🔙 Quay lại trang tải sách pdf ebook Sách Giáo Khoa Hình Học 11 Nâng Cao Ebooks Nhóm Zalo s0 ctao DUC vA DAo rAo DOAN QtIyI.{H (T6ng Chir bion) - VAN NHU CUONG (Chir bion) PHAM rnAc BAN - ra uAN ,:1.. -. Nnn xuAr naN ctAo ouc NHI1NG DrEu Hgc srNH cAN cuu f xnr sr] DUNG sAcx erAo KHoA 1. Khi nghe tndy c6 gi6o gi6ng bdi, lu6n lu6n c6 SGK tnr'6c mdt. Tuy nhi6n kh6ng vi6t, v6 th6m vio SGK dd nim sau c6c ban kh6c c6 thd dDng tfrrgc. 2. Vd trinh bdy, s6ch gi5o khoa c6 hai ming : mAng chinh vir ming phr;. MAng chlnh g6m c6c dinh nghTa, dinh li, tinh chdt,... vi thudng dugc d6ng khung hoic c6 dudng vidn 6 m6p tr5i. MAng ndy duoc in liri vio trong. 3. Khi glp Ciu h6i f], cdn phAisuy nghi tra ldi nhanh vi d0ng. 4. Khi gf,p Hoqt dQng A, pnAi Onng b0t vir gidy nh5p tld tnrlc hiQn nh0ng y6u cdu mir hoat dQng doi h6i. Ban quy€n thuQc Nhh xudt bin Gi6o dgc - BQ Gi6o dgc vi Dio t4o. 692-2006 I qB I s3 6 - 1 s 3 O/GD Md s6: NH102M7 PHEP oot HinH vA puEp sdr,rc DANG TRoNG mAr pniruc Bitc tranh cila hoa si Hd Lan Et-se (M.C. Escher) gdm nhilng hinh bdng nhau mA td cdc chi€h binh tr€n lung ngua. Cdc htnh ndy phil kin mdt phdng. Hai chi€h binh vd ngaa cilng mdu (tdng hodc den) tuong tng vdi nhau qua m1t phip tinh ti4h. Hai chi€h binh vd ngua khdc mdu thi tuong ilng voi nhau qua mAt phip d1'i xilmg tuc vd ti€p theo ld m6t phip tinh ti€h. Ngh€ thudt dilng nhfrng hinh bdng nhau dd ldp d,iy mdt phdng duoc phdt tridn mqnh md vdo the'ki Xil d nudc l-ta-li-a. Chuong niy n6i vd c6c .phep doi hinh vd il6ng dang trong mdt phEng. Hoc sinh s6 ldm quen v6i ph6p tinh ti5n, ph6p ddi x0ng truc, ph6p quay, ph6p v1 trl, ... vd sd hidu th6 nio ld hai hinh bing nhau, thd niro td hai hinh ddng dang mOt c5ch tdng qu6t. Hoc sinh cdn n6m dtrgc Clinh nghia c0a c6rc ph6p n6i tr6n vd c6 thd 5p dung ch0ng dd giAi c6c biri to5n kh6ng qu6 ph0c tap. ? MO 1. Ph6p bi6n hinh DAU Vfi PHEP BIEN HiNH ' Trong Dai sd, ta dfl bidt mot khdi niOm quan trong : kh6i niem "him sd". Ta nh6c iai : Ndu c6 mOt quy t5c dd v6i m6i sd x € IR, x6c dinh duoc mOt sd duy nhdt y e IR thi quy t6c d6 goi ld m1t hdm sd xdc dinh tr€n tdp sd thuc R. BAy gid, trong mOnh dd tron tathay sd thtc bdng didm thuQc mfi phdng th\ ta dtloc khdi niOm vd ph6p bidn hinh trong mit phing. Cu thd la Ndu c6 mOt quy t6c dd vdi m6i didm M thu6c m[t ph&ng, x6c dinh duo.c mOt didm drty nhat M' thu6c m[t phing Ay thi quy t6c d6 goi ld m1t ph€p bi€n hinh (trong mdt phdng). VAy ta c6 DINH NGHIA ll fh1p UAn ninn (ffong mdt phdng) ld mlt quy tdc dd vdi mdi ll didm M thubc m\t phdng, xdc dinh duoc mdt didm duy nhdt ll t t' thulc mdt phdng dy. Didm M' gQi ld rinh cila didm M qua phdp bie'n hinh d6. 2. Citc ri OU Vidul Cho dudng thing d. Ydi m6i didm M, ta xr{c dinh M' ld, hinh chidu (vuOng g6c) cfra M ttdn d (h.1) thi ta du-d. c m6t ph6p bidn hinh. Ph6p bidn hinh ndy goi ld phdp chiAu Quilng gdQ kn dudng thdng d. Vidu2 Cho vecto il, v6i m5i didm M ta xdtc dinh didm theo quy tirc Mfr = i (h.2). Nhu vAy ta cfrng c6 mOt ph6p bidn hinh. Ph6p bidn hinh dp gqi ld phdp tinh fiA'n theo vecto il. 4 Hinh I T t -2M' ,-/ Hinh 2 Vidu3 V6i m6i didm M, ta xdc dinh didm M' tring vli M thi ta cflng duoc mOt ph6p bidn hinh. Ph6p bidn hinh d6 goi ld phdp d6ng nhdt. 3. Ki hi6u vd thudt ngfr NCu ta ki hiOu mOt ph6p bi6n h)nh nlo d6 ld F vd didm M' ld 6nh cia didm M qua ph6p bidn hinh F thi ta vi6t M'- F(M),hoFrc F(M) - M',. Khi d6, ta cdn n6iphdp biAn hinh F biAn didm M thdnh didm M'. V6i m6i h\nhh(,ta goi hhhly( 'gdm c6c didm M'= F(M),trong d6 M e :4lit dnh cfia l(qua phdp bidn hinh F, vd viOt ,/( ' = F(g( ). 1) Hey v6 mdt dtrdng trdn vd m6t duong thtng d rdi v6 Anh c0a drrong trdn qua ph6p chidu.l6n d. 2) Hey v6 m6t vecto il vd m6t tam gi6c ABC rdi tdn luot ve Anh A', B', C' cIac5c dinh A, B, C quaph6p tinh tidn theo vecto /. C6 nhAn x6t gi vd hai tam gi6c ABC vd A'B'C' ? PHEP TINH UETV VA PHEP DOT HINH Dinh nghia ph6p tinh ti6'n Ta nh6c 1ai dinh nghia ph6p tinh tidn dd n6i & Vi du 2 $ I : ll rnAf finh fiAn theo vecto il ld m\t phdp biAn hinh bi€n didm ll ru tnann didm M' sao cho Mfr = il . Ph6p tinh tidn theo. vectd / thudng duoc kf hiOu li f hoac Ti. Yecto il duoc goi ld vecto tinh tiLn. fl fnap ddng nhdt c6 phdi ld phdp tinh ti€n khong ? 2. Circ tinh chf,t cria ph6p tlnh ti6n A.r ffi ",U sir ph6p tinh tiSn theo vectd / nidn hai didm M, N ldnludt thenh hai di6m M', N'. C6 nh6n x6t givd haivecto ffi va tWN ? So s6nh dO dai haivecto tl5. VQy ta c6 dinh li DINH LI 1 NAu phip tinh tidn bi€h hai didm M vd N ldn luot thdnh hai didm M'vd N'thi M'N'= MN. Ngudi ta diOn thtinh chdt trcn cira ph6p tinh tidn lir: Phip tinh fieh khdng ldm thay ddi khodng cdch gifra hai didm bdt ki. DINH Lf 2 Phdp tinh ttdn bi€n ba didm thdng hdng thdnh ba didm thdng hdng vd kh1ng ldm thay a& tnt ta ba didm d6. Chfing minh Gii srl ph6p tinh tidn bidn ba didm A, B, C thdnh ba didm A', B',C'. Theo dinh lf l, ta c6 A'B'= AB. B'C'= BC vd A'C'= AC. NOu A, B, C thhng hdng, B nim giita A vI C thi AB + BC = AC. Do d6 ta cfrng c6 A'B' + B'C' = A'C', titc ld A', B', C'thing hhng, trong d6 B'nlm gitra A'vh C'. Ti dinh li trOn, ta d6 dhng suy ra h0 qui sau d0y HE QUA Phdp rinh tidn bidn dudng thdng thdnh dudng thdng, biAh tia thdnh tia, bidn dogn thdng thdnh doan thdng bdng n6, bi€h tam gidc thdnh tam gidc bdng n6, bi1h dudng tdn thdnh dudng trdn c6 cilng bdn kinh, bidn g6c thdnh g6c bdng n6. 3. Bidu thr?c toq d9 cria ph6p tlnh tidn Trong mat phing vdi h0 truc toa dQ Oxy.cho ph6p tinh tidn theo vecto /. Bidt toa dQ cira i ld (a ; b). Gih srl didm M@; y) ' biOn thdnh didm M'(x'; )) (h.3). [r'=x+a ly':y+b. Khi d6 ta c6 Hinh 3 COng thrlc tr0n goi ld bidu thtc toq dQ cfia phdp tinh ti€n theo vecto il(a; b). 2 H6y giSi thich vi sao c6 c6ng thrlc tr6n. 4. Ung dqng cria ph6p tinh ti6n Cho hai didm B, C cd dinh ffen drtdng trdn (O ; R) vd mQt didm A thay ddi trAn drdng trdn d6. ChrtryS minh rdng truc tdm tam gidc ABC ndm tr\n m1t drdng tdn cd dlnh. Gidi Ndu BC li duong kinh thi truc tam H cta tam gi6c ABC chinh ld A. Vay H nam trOn dudng trdn cd dinh (O; R). NOu BC khOng ph6i ln duong kfnh, vE dudng kfnh BB'c:iua duong trdn (h.4). D6 thay rang ndu FI li trgc tAm cira tam gi6c ABC B th\ m =Et (tren hinh 4, didu d6 suy tt nhan x6t tfi gi6c AHCB'li hinh binh hinh). Hinh 4 Nhu vAy, ph6p tinh tiOn theo vecto cd dinh B'C bi6n didm A thinh didm H. Do d6, khi A thay ddi trOn (O ; R) thi truc tAm H luon ndm tr€n du&rg trdn cd dinh li 6nh cfia dudng trdn (O ; R) qua ph6p tinh ti€n n6i trOn. n Bii to6n 2 Hai thbn ndm o hai vi tt A vd B cdch nhau mdt con sbng (xem rdng hai bd sing ld hai drdng thdng song song) (h.5). N7tdi ta dq dlnh xdy mOt chiAt cdu MN bdc qua sdng (c6'nhi1n cdu phdi vuOng g6c voi bd sbng) vd ldm hai doan dudng thdng til A d1n M vd rrt B ddn N. Hdy xdc dinh vi tr{ chi€t cdu MN sao cho AM + BN ngdn nhdt. Hinh 5 NhSn xdt Bii to6n sO rdt don giAn ndu con s6ng rdt hgp, hgp ddn mrlc hai bd sOng a vI b xem nhu trirng vdi nhau. 3 Hiy giAi bdri to6n trong trrlong hop dflc biCt d6. Trudng ho-p tdng qu6t (h.5) c6 thd dua vd trudng hqp trOn bang m6t ph6p tinh tidn theo vecto ntfr ae a ffing b. Khi d6 didm A bien thinh didm A' sao cho Ti = Mfi vitdo d6 A'N = AM. 4 Tu ggi f d6, hdy giAi bni to6n trong trrrdng hop tdng qu6t. 5. Ph6p ddi hinh Kh6ng phii chi c6 ph6p tinh tidn "kh6ng lim thay adi ttroAng c6ch gifra hai didm" md cdn nhidu ph6p bidn hinh khdc cfrng c6 tinh ch6t d6 (tinh chat nly cdn duo.c goi 1I tinh chAt bdo todn khodng cdch girtahai didm). Ngudi ta goi c6c ph6p bidn hinh nhu vay ln ph6p ddi hinh. DINH NGHIA ll rnap ddi hinh ld phdp bi€n hinh khang tdm thay ddi khodng ll c,h'ch gifra hai didm bdt ki. Chf y rAng c6c tinh chdt d6 n€u cira ph6p tinh tidn dugc chrlng minh chi dga vio tfnh chdt "kh1ng ldm thay ddi khodng cdch.gifra hai didm".Ba vdy, cic ph6p ddi hinh cf,ng c6 nhfrng t(nh chlt d6. CU thd ta c6 DINH LI Phdp ddi hinh bidn ba didm thdng hdng thdnh ba didm thang hdng vd kh1ng ldm thay ddi tht tt ba didm d6, biAn dudng thdng thdnh dudng thdng, bi€n tia thdnh tia, biAh doqn thdng thdixh doan thdng bdng n6, bi€h tam gidc thdnh tam gidc bdng n6, biAh dudng trdn thdnh duong tdn cd cilng bdn k{nh, biah g6c thdnh g6c bdng n6. c6u n6i vd bdi t6p 1. Qua ph6p tinh tidn 7 theo vecto il +d, dudng th&ng d bidn thdnh duong thing d'.Trongtrudng hqp nio thi : d tring d'? d song song va d'? d cat d'? 