tiep h i n h chop d6 biet : S A = A B = A D = 3a Giai Ba n g dinh ly ba di^dng vuong goc va tinh chat cua dudng t h i n g vuong goc v(Ji mat phang, ta co: SXt = S D C = S B C = ° => A ; D ; B thuoc mat cau (S) difdng kinh S C • T a m cua (S) la trung diem I cua S C • B a n k i n h R cua (S) la R = - S C T a co: A C ^ = AD^ + A B ^ = 3a + (3a)2 = 9a ^ + 9a^ = 45a'' 12 j 94 = SA^ + AC^ = (3a)^ ^ ^ =^ = 9aU ^ 81a ^ ^ R= isC= i ^a =- a 2 Vay h i n h ch6p S A B C D n p i t i e p t r o n g m&t cau (S) t f i m I va c6 b d n k i n h R = - a (ycbt) t«ii a : XAC DINH M A T c A u NGOAI T l £ p fllNfl CHOP B A N G T R y C DUdNG T R N Y A M A T T R U N G T R l J C LPHITONGPHAP^ Co so cua phuang p h a p can thiic h i e n bon budc ccf b a n : n Bi : Dung true difcfng t r n d ngoai t i e p da giac day, thong thifdng (d) d i qua d i n h hinh chop (neu (d) khong qua d i n h , ngoai cAc b i e n the ciia d a n g t o a n c6 the xem t h e m phUang phap d a n g h a i true dudng t r o n d dang t o a n sau) • B : Dung m a t phSng t r u n g triTc (a) cua m o t canh ben tiiy y eo t i n h OU v i e t eho gia t h i e t ( t h u d n g chon canh ben vuong goc v d i day la t o t n h a t ) • B3: T i m : (d) o (a) = I => I l a t a r n m a t cau • B : Khoang each t i f tarn den m o t d i n h t u y y ciia h i n h chop la ban k i n h mSt cau: (IS = l A = I B = I C = R) BAI TOAM C O BAN n c Ac Bai 169 Tim tarn G v a b i n t i n h k i n h m o t m a t cau ngoai t i e p tuf giac deu Giai Goi G la t r o n g t a r n A B C D deu canh a va M l a t r u n g d i e m A B , ta c6 |GB = GC = GD [AB = AC = AD => AG la true d u d n g t r n ngoai t i e p A B C D Dung m a t t r u n g trifc (a) qua t r u n g d i e m M cua c a n h A B , luc AG n (a) = I , t a c6 : (I AG => IB = IC = I D | l € (a) l A = IB => l A = I B = I C = I D _^ 11 la tarn mat cau ngoai tiep tuT cii?n ABCD • ban kinh mat cau la R = l A Ta CO : A A M I t / J A A G B IA= lA AM AB AG V6 A B ^ = a AG Vay ban k i n h m a t cau ngoai t i e p tuf d i e n deu 1^ R = — a (ycbt) 95 Bai 170 C h o tijf d i e n A B C D c A A B C d e u c a n h a ; D A = 2a v a D A ± ( AB C) D i n h t a m v a ti n h ban kin h m a t cau ngo tie p til die n Giai Go i G l a t r o n g t a m c u n g l a t a m d ifd n g t r o n n g o a i t i e p t a m giac d e u A B C v a de y d e n : A D ( AB C) => T r u e d c i i a d u c f n g t r o n ( A B C ) q u a G v a s o n g s o n g A D D u n g m a t t r u n g trUc ( a ) c u s A D q u a t r u n g d i e m J c u a A D => d o ( a ) = I fled = I B I C I D ^ II ( a ) lA = ID IA = IB = IC = ID B => I l a t a m m S t c a u n g o a i t i e p t i J d i e n A B C D ( y c b t ) Ban kinh cua m at cau la : R = l A = VJ A^ + AG ^ = 8^ + 12a ^ 3a ^ a V3 2V3 a (ycbt) Bai 171 D i n h t a m v a t i n h b a n k i n h m at c a u n go tie p hin h ASt = - a V2 D i e m H e A C s a o c h o A H = a V2 L a y d i m S Ch o h i n h vu o n g A B C D , t a m O c a n h t r ei n H x ( H x ±I ( A B C D ) ) B i e t chop S AB CD Htfdrng dSn D i Tn g O y / / H x t h i O y l a t r u e d U d n g t r o n n g o a i ti e p h i n h v u o n g A B C D D ifn g m a t t r u n g trUc d o a n S C la (a) => ( a ) n O y s I => I la tam m a t e au ngoai tie p hinh chop S AB CD vdi I C = R l a b a n kin h r-:>D D o c g i a tif t i n h R th e o