2. Cho hai duong thing song song a vd a'. Tim tflt ci nhfrng ph6p tinh tidn bidn a thdnh a'. 3. Cho hai ph6p tinh tidn Ti ve &., Vdi didm M bitt k\, T; bi€n M thdnh didm M', T; bidnM'thlnh didm M". Chfng t6 rlng ph6p bidn hinh bi0n M thhnh M" ld mOt ph6p tinh tidn. . 4. Cho dudng trdn (O) vd hai didm A, B. Mot didm M thay ddi tren dudng trdn (O). Tim qu! tfch didm M' sao cho Mff' + Ul = ME. 5. Trong m6t phing toa dO Oxy, vdi d, a, b ld nhfrng sd cho trr1c, x6t ph6p biOn hinh F bidn mdi didm M@; y) thdnh didm M'(x'; y), trong d6 fxr=xcosa-ysinrz+a )"ly'= xsina + ycosa + b. a) Cho hai didm M(x1; y1), N(x2; yz) vit gIi M', N'lAn luot lI hnh cia M, N qua ph6p F. Hdy tim toa dQ cia M'vi N'. : b) Tfnh khoing cdch d gitra M vd N ; khoing c6ch d' giita M' vi N'. c) Ph6p F c6 phii li ph6p ddi hinh hay kh6ng ? d) Khi d = 0, chfng t6 rlng F ld ph6p tinh tidn. 6. Trong mlt phing toa d0 Oxy, xdt c6c ph6p biOh hinh sau day : - Ph6p biOn hinh P1 bi0n m6i didm M(x; y) thlnh didm M'(y ; -x); Ph6p bidn hinh F2 bidn m6i didm M(x; y) thdnh di6m M'(2x; y). Trong hai ph6p bidn hinh trOn, ph6p nlo li ph6p ddi hinh ? :.. r:gryaq PHEP DdI XONG TRUC 1. Dlnh nghia ph6p ddi xfng trr,rc Ta nh6c lai : Didm M' gqi ld doi xfing vdi didm M qua dudng thdng a nAlu a ld dadng trung truc cila doan thdng MM' (h.6). Ne'u M ndm tr€n a thi ta xem M ddi xfing vdi chfnh n6 qua a. Ph6p ddi xrlng qua duong thing a duo. c dinh nghia nhu sau DINH NGHIA 1 ll fnUp ildi xttng qua itudng thdng a ld phip bi€n hinh bi€n mdi ll aidm M thdnh didm M' ddi xtng vdi M qua a. Ki hi0u vi thuflt ngit Ph6p ddi xrlng qua duong thing a thudng duo. c ki hieu ld D o. Ph6p ddi xrlng qua dudng thing cdn goi don giAn lit phdp ildi x{rng truc. Duong thing a ggi li trryc cfi.a phdp ddi x,hng, hay don giin ld trqc ddi x,frng. @ Quo phip ddi xrlng trvlc Do, nhfrng didm ndo bian thdnh chtnh n6 ? @ Ne'u phip ddi xrntg ruc Da biah didm M'thdnh,didm M' thi n6 biidh didm M' thdnh didm ndo ? Ndu n6 bi\n hinh g(thdnh hinh U(' thi n6 bidn hinh &f ' thdnh hinh ndo ? 2. Dinh li Phdp ddi xfing trryc ld milt phip ddi hinh. I (Dd chung minh dinh li') GiA st D oli ph6p d(ii xung qua dudng thEng a. Ta chon hQ truc toq tlQ Ory md Ox lA drrdng th8ng a (h.7). 10 Ldy hai clidm tu!' tl A(xe;ti vA B(xs;1il, hdy vidt toq dQ cta A' : D"(A) vd B' = D"(B) rdi ddng c6ng thr?c tinh khoAng c5ch tld chrlng minh A'B': AB. F,: \s cHU V Qua hoat dOng trOn, ta thdy ndu ph6p ddi xrlng qua truc Ox bidn didm M(x; y) thenh didm M'(x' ; y) rhi Htnh 7 COng thrlc tr0n.goi li bidu thirc tog iIQ crta phdp ddi xitng qua truc Ox. @ fnAp ddi xfing qua trryc Oy c6 bidu thttc toq dQ nhu th€'ndo ? 3. Trgc ddi xftg cria m6t hinh Chring ta h6y quan s6t bdn hinh sau dAy (m6i chfr c6i li m6t hinh) : ADPO Ngudi ra n6i hinh thri nhdt vi hinh thf hai c6 tinh "cin xfng" vi vdi m6i hinh, c6 thd tim thdy mOt duong thing sao cho ph6p d0i xrlng qua dudng thing d6 bidn hinh 0y thlnh chinh n6. Cdc duong thing d6 goi th truc ddi xring cira m6i hinh. Hai hinh cdn lai khOng "c6n xfng" vi chring khOng c6 nhfrng dudng thing nhu v0y. DINH NGHIA 2 ll Dudng thdng d goi ld truc ildi xirng crta hinh {/( n€'u phip ddi ll *ns ruc D4 bith J(thdnh chinh n6, trtc h Dd@() = il MQt hinh c6 thd kh6ng c6 truc ddi xfng, cfrng c6 thd c6 m6t hay nhi0u truc doi xurng. 11 Vll Trong cdc hinh sau ddy, hinh ndo c6 truc ddi xirng vd c6 mdy trryc ? (M6i chrt cdi ld m1t hinh) ABCDDEGHIKL MNOPORSTUVXYZ Hey lim thrt ! Cdt em hdy nhd mQt giot muc l1n mAt fi gidy trdng, rdi gdp td gidy theo mQt dtrdng thdng di qua giot mqc d6. Ap hai phrin cila td gidy sdt vdo nhau r6i md ra. Cdc em s€ daoc nhfrng hinh c6 truc ddi xrtng khd ki thti ! Dudi ddy gioi thi€u vdi cdc em m1t sd hinh nha vdy. $ a.Ap dgng Ngudi ta td chtlc mQt cuQc ch4Y thi trOn b6i bidn v6i didu kiOn sau : C6c vAn dQng viOn xuAt ph6t tt dia didm A vd dich li dia didm B, nhtrng trudc khi ddn B phii nhring minh vio nudc bidn (ta gii st rang m6P nu6c bidn li mdt dudng thing) (h.8). fr:: ,-J aJ Dd chidn th6ng trong cu6c chay dua 'J ndy, ngoii tOc OQ ch4y, cdn c6 mQt Hinh 8 ydu td quan trgng li v0n dOng viOn phAi x6c dinh vi tri M & m6p nudc mi minh ph6i ch4y tt A tdi dd, nhring minh vdo nudc bidn, r6i tt d6 chay ddn B sao cho qu6ng duong phii chay li ngan nhdt. t2 Nhu v0y, bhi to6n c5 thd ph6t bidu du6i dang to6n hoc thudn tuy sau dAy Cho hai didm A vd B ndm v€' mdt phia cfia dudng rhdng d (h.g). Hdy xdc dinh didm M r€i,n d sao cho AM + MB be nhdt. tA M Hinh 9 @ Neu hai didm A vd B ndm vd hai ph{a cila dudng thdng d thi ldi gi(ii bdi todn tr€n rdt don gidn.Trong trxdng ho.p d6, didm M cdn tim ld didm ndo ? Bay gid x6r trudng hgp A. ^B nam vd m6t phfa ctra d. Hdy lay didm A' d6i xrlng vdi A qua d, vd. ch:6 f rang : AM + MB = A'M + MB. 2 Voi goi 1[ tr6n dAy, hdy n6u ldi giai crja bdi to6n. cou h6i vd bdi t6p 7. Qua ph6p ddi xrlng truc Do (a li truc d6i xfng), dulng thing dbidn thdnh duong thing d'. Hdy tri ldi c6c c0u h6i sau : a) Khi nio thi d song song vfit d' ? b) Khi nho thi d tring v6i d'? c) Khi nio thi d cit d'? Giao didm cira d vd d'c6 tinh chdt gi ? d) Khi ndo d vuong g6c vdi d'? Trong mat phing toa dQ Oxy, cho cdc dudng trdn (61) vd (G) ldn luot c6 phuong trinh : (V1l : *2 + y2 - 4x +5y+ I =0 ; (Gz): *2 *y2 +10y-5=0. Vidt phuong trinh inh cria m6i duong trdn trOn qua ph6p d6i xrmg c6 truc Oy. Cho g6c nhon xoy vi m6t didm A ndm trong g6c d6. Hdy xdc dinh didm B 9. tr)n Ox vh didm C tran Oy sao cho tam gi6c ABC cd chu vi nh6 nhAt. . Cho hai didm B, C cd dinh ndm trOn duong trdn (O ; R) vI didm A thay ddi 10. tr0n dudng trdn d6. Hdy ding ph6p ddi xrlng truc dd chtmg minh rf,ng truc tlrrn H cira tam gi6c ABC nf,m trOn m6t dudng trdn cO dinh. Hudng ddn.Khi BC kh6ng phii ld dudng ktuh, goi H'ld. giao didm cira duong thing AH va duong trdn (O ; R). Chung minh rang H d6i xirng va H' qua dudng thhng BC. 13, 11. a) Chi ra truc ddi xtlng (ndu c6) cira m6i hinh sau dAy (m6i hinh th mQt tt bao gdm mOt sd chfi c6i) : MAM, HOC, NHANH, HE, SHE, COACH, lS, lr, SOS, CHEO b) Chfrrg minh rlng dd thi cfia hdm sd ch6n luOn c6 truc ddi xfmg. PHfP QUAY VA PHfP D6I XUNG IAVT 1. Dlnh nghia ph6p quay ll f rong mat phdng cho milt didm O cd dinh vd g6c ltong gidc A ll kna"g d&. rhep bian hinh bi€h didm o thdnh didm o, biah ll *ai didm M khdc O thdnh didm M' sao cho OM = OM' vd ll (Ottt, OM) = rp duqc goi ld phdp quay tdm O g6c quay Q. Ph6p quay thuong dugc k( hieu lI Q, vd ndu mudn chi 16 tdm quay O vd g6c quay cpth\taki hiQu ph6p quay d6ld Q@, O. o ' Hinh 10 Hinh 10 cho ra thay ph6p quay tam O g6c qay ;bien didm M thanh didm M',bi€nl6 cit G thitnhlS cd 6 '. fl rnep ddng nhdt c6 phdi ld phip quay hay khAng ? t4 . 2. Dinh li Phip quay ld mdt phip ddi hinh. Chfing minh Gii sir ph6p quay Qro,r, bidn didm M thlnh M'vitbion didm N thlmh N', trong d6 O,' M, N khbng ifring hdng (h.11). Theo dinh nghia cria ph6p 9uay, ta c6 OM = OM', ON = ON' vi (OM, OM) = (ON, ON) = e. Theo h0 thrlc Sa{o vd g6c luong giilc, ta c6 (OM, ON) = (OM, OM) + (OM', ON) : (ON, ON') + (OM', ON) = (oM,, oN). o Suy ra MON : M'ON'. Nhu v0y hai tam gi6c MON vd,M'ON' bf,ng nhau, do d6 M'N'= MN. Trudng hqp O, M, N thing hing, ta thdy ngay M'N' 1 Cho hinh ng0 gi6c ddu ABCDE ttm O (h.12). Hay chi ra m6t sd ph6p quay bidn ng0 gidc d6 thirnh chinh n6. 3. Ph6p ddi xrirrg t6m Hinh 1l = MN. B E Hinh t2 u MOt trudng hqp dlc bi0t cria ph6p quay ld ph6p quay vdi g6c quay r. Khi d6, nOu O ld tam quay thi m6i didm M dugc biOn thdnh didm M'sao cho O ld trung didm cira MM'. Boi vAy, ph6p quay d6 cdn c6 tOn goi lh ph6p d6i xrlng qua didm O. Ph6p dOi xung qua didm O cdn c6 thd duo. c dinh nghia nhu sau : Phdp.ilili xrtng qua didm O ld m1t phdp bi€h hinh biAh mdi didm M thdnh didm M' ddi xftng voi M qua O, cb nghTa ld OM + OM' -- 0. -{ (Jl o 15 Ki hiQu vir thu4t ngir Phdp ddi xrrng qua didm O thuong dugc kf hiQu li Ds.Ph6p ddi xrlng qua mQt didm cdn goi don gi6n ld phdp ildi x{tttg tdm.' Didm O eqildtdm crta phdp ddi xirng, hay don gi6n li tdm ildi xitng. Biiu thrtc toa d0 Trong hQ toa dQ Oxy cho didm I(a; b). N€u ph6p doi xfrng tAmDTbidn didm M(x; y) thlnh didm M'(x'; Y) thi [*'=2a-x lv'=2b-Y. COng thtlc trOn goi Ld bidu thtlc toa dQ cila phdp ddi xurtg tdm Dp 2 Hdy giSith(ch tqi sao c6 c6ng th0c tr6n' TAm ddi xfng cfia mQt hinh Chring tahdy quan s6t c6c hinh bidu thi c6c chfr c6i sau day z SN Tuy c6c hinh d6 kh6ng c6 truc ddi xrlng nhlng chfng cfrng ^c6 tinh "can *dg" nLo d6. Lf do ld vdi m6i hinh, ta c6 thd tim thdy m6t didm O sao cho ph6p ddi xrlng t0m D1bidnhinh d6 thdnh chfnh n6. @ Oidm O nhu tha'cila mdi hinh ran ddy ld didm ndo ? C6c didm O nhu v{y dugc gqi li tam ddi xrlng cfra m6i hinh. ll Oid* O Sqi td tdm ddi xirng cfia milt hinh 1( ndu phip ddi ll *,t"g tAm Ds biAn hinh {/(thdnh chinh n6, ttc ld Do(gf ) = il' VTi Trong bdng chfr cdi in hoa, nhfrng chfr ndo c6 tdm ddi xirng ? Nhfrng chfi VA Trong cdc hinh sau ddy, hinh ndo cb tdm ddlxtng ? OES* ndo cb tdm ddi xtmg nhtng kh1ng cd trryc ddi xfing 7 @rO I 16 4. Ung dqng cfra pht6p quay . Bii to6n 1 Cho hai tam gidc ddu OAB vd OA'B' nhr hinh 13. Gqi C vd D ldn ltro. t td trung didm cria cdc doqn thdng A' AA'vd BB'. Ch*ng minh rdng OCD ld tam gidc ddu. Gidi X6t ph6p qtay Q tdm O v6i g6c quay bang mor g6c o luqtng gi6c (OA, OB). R6 ritng Q biOn A thenh B vh biOn B Hinh 13 'A'th)nh B', nOn Qbi6n doan thing,4/'thinh doan thang BB'. TU d6 suy ra e bidn trung didm C ctra AA' thanh trung didm D ctlr- BB'. Do d6 OC = OD vi, edD = 6o0. V4y ocDliltam gi6c ddu. tr Bii torin 2 Cho dadng trdn (O ; R) vd hai didm A, B cd dinh. Vdi mdt didm M, ta xdc dinh didm M' sao c,ho Mfr = MA + tWE,. Tim quj tich didm M' khi didm M chay tr€n (O ; R). Gitii (h.14) Goi 1 lh trung didm ctra af tfri / cd dinh vi MA+MB=LMI . ,,'II BAi vay, MM' = MA + MB khi vi chi khi \ c \ MM'=2M1, trlc li MM' nhdn 1 lim trung 'r_ didm hay ph6p d0i xrlng tAm D1bi6n diim M M' thdnh M'. Y4y khi M chay tr0n dudng trdn Hinh l4 (O ; R) thi qu! tich M' lA anh ctra dudng trdn d6 quaDp NOu ta g2i O' li didm dOi xrrng crta O qua didm l thi quy tich cria M'lh dudng trgn (O' ; R\. tr Bii to6n 3 Cho hai duong trin (O ; R) vd (Or ; Rr) ciit nhau tai hai didm A, B. Hdy dung m\t dudng thdng d di qua A cdt (O ; R) vd (O1; R) l,in luot tai M vd Ml sao cho A ld trung didm ctia MMt. Gilii (h.ts) DAlit ph6p d6i xrlng qua Ath\ Dlbidn didm M thinh didnt ttrt, vi bidn N= (,lD z. ttit,tHttruc-e T7 dudng trdn (O ; R) thdnh duong trdn (O';rR). Vi M ndm trOn (O ;R) n€n Ml nim trOn (O' ; R). M4t kh6c M1l1i nam trOn (Or ; Rr) nan Mllit giao didm khdc A cira hai duong trdn (O'; R) vh (O1 ; rRr). Tt d6 suy ra c6ch drmg : . DUng duong trdn (Ol ; R) ddi xung vdi (O ; R) qua didm A (O'lit didm dOi xfng ctra O qua A). Hinh 15 . Ldy giao didm M 1 crtahai duong trdn (O1 ; R1) vi (O' ; R), Mlkhdc A. . Dudng thing d li duong thing di qua A vd My EE t i sao d thod mdn didu kiQn ctia bdi todn ? Cdu nfi vd bdi tgp 12. Cho ph6p quay Q tam O vdig6c quay p vd cho dudng thing d. Hdyn6u c6ch durg inh d' cia d qua ph6p quay Q. L3. Cho hai tam gi6c vuOng ctn OAB vd OA'B' c6 chung dinh O sao cho O nam trOn doan th&ng AB' vd, nam ngodi doan thhng A'B (h.16). Gqi G vd G'ldn lugt ld trong t4m cdc tam gi6c OAA' vd OBB'. Chrlng minh GOG'li tam gi6c vuOng cAn. t4. Gia srl ph6p ddi xrmg fftm Ds biOn dudng B' thing d thdnh ducrng thing di Chrlng minh OA Hinh 16 a) Ndu d kh6ng di qua mm ddi x(mg O tti d'song song vdi d, O cdch ddu d vb. d' : b) Hai ducrng thing d vd, d'trtng nhau khi vi chi khi d di qua O. 15. Cho ph6p ddi xrlng t6m D1vdduong thing dkhdng di qua O.Hdy nOu c6ch dr;ng anh d' cla duong thing d qrua Dp. Tim c6ch dqng d'md, chi srl dgng compa mQt ldn vi thu6c thing ba tdn. 18 z. HINHttNcra Chi ra cdc tdm d6i xung cliua cdc hinh sau dAy : 16. a) Ffinh gdm hai duong thing c6t nhau ; b) Flinh gdm hai dudng thing song song ; c) Flinh gdm hai duong trdn bang nhau ; d) Dudng elip I e) Duclrg hypebol. Cho hai didm B. C cd dinh tr€n dudng trdn (O : R) vd m6t didm A thay L7. ddi tren dudng trbn d6. Hdy ding ph6p ddi xfng t6m dd chfng minh r6ng truc tam H cia tam gi6c ABC nim trOn m6t dudng trdn cO dinh. Hadng ddn. Goi 1 lI trung didm cira BC. Hdy vO dudng kinh AM ctra dudng trdn r6i chrlng minh rlng 1ld trung didm cira doanthhng HM. Cho dulng trdn (O ; ,R), dudng thing A vd didm /. Tim didm A tren 18. (O ; R) vi didm B trOn A sao cho 1ld trung didm cria doan th&ng AB. Trong mltphing toadO Oxy,cho dudng thing L: ax+by + c= 0vddidm 19. 1(xs ;lo). Ph6p d6i xrlng t0m Dlbidn dudng thing A thinh duong thing A'. Vidt phuong trinh cira A'. HAI HiNH BANG NHAU Chring tabidt rang ph6p ddi hinh bidn tam gir{c th}nh tam giSc blng n6. Bay gid a dil vAn dd : Cho hai tam gi6c bang nhau thi c6 hay kh6ng m6t ph6p ddi hinh bien tam gi6c nhy thhnh tam gi6c kia ? 1. Dinh li Ndu ABC vd A'B'C' ld hai tam gidc bdng nhau thi c6 phdp doi hinh bi€h tam gidc ABC thdnh tam gidc A'B'C'. Chitng minh Ta x6c dinh mQt ph6p bidn hinh F nhu sau : F bien m6i didm M thinlh di€m M' sao cho ndu CM : pCA + qCB (h.17). (p . lR, q € R) tli C'M' = pC'A' + qC'B' 19 Ta chrlng minh F li ph6p ddi hinh. ThAt vfly, gi& srl c6 th€m didm N vd F bidn N thanh N', trlc Id ndu Cfr = nCA + rcE ffi =CN -CItl = (k - dcL+ (/ * deE. Suy ra._2 MN" = MN ., ^ a __,) = (k - p)'cA' + (/ - q)'cB' . +2(k-p)(t-dcA.cd' Hodn toin tuong tu, ta cfrng c6 M'N'2 = Mfrz B' Hinh 17 = (k - p)2C'A'2 + (l - q)2C'B'2 +2(k - p)(l - dei"ed' Vi hai tam gi6tc ABC vit A'B'C'bdng nhau nQn CA = C'A" CB = C'B' vd jA.CE =Cfr .ed'. B&i vQy, ta suy ra MN = M'N'hay F ln ph6p ddi hinh' R6 rhng ph6p ddi hinh d6 bidn A, B, c ldn luot thdnh A" B" c" t(tc 1I bicn tam gi6c afb tnann am gr6c A'B'C'' tr 2. Thd nio ldr hai hinh bXng nhau ? tf k':hi Tt dinh lf tron ta c6 thd ph6t bidu i "Hai tam gidc bdng nhau bhi vd' cl c6 phdp ddi hinh biAn iam gidc ndy thdnh tam gidc ftia". Nhu vfy, khdi ,iern ;[ang nhau" ctra hai tam gi6c c6 thd dugc dinh nghia bing hai c6ch tuong duong sau d8Y 1) Hai tam gidc goi td. bdng nhau nA'u chting c6 cdc canh tuong fing bdng nhauvd cdc gbc ttong fing bdng nhau' 2) Hai tam gidc gqi ld bdng nhau ndu c6 ph6p ddi hinh bi€n tam gidc ndy thdnh tam gidc kia. D6i vdi su b[ng.nhau ctra c6c hinh n6i chung, ngudi ta ding c6ch dinh ngtria thrl hai. Vfly tu c6 dinh nghia tdng qu6t sau d&y ll nai hinh goi ld bdng nhau n€'u cd phip ddi hinh bi€n hinh ll "aY thdnh htnh kia' 20 Til dinh nghla tren ta suy ra N€u hinh Jq bdng hinh J$ vd hinh ,7Q bdng hinh ,7fi thi hinh s(1bdng hinh 0Q. ThAt viy, v\ gq bdng 0$ nOn c6 ph6p ddi hinh F bi€n J(1 bingtQ nOn c6 ph6p ddi hinh G bi€n1$thinhtQ. NOu ta thuc hiOn liOn tiOp ph6p ddi hinh F vi ph6p ddi hinh G thi hidn nhiOn ta duoc ph6p ddi hinh bidn gfi thdnh 44.YLy uLbai,ng s4. Chang han. trOn hinh 18, h\nh Jfi bing hinh Z$ v\ c6 ph6p tinh tiOn bicn Afi thinh ,7$ ; h\nh ,7Q beng h\nh $Q vi c6 ph6p dtii xrlng truc thlnh &6, v\ Uh r(, I-_l r--o oo4 L_l bidn $$thdnh &6. Y 4y hai h\nh &fi vd JQ bang nhau. go thd *"*?i ehua bidf em I Hinh 18 Tir xa xua, ngudi ta 65 bidt trang tr[ brlc tUdng, d6t th6u thim hoa, l6t ndn nhd, ... bing nh0rrg hlnh v6, nhinrg vi6n gach bing nhau v6i cfuc hoa vdn gidng nhau, ... {&r-r- *IIJL*t- {€ -tF t * -r-tt$ -r'{T --r-tlF {('r-r * 1 * { JL * _HF [- * -l JL*H {Bt- l-l {FL-r -r-Lif * * { * { * -r+T -1 t- -rrtF -r{F + J*-#{T L tC-t-T- -r{F {Tr-r *r-r-- * * { C6c m5u hinh v6, hoa vdn, ... c6 thd rdt fnec nhau nhung ngudi ta chfng minh tlrroc ring thrlc ra ch? c6 17 cAch s5p xdp ldp di I6p lai c6c hinh rihu thd dd let khSp mdt phEng. Ndu chi dDng c6c ph6p tinh ti6n vir ph6p quay dd UiSn m6t vi6n gach niry thirnh m6t vi6n gach kh6rc thi c6 5 cich l6t : Cdn ndu dirng th6m cA ph6p ddi xurng truc thi c6 th6m 12 clch l6t nfra : J-J C- r g- 7- J-J L 7 L 7- J-J g. r - f. 7 22 Trong 17 cAch l6t tr6n, ngrtoi ta d5 tim thdy 11 c6ch l6t 6 ddn Alhambra thinh phd Granada (TAy Ban Nha), 5 c6ch kh5c di tim thdy 6 chdu Phi, c6ch cdn lai cOng dd tim thdy trong m6t trang trl cd 6 Trung Qudc. cou n6i vd bdi tQp Chtmg t6 rang hai hinh chfr nhAt cing kich thudc (ctng chidu dii vd chidu 20. rOng) th) bang nhau. a) Chrlng minh ring hai tf gi6c ldi c6 cdc cdp canh tuong rlng bdng nhau 2L. vd m6t cap dudng ch6o tuong tmg bang nhau thi bing nhau. b) Chung minh rang hai tf gi6c