yc b t W i : XAC D m B AN G H AI M AT cAu NGOAI Ttt? HlNfl CHOP TRgC DUCJNG TRON PHAN B itT L PH irON G P H A P , Co s a c u a p h u a n g p h a p de x a c d i n h m a t c a u n go tie p c a n t h i f c h i e n b u d c ccf b a n : n B i : D u n g ( d j ) l a t r u e d U c fn g t r o n n g o a i t i e p d a g i a c dAy (thong thUdng di khong qua dinh hinh chop) • B : D u n g ( d ) l a t r u e d U c J n g t r o n n g o a i t i e p v I A - I B = IC ^ IC = IB = IC = IS = R => I l a ta m h i n h cau ngoai tiep h in h chop S.ABC a V5 ASAB deu ca nh 2a r=> S M = a V s => C M = - S M = De y den l O = G M = a^/ 3a ^ R = IC = VlO^ + OC^ = 12a 2Sa (ycbt) j Bai 173 Cho h in h chop S.AB CD vdi A B C D la h in h vu ong canh 2a G oi H 1^ tr u n g d iem A B va SH = a V s la da i du dng cao h in h chop D i n h ta m va b an k i n h ma t cau ngoai tiep h in h chop S.ABCD HU dng dan fd, l a true dudng tron (ABCD) Dung : < ^ l a true diicrng tron (SAB) (di) Trong (S HK) : (d,) o (d^) = I => I la ta m ma t cau ngoai tiep h in h chop S.AB CD GjA Ban k i n h ma t cau: (d2) K/ l R= >/iOI- '+OB^ = -.aV3 V3 f2aV2' R = (ycbt) d -V ( 97 txHii : DitN TiCfl t A HlNfl CHliU L PHirONG P H A P T r o n g m o t so t r i f d n g hap viec ti'nh dien t i c h m o t da giac la kho k h d n Ta dp dung d i n h l y dien t i c h va h i n h chieu de t i n h no Viec chon di§n t i c h can t i m la S hay S' can te' n h i va nen d U a vao goc chieu ip n h u sau : S = S' cos (p K h i t i n h g i a n t i e p goc cua n h i dien hay goc cua h a i m a t phSng t a suf d u n g cong thijfc : S cos = S' O G h i c h i i : Mot goc vuong it nhdt mot canh goc chieu dilgc bdo loan bi chieu phdi song song vai cqnh tuang ling cua goc thi mot chieu Xem hinh thdy : \ Ti/ang n c A c tu fA2 A3 K A2 A3 ! A ^ A ? ^ * 90° B A I T O A N C O A ^ A ^ = 90" B A N B a i 174 Cho tiJ dien deu canh a T i n h goc p h i n g cua n h i dien bat k y tao bai cac m a t cua n h i dien Giai A Goi G la t r o n g t a m AI3CD deu va A I l a t r u n g t u y e n AACD Ta C O :

FCS;^= 90° PK = AC =>PM, FE khong vuong goc Doc gia tif chilng minh c^c doi dudng cheo lai doi mot khong vuong goc => dpcm Bai 177 Cho hinh hop chfl nhat ABCD.ABC'D' vdi cac canh la : AB = a; AD = b; AA' = c Goi I, J, Kla trung diemciia AB, AD, DD' 1/ Dung thiet dien cua mp(IJK) vdri hinh hop 21 Tinh dien tich thiet dien d6 Giai [Dudng thing IJ cSt BC; CDtai Eva F 1/ Difng : •^.Dirdngthang FKcat D'C'va CCtai Lva S Noi SEcat B'CtaiMva BB'tai N Noi lien tiep cac giao tuyen thi thiet dien la luc giac IJKLMN (ycbt) 2/ Chieu thiet dien xuong mSt day ABCD ta duoc luc giac IJDL'M'B, L', M' la trung diem cua CD va CB Dien tich ciia hinh chieu : S'= dt (ABCD) - dt AAIJ = ab - i ^ = ^ 4 99 ... 500 BAI TOAN HJNH KHONG GIAN CHON LOG Ma so: 1L - 19 5OH2 014 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 - 2 014 / CXB/ 01- 127/ OHQGHN 10 /... QUOC GIA HA NQI 16 Hang Chuoi - Hai Ba Trcfng - Ha Npi Dien thoai: Bien tap - Che ban: (04) 39 714 896 Hanln chinli: (04) 39 714 899; Tong Bien tap: (04) 39 715 011 • Fax: (04) 39 714 899 * Chiu Gidm... 2OB.OCcos60° BC = « BC'' = + i « Do : ~ 2 .1. -.i = AC = B C = Tifong t u : « - (1) AB^ = O A'' + OB^ - 2OA.OB cosl20'''' = + - 2 .1. 1( -1) = AB = Va (2) Ttf (1) va (2) ta difgc : CA + CB = AB C e