l6i c6 cdc cip canh tuong rlng b[ng nhau vh mOt cip g6c tuong rmg bAng nhau thi bang nhau c) Hai tir gi6c 16i c6 cdc cdp canh tuong rlng bang nhau thi c6 bing nhau hay kh6ng ? Da grdc l6i n canh goi ld n-gi6c ddu ndu t* cecdc canh cira n6 bing nhau vd tdt cil cdc g6c ctra n6 blng nhau. Chung t6 rang hai n-gi6c ddu bang nhau khi vd chi khi chring c6 canh bdng nhau Ifrnh 0(1gdm ba duong trdn (O1 ; r1), (Oz; 12) vd (O3 ; ry) doi mot tiep xric 23. ngodi vdi nhau. ltrnh 5$ g6m ba dudng trdn (11 ; r); (lz; 12) vd (13 ; ry) ddi m6t tidp xric ngoii v6i nhau. Chrmg t6 rang hai hinh 4rirh$bangnhau. Cho hai hinh binh hdnh. Hdy vd m6t dudng thing chia m6i hinh binh hdnh 24. d6 thinh hai h)nh blng nhau. 23 Hin-be (Hilbert) Chdng ta hdy quan s6t hai bfc ch0n dung 6 hinh vE tren. Tuy kich thudc ctra chring khdc nhau nhtmg hinh dang cira chring rdt "giOng nhau" (ta n6i chring "ddng d4ng" vdi nhau). Vi brlc nh6 hon lh chdn dung cira nhh to6n hgc Hin-be, , nOn brlc l&r hon cflng li chin dung cria nhi to6n hoc d6. Sau d0y, chring ta sE n6i vd c6c ph6p bidn hinh kh6ng lim thay ddi hinh dang ctra hinh. Trudc h0t, trong bii nly, ta n6i ddn ph6p vi tr,r, m6t tru&rg hqp riOng cira nhfrng ph6p bien hinh nhu thd. 1. Dlnh nghia Cho milt didm O cd dinh vd mdt sd k kh6ng ddi, k + 0. Phip bieh hinh bi1n mdi didm M thdnh didm M' sao cho + OM' = kOM duqc sqi ld phdp vi tu tdm O rt sd k. Ta thuong kf hiOu ph6p vi tU boi chfr y, ndu cAn n6i 16 tdm O vd ti s6 k cha n6 thi ta ki hiQu ldvp, p7. Hinh 19 cho ta thdy ph6p vi tu t6m O ti sfi k = 2 vh,ph6p vi tu tam 01 ti sd I ,c1 ,2- -- biOn hinh U( thitnh c6c hinh nhu th€ nho. t- _ Hinh 19 24 2. Citc tfnh chdt cfra ph6p v! ttr DINH LI 1 Ndu phip vi tttr ti sd k bieh hai didm M vd N ldn luot thdnh hai didm M'ri N'thi Mfr = kMfi' vd M,N, =ltlUN. Chitng minh Ndu O lh tam cira ph6p vi tu thi theo dinh nghia, ta c6 Ofr = kofr, ON' = kON. v4y rwfr : ofr - ofr = kofi - kofr = kert -m) = kMfi. Tt d6 suy ra M'N' =l*l1/'ttt. n DINH Li2 Phip vi tu bi€h ba didm thdng hdng thdnh ba didm thdng hdng vd khAng ldm thay ddi tht M cfia ba didm thdng hdng d6. Chirng minh Gii sfr ba didm A, B, C thing hdng mI I nam giita A vh C, tfc li EA = *Ed vdi m< 0. Neu ph6p vi tu ti sd k bidn A, B, C ldn luot thd nh A', B', C' thi theo dinh li l,tac6 ET = kdi, yd : kde . .'' Tt d6 suy ra B'A' = kBA = k(mBC) = m(kBC) = mB'C', trlc li ba didm A', B', C thing hing vdi B'nam glrta A' vd C'. tr nE ouA Phip vi tu ti sd k bidn dudns thdng thdnh dudng thdng song song (ho\c trilng) vdi dudng thdng d6, bi€n tia thdnh tia, biAn doqn thdng thdnh doqn thdng md dQ ddi dtoc nhdn l€n vA lkl, bidn tam gidc thdnh tam gidc ddng dang vdi ti sd ddng dang ld lkl, biah g6c thdnh g6c bd:ng n6. fl Nnnng dudng thdng ndo bidn thdnh chinh n6 qua phdp v! ttt voi ri sd k + I ? Nhfing dtdng trdn ndo bidn thdnh chinh nb qua phdp v! M vdi ti sd k * I ? ,25 3. Anh cia dudng trdn qua ph6p vi tU DINH LI3 Phdp vi tu ti sd k biah dudng tdn c6 bdn kinh R thdnh dudng trdn c6 Odn ktnh lklR. Chitng minh (h.20) Gie sir V le phdp vi tu tdm O ti sd ft vd (/ ; R) ld duong trdn dd cho. Goi 1'ld inh cria l vd M'ld hnh cfia didm M btitki thi ta c6 I,M,=ltltu. B&i vfly IM = R khi vd chi khi 1'114'=lklR hay le M' thuQc duong trdn (1'; R) vdi fi' = ltlR. D6 chinh le inh cira dudng o trdn (/ ; R) qua ph6p vi tu V. n 1 Hinh 20 TrOn hinh 2O,hdy v6 m6t duong thtng d qua tdm vi ttJ O, c5t duong trdn (1 ; R) tai A vir B, cftl dudng trdn (.I'; R) tai C vd D. HEy n6i 16 c6c didm A vir B ctrroc bidn thdrnh nhilng tlidm nio qua ph6p vi trl d6, vi giAi thfch tai sao. Ndu dudng thtng d n6i tr6n ti6p x0c v6i durdng trdn (1 ; R) thi d c6 ti6p xrlc v6i dudng trdn (/'; R) hay kh6ng ? Nhdn x6t gi vd c5c ti6p didm ? 4. Tim v! tU c0a hai dudng trdn Ta d6 bidt rang ph6p vi tg bidn ducrng trdn thinh duong trdn. BAy gid ta x6t bii todn nguo. c lai. Biri toSn I Cho hai dudng trdn (l ; R) vd (l' ; R) phan biot. Hdy tim cdc ph€p vi tt1 bi€n dudng trdn (l ; R) thdnh dudng trdn (l' ; R'). Gitii Trudc hdt, ta c6 nhfln x6t : Ndu ph6p vi tU tam O ti sd ft bidn (/ ; ft) thinh (I' ; R') thi lkl = R' huy k = t+ vit oF = koi. Tu d6 ta xdc dinh dugc R --_J -R cdc ph6p vi tg md bii to6n y€u cdu. Cu thd li : 26 Trudng hqp hai dudng trdn (l ; R) vd (l' ; R) ddng tdm, R + R', hidn nhiOn khi d6 O trirng v6i 1. VAy ta c6 hai ph6p vi tg : ph6p vi tg Iz1 ttdm I ti sO { vd ph6p vi tV vzram 1 ti so -{. (Tron R h'inh 21, ph6p vi tu V1 bidn M-thinh M'1vd ph6p vi tuV2bidn M thinth M'). Trtdng hW I khbng trilng vdi I' nhung R = R',.trtC h k:*1, khi d6 didm O phii thoi mdn didu kicn D7' = koi nOn t chi c6 thd beng -1, viL O ld trung didm cira doan thing II'. YQy trong trudng hgp niy chi c6 mOt ph6p vi tu : tam O, ti sd -1, d6 cflng chinh ln ph6p d6i xung qua didm O (h.22). Hinh 21 Hinh 22 Trtdng hW I khilng trilng 1'vi R * R', ta c6 thd x6c dinh cdc ph6p vi tu nhu sau (h.23) : Hinh 23 Ta ldy M'rM'zld mOt duong kirih cria Q' ; R) vd IM ld m6t b6n kinh cira (1;R) sao cho hai vecto Vfr1 vi, ffi cung hu6ng. Duirng thing II' c6t MM'tvd MM'2 ldn lugt tai Olvd O2. Khi d6 ph6p vi tU yl m{n O1 ti sd t, = n vd ph6p vi tu V2 ttm O2 ti s6 ^ k, = -{ ddu bien duong trdn (1;R) thenh dudng trdn (1'; R). tr .R ThuAt ngif Ndu c6 ph6p vf tq t6m O bien duong trdn niy thinh duong trdn kia thi O dugc ggi lit tdm vi ttt cfia hai iludng trdn d6. , NOu ph6p vi tu d6 c6 ti sO duong thi didm O eqi lit tdm vi tq ngodi, ndu ph6p vi tg d6 c6 ti sd 0m thi didm O ggi ld tdm vi tu trong. TrOn hinh 23,haiduong trdn (1 ; R) vd (l' ; R) c6 O1li tAm vi tu ngoii, Oz ldtdm vi tu trong. 5. Ung dgng cfra ph6p v! tU Biri to6n 2 Tam gidc ABC c6 hai dinh B, C cd dinh cdn dinh A chay tr1n mQt dudng trdn (O ; R) cd dlnh kh1ng c6 didm chung voi dudng thdng BC.Tim quj tich trqng tdmG cila tam gidc ABC. Gidi (h.24) Gqi 1 li trung didm cira BC thi 1cd dinh. Didm G li trong tAm tam gi6c ABC khi vd chi khi 1- IG = 1IA. 3 Htnh 24. Nhu vAy ph6p vi tu V tam 1ti so 1 Uign didm A thdnh didm G. Tt d6 suy ra 3 khi A chay trOn duong trdn (O ; R) thi qu! tich G Id inh cira dudng trdn d6 qua ph6p vi tu v, trlc li duong trdn (o' ; R') md 7d' = lt| ,u R' = 1R. E JJ Biri toin 3 Cho tam gidc ABC voi trong td.m G, trac tdm H vd tdm dudng trdn ngoqi tidp O. Chfing minh rdng GH = -2GO (nhu vQy khi ba didm G, H, O khilng trr)ng nhau thi chthng ct)ng ndm tr€n mQt dtdng thdng, dttoc gqi td drdng thdng O-le). 28 2 (Dd giSi biri to6n 3) Goi A', B', C' ldn ludt la trung didm c6c canh BC, CA, AB c0a tam gi6rc ABC (h.25). 1) Hey chr?ng minh ring O ld truc tdm c0a tam gi6c A'B'C'. B 2) Goi 7 ld ph6p vi tu tAm G, t? sd -2. Hdy tim 6nh c0a tam gi5c A'B'C'quaV. 3) Qua ph6p vi ttJ 7, didm O bi6n th?rnh didm ndo ? M sao ? TiJ d6 suy ra kdt luan c0a bdi to6n. Hinh 25 @ Cqi O' ld tdm dudng trdn ngoai ti€p tam gidc A'B'C'. eua ph6p vi tuV n6i tr€n, didm O' bieh thdnh didm ndo ? cou n6i vd bdilQp 25. 27. It, Cdc ph6p sau dAy c6 phii ld ph6p vi tu hay kh6ng : ph6p ddi xrlng tAm, ph6p dtii xfng truc, ph6p ddng nhat, ph6p tinh tidn theo vecto ktr6c d ? C6c khing dinh sau ddy c6 dring kh6ng ? a) Ph6p vi tu 1u6n c6 didm b6t d6ng (tfc le didm bidn thlnh chinh n6). b) Ph6p vi tu khOng thd c6 qu6 m6t didm bat dQng. c) Ndu ph6p vi tu c6 hai didm bat dong phan bi0t thi mgi didm ddu bdr d0ng. X6c dinh tam vi tu trong vi tam vi tu ngoii ctra hai dudng trdn trong c6c trudng hqp sau : a) Hai duong trdn tidp xfc ngodi vdi nhau. b) Hai duong trdn ti6p xric trong vdi nhau. c) M6t dudng trdn chrla duong trdn kia. Cho hai duong trdn (O) va (O) c5t nhau tai A vI B. Hdy dung qua A mOr duong thing d cat (O) b M vir c1l- (O)O N sao cho M li trung didm cira AN. I Cho duong trdn (O ; R) vd didm l cO dinh khdc O. MOt didm M thay ddi 29. t trOn du&rg trdn. Tia phfln gi6c ctra g6c MOI c6t IM tai N. Tim quf r(ch I didm N. Et Cho hai dulng trdn (O) vd (O) c6 b6n kinh kh6c nhau, tiOp xric ngodi vdi 30. nhau tai A.,MOt duong trdn (O") thay ddi, lu6n luOn tidp xric ngoii v6i (O) vn (O) ldn luot tai B vi C. Chung minh rdng duong thfrng BC 1u6n di qua mOt didm cd dinh. ZA PHEP DONG DANG 1. Dlnh nghla ph6p ddng d?ng Phip bidn hinh F gqi ld phdp ddng dqng ti sd k (k > 0) neu vdi hai didm bdt ki M, N vd dnh M', N'cfia chfing, ta c6 M'N'= kMN. fl rnAp ddi hinh vd phdp vi ta c6 phdi ld nhrtng phip ddng dang hay khbng ? N€'u c6 thi ti sd ddng dqng ld bao nhiAu ? Ggi 7 ln ph6p vi tu t0m O ti sd k vit D ld m6t ph6p ddi hinh. V6i mdi didm M bdtki, 7 bidn di6m M thinh didm Ml vit D bidn tlidm M, thirnh didm M'. NhtJ v6y ta c6 mQt ph6p bidn hinh F bidn di6m M thdnh didm M'. C6 thd n6i F c6 dugc bing c6ch thr/c hiQn li6n tidp hai ph6p bidn h)nh V vit D. Ngudi ta cdn n6i ring F lit phdp hW thenh c0a hai ph6p bidn hinh V vir D. Hdy churng t6 ring F ii m6t ph6p cl6ng dang ti sd lfl. Nhu vay, nou thuc hiOn liOn tiep mot ph6p vi tu vI mQt ph6p ddi hinh thi kdt qui li mOt ph6p ddng d?ng. Didu nguoc lai c0ng dring. Ta c6 thd chrlng minh duoc dinh li sau dAy. 2. Dlnh li Mqi phip ddng dang F tl sd k ddu ld ho,p thdnh cfia mQt phdp vi tttrV ti sd kvd m1t phdp ddi hinh D. Hg OuA (tfnh chat ctra ph6p ddng dang) Phdp ddng dang bi|n ba didm thdng hdng thdnh ba didm thdng hdng (vd khbng ldm thay ddi thfi tt ba didm db), bidn dudng thdng thdnh dudng thdng, bieh fia thdnh tia, bi€h doan thdng thdnh doan thdng md dQ ddi duoc nhdn l€n voi k (k ld ti sd cila phdp ddng dqnd, bi€n tam gidc thdnh tam gidc ddng dqng vdi ti sd k, biAh dudng trdn c6 bdn ktnh R thdnh dadng trdn c6 bdn kinh kR, bi€h g6c thdnh gdc bdng n6. 30 @ CA phdi moi phdp ddng dang ddu bi€h dadng thdng thdnh dudng thdng song song hodc trilng voi n6 hay khong ? 3. Hai hinh ddng deng Tr0n hinh 26. tac6 hai h\nh,1(vir J(r uiru vdi nhau (nghIa rd c6 ph6p vi tu v bidn hinh gfthinlhh\nhff1). hai hinh 14uir5('bdngnhau (nghia ld c6 ph6p ddi hinh D biOn h\nh&fi thhnh h\nh#('). o' inh 26 Ndu goi F li ph6p hqp thinh ciav vi D thi F li ph6p ddng dang bienh\nhz( thdnh h\nhJ('. Ta n6i rlng hai hinh fihd /(' ddng dang voi nhau. Nhu v0y ta c6 : DINH NGHIA Ill Aoi hinh goi td il6ng dgng voi nhau n\'u c6 phtip d6ng dang ll Oien hinh ndy thdnh hinh kia. e CHU'i O l6p 8, ta da bidt thd nio ld hai tam gi6c ddng d+ng. Kh6i niem d6 phn hqp vdi dinh nghia trOn. c6u h6i vd bdi tQp Chung t6. r[ng ndu ph6p ddng dang F bidn tam giilc ,q,nC tninh tam gi6c 31. A'B'C'thi trong tam, truc tAm, tAm dudng trdn ngoai tiOp tam gi6c ABC ldn luot bidn thinh trong tAm, truc t0m, t6m dudng trdn ngoai ti0p tam gi6c A'B'C'. Chtmg t6.r6ng cdc da giSc d6u c6 cirng s6 canh thi ddng dang vdi nhau. 32. 3l 33. Dmg tam gi6c ABC ndu bidt hai g6c 'B = B, C - f viL mOt trong c6c ydu td sau : a) Duong cao AH = h ; b) Duong trung tuYdn AM = m; c) B5n kfnh R ctra dudng trdn ngoai tidp. ON TAP CHUONG I | - T6m tit ntrfrng kidn thrlc cdn nh6 1. Ph6p ddi hinh ld ph6p bien h)nh kh6ng llm thay dtii ktroing c6ch gifra hai didm bat ki, nghia li ndu ph6p ddi hinh biOn hai didm M, N ldn luot thdnh hai didm M', N'th\M'N'= MN. 2. Cdc tinh chat c[ra ph6p ddi hinh : biOn ba didm thing hing thinh ba didm thing hlng vi khdng lim thay ddi thrl tu ba didm d6, biOn duong th&ng thdnh duong thing, bidn tia thinh tia, bidn doan thEng thdnh doan thing bang n6, bi0n g6c thinh g6c blng n6, biOn tam grdc thdnh tam gi6c blng n6, bidn duong trdn thinh duong trdn c6 cing bdn kinh. 3. C6c ph6p ddi hinh cu thd : a) Ph6p tinh ti6n T; (theo vectd rt) bi6n m6i didm M thlnh didm M'sao cho MM' = il. b) Ph6p doi xtmg trlc D4 (truc ld du&rg thing d) bidn m6i didm M thdnh didm M'ddi xring vdi M qta d. c) Ph6p guay Q@, q1 $rlrn O, g6c q)ay q) bi6n Othenh O, bidn m6i didm M kh6c O thdnh didm M'sao cho OM = OM' vdg6c luong giSc (OM, OM)bFng 9. d) Ph6p doi xring tum Ds(mm li didm O) bidn m6i didm M thlinh didm M' doi xtlng v6i M qua O. 4. Dinh nghia vd hai hinh bing nhau : Hai hinh goi li blng nhau nOu c6 ph6p ddi hinh biOn hinh ndy thinh hinh kia. 5. Ph6p ddng dang ti sd k (k > 0) 1A ph6p bion hinh bion m5i cip didm M, N thinh clp didm M', N'sao cho M'N' = kMN. 6. Ph6p ddng dang c6 cdc tinh chdt : biOn ba didm thing hdng thdnh ba didm , ttring trang fva thdng lim thay ddi thrl tu ba didm d6),-bi0n dudng thing 32 thinh duong thtng, bidn tia thinh tia, bidn do4n thing thinh doqn thlng mi do dii duo. c nhan .l€n v6i k (t li ti sd cta ph6p ttdng d?ng), bi6n tam giSc thanh tam gidc ddng dang vdi ti s6 t, bi6n mOi g6c it a"n g6c c6 cing s6 do, bidn duong rrdn b6n k(nh R thinh duong trdn C6 b6n kinh tR. 7- Ph6p vi ty vp, q tdm o ti sd k (k + 0) bidn m6i didm M thinh didm M, sao cho Ofr = kOil. 8. cdc tinh chdt cira,ph6p vi tqr : ph6p vi tu tam o ti sd t li mor ph6p ddng dang ti sd lti nOn c6 cdc tinh chdt cira ph6p ddng dang. Ngoii ra, ph6p vi tu c6 tinh chdt dac biOt sau : du&rg thing n6i mot didm vn 6nh cira n6 lron luon di qua o ; hnh d' cria du&rg thing d lu0n song song hoac tring va d. 9. M6i ph6p ddng dang bao gid cfrng c6 thd xem lh hqp thanh cta mOt ph6p vi tu vi rnOt ph6p ddi hinh. 10. Dinh nghia vd hai hinh ddng d4ng : Hai hinh duo. c ggi li d6ng d4ng v6i nhau n€u c6 ph6p ddng dang bidn hinh niy thdnh hinh kia ll - C5c ciu h6i tr,r kidm tra 1. Cdc khtng dinh sau ddy c6 dring khOng ? a) Phdp ddng nhAt li m6r ph6p tinh tidn ; b) Ph6p ddng nh6t li mOt ph6p quay ; c) Ph6p d6ng nh0t li mOt ph6p dtii xrlng tAm ; d) Ph6p ddi xfng tam li m6t ph6p vi tq; e) Ph6p quay ld m6t ph6p ddng dang ; f1 Ph6p vi tu li m6t ph6p ddi hinh. 2. Cho hai didm A, B ph0n biOt. cdc khirg dinh sau dfly c6 dring kh6ng ? a) C6 duy nhdt m6t ph6p dtii xrfng truc bi€n A thenh B ; b) C6 duy nhAt m6t ph6p dtii xrlng tAm bi€n A thlnh B ; c) C6 duy nhdt mQt ph6p tinh tidn bien A thinh B ; d) C6 duy nhdt mQt ph6p quay bi6n A thanh B ; e) C6 duy nhAt mOr ph6p vi tu bidn A thhnh B. 3. H[y chi ra mOt sO hinh c6 m6t trong c6c tinhchat du6i day : a) C6 v0 s6 truc ddi *fog ; b) C6 v0 sd mm d6i *tog; c) C6 ding n tr\rc ddi xtlng. g. uiruurrruc-n ilr - Bii t?p \ f . Cho hai duong trdn (O ; R), (O'; R) vi m6t dudng thhng d. 1 af Tim hai didmM, N ldn luot nAm tr€n hai duong trdn d6 sao cho dli I .1 i Auong trung truc ctta doan thing MN. b) X5c dinh didm 1 tran d sao cho tiep tuydn IT crta Q ; R) vh tiep tuy€n IT' cta (o' ; R') hqp thlnh c|c g6c mi d li m6t trong c6c du&rg ph6n gi6c cliua cdc g6c d6. 2. Chtmg minh rang neu m6t hinh ndo d6 c6 hai truc ddi xung vu6ng g6c v6i nhau thi hinh d6 c6 tam d6i xfug, 3. Cho dudng thing d di quahai didm phan biet P, Q vh, hai didm O4"eatl mor phia doi v6i d. Hdy x6c dinh ffan d hai didm M, N sao cho MN = PQ vit AM + BN b6 nh0t. 4. Cho vecto il vd m6t didm O. Vdi didm M xrlng v6i M qaa O vd M'ta a b ldn luot thinh a,vd. b,? (A) Khong c6 ph6p tinh tidn nio ; (B) c6 duy nhdt mOt ph6p tinh tien ; (c) chi c6 hai ph6p tinh tidn ; (D) c6 rdt nhidu ph6p tinh tidn. 3. Cho hai ducrng thing c6t nhau d vd d'. C6 bao nhiOu bicn d thdnh d'? (A) Kh0ng c6 ph6p ddi xtrng tru0 nlo ; (B) C6 duy nhdt mOr ph6p ddi xrlng truc ; (C) Chi c6 hai ph6p ddi xrlng truc ; (D) C6 rAt nhidu ph6p ddi xrlng rruc. ph6p d6i xfng truc 4. Trong c6c hinh sau dAy, hinh nio c6 bdn rruc dOi xfng ? (A) Flinh binh hdnh ; (C) t{inh thoi ; (B) Hinh chfr nhAt ; (D) Flinh vu6ng. 5. Trong c6c mOnh dd sau, mOnh dd nio sai ? (A) Flinh gdm hai dulng rrdn kh6ng bing nhau c6 truc ddi xrlng ; (B) llinh gdm mor duong trdn vi m6t doan thing tu] f c6 truc ddi xring ; (C) Flinh gdm mot dudng rrdn vi,mOt dudng thing tu! f c6 rr\rc d6i *fog ; (D) Flinh gdm mOt tam gidc can vI duong trdn ngoai ti6p tam gi6c d6 c6 truc d6i xring. 6. Trong cdc hinh sau dAy, hinh nho kh6ng c6 tlm dOi xung ? (A) Flinh gdm mot duong trdn vi mQt hinh chfr nhat n6i tiep ; 35 (B) Ffinh g6m m6t dudng trbn vi mOt tam gi6c ddu nOi tidp ; (C) Hinh iuc gi6c ddu ; (D) Flinh gdm m6t hinh vuOng vi duong trdn n6i ti€p. 7. Cho hinh vuong ABCD tarm O. X6t ph6p qray Q c6 tam quay,O vi g6c qxlay g. v6i gi6 tri ndo sau d6y c]idla g, ph6p quay Qbidn hinh vuong ABCD thinh chinh n6 ? (c) rp =[, (A)cP=!; @) a =;. o (B) p =f,, 8. Cho hai duong thing song song d vd d'. C6 bao nhieu ph6p vi tg v6i ti s6 ft = 100 bi6n d thdnh d' ? (A) Khong c6 ph6P ndo ; (C) Chi c6 hai ph6P ; (B) C6 duy nh6t'm0j ph6p ; (D) C6 rat nhidu ph6p. g. Cho duong trdn (O ; R). Tim mOnh dd sai trong c6c menh dd sau day : (A) C6 ph6p tinh tiOn bidn (O ; R) thinh chinh n6 ; (B) C6 hai ph6p vi tu biOn (O ; R) thinh chinh n6 ; (C) C6 ph6p ddi xrlng truc bi6n (O ; R) thenh chfnh n6 ; (D) Trong ba menh dd A, B, C, c6 itnhdt mot mOnh dd sai' 10. Trong ci{c mOnh dd sau dAy, m0nh dd nho sai ? (A) Tam vi tu ngoii cira hai dudng trdn nam ngoli hai dudng trdn d6 ; G) Tam vi tu ngoli ctra hai duong trbn khong ndm gifra hai tam cta hai duong trdn d6 ; (O Tam vi tu trong cta hai dudng trdn lu6n thu6c doan thEng n6i tam hai dudng trdn d6 ; (D) Tam vi tu cira hai dudng trdn c6 thd le didm chung ctra ci hai duong trdn d6. 11. Ph6p biOn hinh ndo sau dAy kh0ng'c6 tfnh chi't : "Bidn mOt dudng thing thdnh dudng thing song song hoic trtng vdi n6" ? (A) Ph6p tinh tien ; (C) Ph6p ddi xfng tryc ; (B) Ph6p ddi xrlng tlm ; (D) Ph6p vi tu. L2. Trong c6c mOnh dd sau d0y, mQnh dd ndo sai ? (A) Ph6p ddi hinh ld mOt ph6p d6ng d?ng ; (B) Ph6p vi tU li mQt Ph6P ddng deng ; (C) Ph6p d6ng d4ng li mQt ph6p ddi hinh ; (D) C6 ph6p vi tu kh6ng phii ld ph6p ddi hinh' 36 Eai dge fhem nNn ru obuc DANG vA nhml Hoc rpAC-T N (rpACrAr) Hinh tro^ng Jni.t pntrlO duoc goi lit hinh tq ct6ng dang ndu m6i mdu nh6 crja n6 ddu :l*.T0l9g fal 9:lq.9qns v6i.!in!.d6, t,rc 6 rrri p-nons to oO pnan n;y G;;gt rt so thich hdp, ta c6 th6 ddt chdng khft t6n hinh dE cho. vi du : doan thEng, hinh tam gi6c ddu, hinh vu6ng la nhfing hinh trr d6ng dang. fnidy h!nf, !y ddng dang duoc xdv dung bing phLrong phdp t{p (xay dung theo trrng bu6c). Vi du : 'T?p cing-to (cantor) : cho m6t cloan thtng. d nuoc m6t, chia doan thtng d6 thdnh ba doan con bing nhau rdi xo6 khoang I gi0a (kh6ng td nai mrit). d-m6i nu6c tidp theo, chia m& doan chua xo5 tnain ui d"il ;;; ;"iil';;;; # I;; kho6ng 6 gi0a (kh6ng td naihr:ty. ca tdm thd mat ini rrinn "on rri tir t6p cdng-to. %WW Xoa thd mdithi phdn cdn tai td ,,tQp Cdng_to,,. :_D_r_d:g J:". Ta".tVon Koch) : Cho m6r doan thtng. d ntr6c m6t, chia doan thing tl6 thdnh ba doan con bing nhau, dr,rng iam giac ddu tr6n doin "on a gi0, r6i xo5.cqnh day.crta tam gi5c d6 thi drroc m6t duong g5p khric. o m6i ouo"iidp theo, chia m6i doan c0a duong gdp. khtic thanh ba ooah con bing nhau, dLrni tam gi6c ddu tren boqn con o g'irli idi xoa "a"rr oav .oliailia" d6. cf ram thd miithi duoc "dudng V6n Kdc". Dung thd mdi thi duoc "dudng V6n K6c,,. '-]fa.m.xfepin-xki (sierpin.ski) : cho m.6t hinh vu6ng. 6 nuoc m6t, chia hinh vu6ng cf6 thinh t hinh vu6ng con b5ng nhau (bing c6c tfoan thtng song'song v6i c6c cant hinh vu6ns) rdi xo5 hinh^vu6ns"cd 6;hi.islo; ar.r,6;;;oX li"'"rrn) thi drrdc hinh gqT .8 hinh vu6ng gon. d bu6c hai, lai chia m6i hinh vu6ng con chua xod niy thinh 9.hinh vu6ng con bing nhau, rdi xo6 hinh vu6ng con 6 chin-h gifra. Crl tdm thd m6i thi hlnh cdn lai li "th6m X6c-pin-xki' Xoa thd mdi thi phdn cdn lqi b "them Xec-pin-xki',. 37 Nhidu hinh trJ d6ng dang phtlc tqp nhrr th6 li nh0ng d6i trlgng nghiOn .,1r-.Ut Hinh hgc frac-tan, mQt m6n hinh hoc duoc khdi ddu nghiEn crru tir cudi tnd fi XX bfii nhi toin hoc Man-den-br6 (Benoit Mandelbrot) nhim m6 tA hinh hoc nhidu cdu tnic g?p gay, 96 ghd, ldi l6m, ki di, h6n dQn, ... c0a nhi6u hi6n trjong vQt li, ttJ nhi6n. ffinn nqifric-tan c6n nghiOn ctru cA nhClng hinh kh6ng tqf d6ng dang nhtr .b6ng tuydt V6n Kdc". . B6ng tuyat V6n Kdc drroc xay drJng bing phtrong phep ldp nhr/ sau : Cho tam gi6c ddu. d Uu6c m6t, chia m6i canh c0a tam gi6c thAnh ba dogn bbng nhau, Iung tm gi6c tl6u tr@n tloan 0 gifra (6 b6n ngodi tam gi6c tlfi cho) rdi xo5 canh d6y c0a tam gi6rc ddu niry thi drldc m6t dtlong gdp kh0c kin. 6 m6l Uu6c tidp theo, chia m6] doan c0a:drrdng gdp kh0c kin tnann ba doan con bing nhau, dr,rng tam gi6c ddu tr6n doan con 6 gifra (6 b6n ngoiii dttdng gdp khfic kin d6) rdi xia cqnh d6y. Cr? lim thd mai thi dudc "b6ng tuydt Von Kdc". Dqng th6 m^i thi duQc'b6ng tuy6t v6n K6c". 38 DUoNG rnAno vn urdr pniruc rRoNc ruOruc GIAN. QUAN SONG SONG oydm, duong tnEng vd mit phEng ld nhrlng khii ni6m quen thuOc trong ddi sdng hang ngdy c0a ch0ng ta. chfng c0ng tir nh0ng d6i trrong co o6n c0a hinh hoc kh6ng gian. Tir ch0ng, ta c6 thd tao n6n nhfng vqt ind tnac nhau nhU: hinh ch6p, hinh ling tru, hinh n6n, ... Hoc xong chrJong niy, hoc sinh cdn n5m vrlng : c6ch x5c dinh mdt phing ; mdi quan hQ gi0g c5c cfrrong thEng, giira c6c mdt ph8ng, giira cdc drrong thtng va m5t phEng, cfic biet ti quan hO song song gifrJc[,ing ;c5ch xdc dinh thidt di6n c0a m6t hinh khi c5t bdi m6t mdt phEng ; c6ch vc ninn oidu oi6n va caciinn chdtc0a hai hinh quan trons ti hinh.[op, ninn;ili;$ -- DAI CUONG Vf uAr psANG DuoNc TH.E NG vA ? ntl6 d:lu vi hinh hgc kh6ng gian Trong chuong trinh hinh hoc 16p 10 vi chuong I cira lop 11, ta chi n6i d6n nhfrng hinh trong mqt ph&ng nhu : tam gi6c, dudng trdn, vecto, ... Chring duo.c goi ld nhtng niin phdng. Nhtmg xung quanh chring ta cdn c6 c5c hinh khOng nam irong mlt phfrng nhu : cay brit chi (h.28), quydn s6ch (h.29), quA b6ng (h.30), ng6i nhi (h.31), ... Hinh 28 Hinh 29 Hinh 30 Hinh 3l MOn hoc nghi€n ctlu cdc tinh chdt cila nhrtng hinh c6 thd khbng cilng ndm trong mbt mdt phdng gqi ld Hinh hoc kh1ng gian' M{t phing Trang gidy, m4t bing den, mdt ru&rg l6p hoc, mat h6 lang gi6, mat ban, tam gudn! phiog, ... cho ta hinh inh mQt phdn m4t phing trong kh6ng gian' Ngudi ta thubng bidu di6n mQt m4t phing bang mQt hinh binh hnnh G.32) vh ding mQt chfr cdi dlt trong ddu ngof,c ( ) dd dat tOn cho m4t phfrng lty. Yi du : mdt phing (P), m{t phing (Q), m4t phing (a), m4t phing (h "' Hinh 32 vi vi6t t6t li mp(P), mp(Q), mp(a), rnp(P) "' holc (P), (Q), @), (b "' Diim thuQc m4t Phing Ta bidt rang khi cho didm A vd duirng thhng a thi ho[c didm 4 thu6c duong thlng a,hoAc didm A kh6ng thuQc dudng thing a. Tuong tu nhu vly, v6i mQt didm A vd mQt mlt phing (P), cfrng c6 hai khi ndng xiy ra : 40 | ::- r: ?ry - Hodc didm A thuQc mp(P), khi d6 ta kf hiQu A e mp(P) hay A € (P)' - Hopc didm A kh6lg rhuQc mp(P), ta cdn n6i didm A & ngohi mp(P) vh kf hiQuAemp(P)hayA e(P). W ruay quan sdt htnh 33. km mdt bdn ld mQt phdn cila mqt phdng (P). Trong cdc didm A, B, C, D, E, F, G, H, I, K, L, didM NdO thu\c mdt phdng (P) vd didtn ndo khilng thu1c mdt phdng (P) ? Khi didm A thuOc m4t Phing (P), ta cdn n6i : "ilidm A ndm trAn mdt Phdng (P)" hay "ilidm A ndm trong m\t Phdng (P\", ho{c cdn n6i"mdt phdng (P) di qua didm A" hay "mfit Phdng (P\ chtha didm A" ' Hinh biiu diOn cira mQt hinh trong khOng gian Htnh 33 Hinh lap phuong ld hinh nlm trong khOng gian, n6 c6 s6u m[t li hinh vu6ng. ffUn tf di0n cf,ng li hinh n6m trong khOng gian, n6 c6 bdn mlt li tam li6c. Dd de hinh dung, ngudi ta tim c6ch vE chfng thinh nhfrng hinh phing, gqi th hinh bidu d.i6n ciua c6c hinh kh6ng gian d6' t' I I.I I I I I I Hai htnh bidu di6n c'rta hinh ldp phuong Hinh 34 I I I t I I Hai htnh bidu di6n crta hhh rA diQn Hinh 35 Hinh lnp phuong, hinh tf dien khong phii li nhfrng hinh phing nhtmg c6c hinh biSudi6n cira ohring duo.c vE trOn mf,t phing. Tuy th€, c6c hinh bidu di6n'cfrng tao cho chring ta cim gi6c nhu dang nhin thlty hinh lQp phuong, hinh tf di0n. Dd vd hinh bidu di6n cira mQt hinh trong khong gian, ngudi ta dua ra nhfrng quy tic thucmg dugc 6P dung nhu: - Dudng thdng duo.c bidu di6n bdi dudng thdng. Doqn thdng duqc bidu di6n bdi doqn thdng 4l - Hai dudng thdng song song &oqc cdt nhau) duoc bidu cti€n boi hai duong thdng song song (hodc cdt nhau). - Didm A thuoc drdng thdng a droc bid"u didn bdi m6t didm A' thu6c dudng thdng a', tong d6 a' bidu di6n cho dudng thdng a. - Dilng nit vd lidn (-) dd bidu didn cho nhfrng drdng tr6ng rhdy vd dting nit drh doqn (- - -) dd bidu di€n cho nhfing dudng bi khudt. Cdc quy tic kh6c, chring ta sE duo. c hoc sau. I & VE hinh bidu di6n c0a mp(P) vi m6t drlong thEng a xuy6n qua n6. & 2 V6 m6t sd hinh bidu diSn c0a hinh tr3 di6n. C6 thd v6 hinh bidu di6n c0a hinh trl di6n mi kh6ng c6 n6t drlt doan niro hay.khOng ? 2. Citc tinh chdt thrla nhQn cfra hinh hgc kh6ng gian Do thuc ti6n, kinh nghi€m vi quan s6t, ngudi ta thta nh6n m6t sd tinh ch1t sau dAy cta hinh hoc kh6ng gian. Tfnh chdt thila nhAn 1 Cd mil vd chi mdt dudng thdng di qua hai didm phdn bi€t cho trudc. Nhu v0y, hai didm phfln bi0t A, B xdc dinh duy nhdt mot duong thing. Duong th8ng d6 duo. c ki hi€u li duong th8ng AB hoac ngin gon liL AB. Tfnh chdt thira nhAn 2 Cd mdt vd chi mdt mdt phdng di qua ba didm kh6ng thdng hdng cho trudc. Nhu vfly, ba didm kh0ng thing hing A, B, C xdc dinh duy nhat m6t mat phing. Mat phing d6 duoc kf hi0u ld mit phing (ABC) hay mp(ABC) hay ngSn gon ld (ABC). Trong thuc t€, kidng ba chan hoic c6c giri dd ba chan khi dat trOn mat dar khong bi c4p k€nh vi theo tinh chdt thta nhdn2, ba didm khOng thing hing ndo c0ng x6c dinh mOt mit phing. 42 Kidng ba chdn Hinh 36 Tinh chdt thila nhAn 3 Gid dd ba chdn Hinh 37 Tdn tai bdn clidm khdng cilng ndm tr€n mil mdt phdng. Nou c6 nhidu didm tl1uoc mot mf,t phing thi ta n6i rlng c6c didm d6 d6ng phdng, cdn ndu khong c6 mat phing nio chrla c6c didm d6 thi ta n6i rang chring khilng il6ng Phdng. Nhu v6y, tfnh ch0t thira nhAn 3 c6 thd duo.c ph6t bidu nhu sa:u : T6n tai bdn didm khOng ddng phdng. 3 Gi6 sfi (P) li mQt m6t phtng ndro d6. Chfng minh ring c6 (t nhdt m6t clidm kh6ng thu6c mp(P). Tinh chdt thira nhAn 4 NA'u hai mdt phdng phdn biAt cb mdt didm chung thi chrtng c6 mOt dudng thdng chung duy nhdt chfta tdt cd cdc didm chung ctia hai mdt Phdng d6. Gii sfr (P) vn (Q) lit hai mit phing phan biot c6 didm chung A. Theo t(nh chdt thla nhan 4 rhi (P) ve (o) c6 dulng thing chung duy nhnt a di qua didm A. Duong thing a d5 duoc goi ld giao tuyan cila hai mil phdng (P) ve (0), cdn n6i hai mf,t phing (P) vn (Q) cdt nhau theo giao tuydn a, k( hi€ua =e)n(Q). 43 @ guydn vd ghi bdi dang d tu6c mdt cdc em (h.3g). Hai bta vd ld hinh dnh cila hai m(t phdng phdn bigt.Vdy giao tuy€h cila chfing ld gi ? Tfnh chdt thila nhfln 5 N Htnh 38 Trong m6i m\t phdng, cdc kAi qud dd bi1't crta hinh hoc phdng ddu dung. Ta sE thdy ring trong)a D', C vd A', D vd, B', vh c6 c6c dudng ch6o ld AC', BD', CA', DB'. Hinh hOp c6 mudi hai canh chia ldm ba nh6m, m6i nh6m gdm c6 bon canh song song vh bang nhau. Hai canh goi ld hai canh ildi iliQn ndu chring song song nhung khOng cing nam trOn bdt ki m6t m4t ndo ctra hinh hOp. r 2 hQp gqi ld haiitinh ddi d.iQn ndu Doan thing ndi hai dinh ddi diOn .Br ,---r--:x:- A"r:!:----)'---. ,;1" '.. Hinh 7l Chrlng t6 ring bdn duong ch6o c0a hinh h6p c5t nhau tai trung Clidm c0a m6i duong. Didm cit nhau d6 goi ld tdm cia hinh h6p. 6. Hinh chop cgt Dinh nghia Cho hinh ch6p S.A1A2...A, vdmQt m4t phang (P) kh6ng qua dinh, song song v6i m{t phang ddy., cAt c6c canh SAl. SA2, ..., SAn lAn luqt tai A'1, A'2, ..., A'n.tfinh hq,p b6i thict dicn A\A'2...A', vd drly A14...A" cfia hinh ch6p cing vdi cdctir gi6c A'1A'24A1, A'zA\44, A'rA'y4yA, goi ld mlt" hinh ch6P cut, ki hiOu 1) A'1A'2...A',,.Ath...An $.72). Ddy cfiatiinn cfr6p goi l) itdy l6n ctra hinh ch6p cut, cdn thiet diOn A'1A'2...A'n goi li 66 A2 A3 Hi.nh 72 5. HiNHIlNC.B ildy nhd cira hinh ch6p cut.' Cdc tfi gi6c A'1A'2,L2A1, A'2A'3A3$, A'rA'1A1An ggi lh cdc mdt bdn cia hinh ch6p cut. C6c do4n thing 4.A'r, A2,A'2, ..., ArA'n goi ld cdc cgnh b4n cia hinh ch6p cut. Tu} theo d6y ld tam gi6c, t(t gi6c, ngfr gi6c, ..., ta c6 htnh chdp cut tam gidc, hinh ch6p cpt tit gidc, hinh chdp cut ngfi gidc, ... Tinh chdt Vi hinh ch6p cut dugc c6t ra tt mqt hinh ch6p nOn ta d6 ding suy ra tfnh chAt sau d0y Hinh ch6p cut c6 : a) Hai ddy ld. hai da gidc c6 canh taong ang song song vd ti sd cdc canh tuong ftng bdng nhau. b) Cdc mfi b2n ld nhrtng hinh thang. c) Cdc dtong thdng chita cdc canh b1n ddng quy tqi mQt didm. cdu n6i vd bdi t6p i, '29. Trong c6c m0nh dd sau, mOnh dd nio dring ? a) Hai mat phing ph0n biQt cing song song v6i mOt duirng thing thi song song v6i nhau ; b) Hai m4t phing phAn biOt cing song song vdi mOt mf,t phing thi song song vdi nhau ; c) Ndu hai m6t phing song song thi mgi dudng thfng nlm trOn m4t ph8ng niy ddu song song vdi m6t phing kia ; d) Ndu hai m4t phfrng song song thi m6i duong thing n[m tr0n m4t phing niy ddu song song vdi mgi duong th8ng n4m tr0n mat phing kia ; e) Ndu hai mf,t phing phAn biOt ldn lrrot di qua hai dulng thing song song thi song song vdi nhau ; f) Ndu mQt ducrng thing c6t mot trong hai mit phlng song song thi c6t mat phing cdn lai. 30. Trong c6c mOnh dd sau, mOnh dd nio dring ? a) llinh hQp lh mQt hinh ling tru ; b) Hinh l[ng try c6 tdt cL cdc c4nh song song ; c) Hinh lf,ng tru c6 tilt cit cdc mit bOn bang nhau ; d) Hinh l[ng trg c6 cdcm4t bOn li hinh binh hinh ; e) Hinh hQp c6 cdc mdt ddi di€n bang nhau 67 .,_. -i-=-,.tail 3i.' Ctro hai'duong thing ch6o nhau. Chtmg minh rang c6 dring hai m6t phing song song vdi nhau ldn lugt di qua hai dudng thing d6. 32, Cho hai dudng thing ch6o nhau a vd, b ldn luot ndm tren hai m[t phing "' ' song song (P) va (0). Chrlng minh rlng ndu didm M kh6ngn[m trOn (P) vi kh6ng ndm trOn (O) thi c6 duy nhAt m6t duong thing di qua M cht ch a vd b. fSS; frong mat ph&ng (P) cho hinh binh hinh ABCD. Qua A, B, C,D tdn lugt vO -' bdn duong thing a, b, c, d d6i m6t song song v6i nhau vh kh6ng nf,m trOn (P). MEt mat phing cit a, b, c, d ldn luot tai bdn didm A', B', C', D'. Chttng 'minh ring A'B'C'D'ldhinh binh hinh. 34. Cho.trl didn ABCD. Ggi M lh, fiung didm cria AB. Hbi mlt phing (P) qua didm M, song song vdi ch AD vd BC c6 di qua trung didm N c:tra CD khOng ? Tai sao ? ( rcall 35. Cho hai didm M, N ldn luot thay ddi trOn hai m4t phing song s'ong (P) vd (O). Tim qp hqp c6c didm l thu6c doan thing MN sao "ho ly = k. k * 0 cho trudc. 36. Cho hinh l[ng tru tam gi6c ABC.A'B'C'. Goi lLI ld trung didm cria canh A'B'. a) Chtmg minh ring duong thfrng CB'song song v6i mp(AHC'). ({c"x) b) I1m giao tuyOn d crta hai mat phing (AB'C) vd (A'BC). Chrlng minh ring d song song v1i mp(BB'C'C). cl iac aint tfriet diOn ctra hinh lang tru ABC.A'BC'khi cit b&i mp(Hi, d). 37. Cho hlnh hOp ABCD.A'B'C'D'. Chung minh rlng a) mp(BDA) ll mp(B'D'C); b) Dudng ch€o'AC'di qua c6c trong tam G1, G2 crta hai tam gi6c BDA' vd B'D'C: c) G1 vd G2 chia doan AC'thinh ba phdn,bang nhau ; d) C6c trung didm cira s6u canh BC, CD, DD', D'A', A'B', B'B cing nam tren m0t mlt phing 38. Chring minh ring tdng binh phucng t* ce c6c duhng ch6o cira mQt hinh hop bang tdng binh phuong tdtca cdc canh cira.hinh hQp d6. 39. Cho hinh ch6p cut ABC.A'B'C' c6 d6y lon ABC vd c6c canh bdn AA', BB', CC'. Goi M, N, P ldn luot lh trung didm cira c6c canh AB; BC, CA vd, M', N', P'ldn luot lb trung didm cira c6c canh A'B', B'C', C'A'. Chfing minh MNP.M'N ?'ld hinh ch6p cut. 68 :. -. =: PHEP CHIEU SONG SONG Dihh nghia ph6p chi6u song song Trong khOng gian cho mat phing (P) vn dudng thing / c6t mp(P). Vdi m6i didm M trong kh6ng gian, vE duong thing di qua M vd, song song hoic trung vdi /. Duong thing niy cat mp(P) tai mQt didm M' nio d6 (h.73). Hinh 73 ll f nep ddt tuong ttng mdi rlidm M ffong khAng gian vdi didm M' ll cila mdt.phdng (P) nhu tr€n gqi ld phdp chiAa song song kn ll m,frt phdng (P) theo phuong l. Mat phing (P) gqi ld mdt phdng chidu, duong th8ng / goi lh phuong chidu ; didm M' g7i ld hinh chidu song song (hoac dnh) crta didm M qua ph6p chiOu song song n6i trOn. Cho hinh gf T4p hW gf ' gdm hinh chidu song song cira t0t ci cdc didm thuQc &( eqi ld hinh chidu song song (hoic dnh) c,fia hinh ,7( qua ph6p chidu n6i trOn. B6ng tr0n mlt ddt phing cira mQt vAt chinh ld hinh chidu song song ctra vAt dy trOn mat dAt (cic tia s6ng mf,t trdi duo. c coi nhu song song vdi nhau). fl NeA didm M thuAc rnfit phdng chidu (P) thi hinh chiAu song song cfia nQ ld didm ndo ? @ Cno duong thdng a song song voi phuong chidu l. Hinh chi€u song song cila a (hoQc m1t phdn crta n6) ld hinh ndo ? 2. Tinh chdt Trong cdc tinh chat du6i dAy cira ph6p chidu song song theo phuong l, ta chi x6t hinh chiOu song song cia cdc doan thdng hodc dxdng thdng kh)ng song song vd kh1ng trilng vdi l. Tinh chdt I Hinh chiAlu song song cfia m1t dudng thdng ld mOt drdng thdng. 69 Chirng minh (h.74) X6t ph6p chidu song song lcn mp(P) theo phuong l. Gia sir a ld mOt dulng th&ng khOng song song vi khOng trDng v6i /. Goi M ld mOt didm bat ki cira a vd M'lh hinh chi0u ctra n6. V\ MM' song song (ho[c tring) vdi I nAn M' nim trOn mp(Q) di qua a vi song song vdi / (ho4c chtla /). MIt kh6c, M' nim trOn mp(P). Y4y M' ndm trOn giao tuy6n a' cita hai mit phing (P) ve (0). Hinh 74 Nguoc lai, d6 thAy m6i didm M'nim trOn a'ld hinh chi0u cira mOt didm M n6m trOn a. Y qy hinh chi0u cia a chfnh li duong thilng a' . n ffi WeA dadng thdng a ndm trong mdt phdng chi€'u (P) thi hinh chia'u song song, cfia a ld hinh ndo ? @ Neu dudng thdng a cdr mdr phdng chidu (P) tqi didm A thi hinh chidu song song cfia a c6 di qua didm A hay khdng ? ue quA Hinh chiA'u song song crta milt doan thdng ld mdt doan thdng, cfia mbt tia ld mbt tia. Ti viOc chfng minh tinh chdt l, ta thAy hinh chi€u song song cira du&rg thing a ld giao tuydn ctra m6t phing chidu (P) vd mp(Q), trong d6 (O) le m[t phing di qua a vd song song vdi / holc chrla /. Do d6 ta c6 Tinh chdt 2 (h.75) Hinh chidu song song cila hai duong thdng song'song ld hai dudng thdng song song hodc trilng nhau. Hinh 75 7A Tfnh ch{t 3 Phip chidu sot'tg song kh1ng ldm thay ddi tl sd crta hai doan thdng ndm tr€n hai dudng thdng song song (hoac tring nhau). Tinh chAt 3 c6 nghia ld : NOu dudng th&ng song song (hoac mp(P) ld A'B'vi C D'thi Hinh 76 minh hoa tinh chot d6. AB vir CD ld hai doan thing nim trOn hai tring nhau) c6 hinh chiOu song song trOn A'B' C'D' C' D' Hinh 76 3. Hinh Uidtr Ai6n cria mOt hinh kh6ng gian 6 g t cira chuong ndy, ta dd n6u ra^ mOt s0 quy t5c dd vO hinh bidu di6n ctra mQt hinh khOnggian trOn m6t ph8ng. C6c quy titc dy dua trcn dinh nghla sau dAy DINH NGHIA ll nion bidu dian crta mot hinh ,fftong khbng gian td hinh ll ,r,nU song song cfia hinh ,ffrr\n mQr mdr phang hodc hinh ll aang dang vdi hinh chi€u d6. Nhu vAy, mudn vE dring hinh bidu di6n, ta phii 6p dung cdc tinh chdt n6i trOn cira ph6p chidu song song. Do d6, ngoii nhfrng quy tic dE hgc 71 trudc d6y (dugc suy ti cic tinh chdt 1 vd, 2), ta cdn luu f them quy tic sau (suy tt t(nh chdt 3) : NAlu tr€n hinh &{ c6 hai dogn thdng ndm tr€n hai dudng thdng song song ftodc trilng nhau) thi chrtng chdng nhfrng duqc bidu dien bdi hai doan thdng ndm trdn hai dutrng thdng song song (hodc trilng nhau), md ti sd cila hai doqn thdng ndy cdn phdi bdng ti sd cfia hai doan thdng Mong ilng ffAn hinh {t @ nmn bidu diAn cila hinh binh hdnh ld hinh gi ? g CHU f Ph6p chidu song song n6i chung khOng gifr nguy0n ti sd cira hai do4n thfng khOng nirrr tren hai duong thing song song (hay kh6ng ctng nim trcn mQt ducng fti"g) vi kh0ng giilnguyOn dO ldn cira mOt g6c. Tt d6 suy ra ndu trOn h\nh#(c6 hai doan th8ng khOng nim trOn hai dudng thing song song thi ti sd ctra chring khdng nhdt thidt ph6i gifr nguyOn ffen hinh bidu di5n. C[ng nhu vQy, dQ 16n cira mOt g6c trOn h\nh&f kh0ng nhdt thiOt duo.c gif, nguy€n rr0n hinh bidu dien. @ ruinn bidu di6n cila hinh thang ld hinh gi ? fl nmn bidu di6n cfia hinh thoi, hinh chrt nhQt, hinh vubng td hinh gi ? W CA phdi mqv tam gidc bdt ki rtdu c6 thd xem ld hinh bidu di6n cila tam gidc cdn, tam gidc vuilng, tam gidc ddu hay khAng ? (h.77). C =C' Hinh 77 Htnh 78 @ nmn bidu di6n ctia mQt trh diQn ttdu cd thd v€ nhu htnh 78 hay khilng ? 72 Hinh biiu diOn cfia mdit dudng trdn Ngudi ta chrlng minh duoc rang : Hinh chiAu song song cfia m1t dadng trdn ld m6t dudng elip hodc m1t dudng trdn, hodc d\c biil c6 thd ld mQt doqn thdng. vi v4y, ta thudng ding duhng elip lim hinh bidu di€n cira dudng rrdn, r6m ctra elip bidu di6n cho t6m cira dudng trdn (h.79, h.S0). Hinh 79 Hinh 80 1 oS "ttam gi6c ABC h hinh bidu di6n crla m6t tam gi6c ddu. HEy durng hinh bidu diSn c0a tdm dr-rong trdn ngoai tidp tam giSc ddu d6. - 2 Cho mQt duong elip lir hinh bidu di6n c0a m6t tfuong trdn. Hdy v6 hinh bidu diSn cfia m6ihinh sau d6y: a) M6t ddy cung vd dudng kinh vu6ng g6c v6i d6y cung d6 c0a dtrdng trdn. b) Hai duong kinh vu6ng g6c c0a dr-rdng trdn. c) MQt tam gi6c ddu n6i ti6p dudng trdn. Vui mQf ehuf ! :-e&- fffiffi& f;G{&E.d'tr \ffi . Hinh sau c6 phAi Id hlnh bidu di6n c0a m6t hinh kh6ng gian hay kh6ng ? Tranh cila Et-se (M.C. Escher) ) . "Chuy€n dOng vTnh crlu ?"; Li6u ntr6c c6 chAy mdi nhrJthd kh6ng ? Tranh cfia Etse (lV.C. Escher) cou n6i vd bdi t6p 40. Trong c6c m€nh dd sau, mOnh dd nio dring ? a) IDnh chi@u song song cira hai duong thing ch6o nhau c6 thd tring nhau ; b) Fltnh chidu song song cira hai dudng th&ng ch6o nhau thi c5t nhau ; c) tlinh chidu song song ctra hai duong thing ch6o nhau c6 thd song song vdi nhau ; d) tfinh chidu song song ctra hai dudng thing ch6o nhau c6 thd c6t nhau, trDng nhau, song song vdi nhau. 41". Trong c6c mOnh dd sau, mOnh dd nio dring ? a) Hinh chiOu song song c[ra hai dudng it irg cit nhau c6 thd song song vdi nhau ; b) tfinh chidu song song cira hai duirng thfrng c6t nhau c6 thd c5t nhau ; c) Ilinh chidu song song cira hai dudng thing cit nhau c6 thd trtng nhau ; d) MQt duong thing c6 thd song song vdi hinh chidu song song ctra n6 ; e) MQt du&rg thing 1u0n cit hinh chidu song song cira n6 ; 0 MOt dudng th8ng c6 thd trtng v6i hinh chidu song song ctra n6. 42. Tarn gi6c ABC c6 hinh chiOu song song th tam gi6c A'B'C'. Chrlng minh rang tro.ng mm tam gi6c ABC c6 hinh chidu song song li trong tam tarrl gi6c A'g',C',. 74 43. Ye hinh bidu di6n cira mQt trl diOn vd trong tAm cria n6. 44. VC hinh bidu diOn ctra m6t tam giSc vudng n6i tidp trong mOt dudng trdn. 45. Ve hinh bidu diOn cira m0t hinh vudng n6i tiOp trong m6t dudng trdn. 46. YC hinh bidu di6n ctra mOt luc gi6c ddu. 47. Arc hinh hQp ABCD.ArBtCtDr.Tim didm 1trOn duong ch€o B1D vh didm "I tren dudng ch€o ACsao cho iJ ll BCr.Tfnh ti ,d *. il ' ii' . { F,l .,, i r. i', . ',, r IBt -'". .r "';l '1' .:.- I - T6m tit nhfrng kidn thrtc cdn nh6 1. MOt m4t ph&ng duoc xiic dinh ndu bidt mOt tror-rg c6c di6u kiOn sau d0y : a) Mit ph&ng d6 di qua ba didm kh6ng thing hlng. b) Mat phing d6 di qua mOt didm vd m6t dudng thing kh0ng chrla didm dy. c) Mit phang d6 di qua hai duong th&ng cit nhau. d) Mat phing d6 di qua hai dudng thing song song. e) Mat phing d6 di qua m6t duong thing vd song song vdi mQt duong thing ch6o vdi duong thing dy. O fufat phing d6 di qua m6t didm vi song song v6i m6t m[t phlng khdng chfa didm dy. 2. Dinh li vd giao tuydn cira ba mat phing : Ndu ba mit phing c6t nhau theo ba giao tuy6n phAn biOt thi ba giao tuyOn d6 ho4c ddng quy holc d6i mOt song song. 3. Ba doan thing ndi trung didm c6c canh ddi diOn cria mOt trtr'di6n ddng quy tai trung didm G ctra m6i doan. Didm G d6 gqi li trong tAm cira trl di6n. 4. Dudng thing vi mf,t phing song song (tric ld chring khong c6 didm chung) : a) Duong thing a (khOng nim trOn mp(P)) song song v6i mp(P)- khi vh chi khi n6 song song v6i mOt dudng thing nam trong (P). b) Ndu mp(p) di qua dudng thing a md d song song vdi mp(P) thi giao tuydn cira mp(P) vi mp(O) (ndu c6) song song vfi a. c) Hai m4t ph&ng c6t nhau cing song.song vdi m6t dudng thfrng thi giao tuydn cria chring song song vdi dutmg thing d6. !r:,f ,,,!,ii.,,..,,.i,.r.i '. ,-. ,1.', , 7,s 5. Hai mf,t phing song song (tfc lb chfng kh6ng c6 didm chung) : a) Ndu mit phing (P) chrla hai dulng thing a, b cht nhau vh ctng song song v6i mp(Q) thi (P) ll Q). b) Ndu hai mdt phing (P) vn (Q) song song thi moi mp(R) dd, cat (P) thi c6t (0) ve c6c giao tuy€n cira chring song song. c) Dinh li Ta-l€t: Ba mit phing dOi m6t song song ch6n ra tr€n hai cdt tuydn bat ki c6c doan thing tuong ung ti le. d) Dinh li Ta-l6t d6o : Gii sr? tren hai duong thing ch6o nhau a vd a' ldn luot lAy cdc didm A, B, C vi A', B', C'sao cho :AB BC= cA.Khi A'B' B'C' C'A' d6, ba dudng thing AA', BB', CC'ldn luot nam trOn ba mlt phing song song, trlc li chfing cing song song v6i m6t m5t ph&ng. 6. Hinh ch6p c6 ddy ld mOt da gi6c vi c6c m6t bOn ddu ld nhfrng tam gi6c c6 chung m6t dinh (dinh cira hinh ch6p) 7. Flinh lang try c6 hai ddy n6m trOn hai mit phing song song ; cdc mdtbOn ddu ld nhfrng hinh binh hinh ; c\c canh bOn bang nhau vi d6i m6t song song. 8. Hinh h6p lh hinh l6ng tru c6 ddy ld hinh binh hdnh ; b6n dudng ch6o cira hinh hOp ddng quy tai trung didm cia m5i duong, didm d6 goi li tam ctra h)nh hOp. 9t. Flinh ch6p cut c6 hai ddy nf,m trOn hai mat ph8ng song song ; cdc mdtbOn ddu le hinh thang ; c6c du&rg th&ng chiracdc canh bOn ddng quy tai mOt didm. 10. Ph6p chidu song song theo phuong /: a) Kh6ng ldm thay ddi s1r thing hdng vi thrl tu ctta cdc didm thing hlng. b) Bidn hai dudng thing song song (nhurg khOng song song vdi l) thhnh hai dudng thing song song ho[c trtng nhau. c) Gifr nguy0n ti sd ctra hai doan thfrng nf,m trOn hai dudng thing song song hof,c cing nAm trOn mOt duong thing. LL. Hinh bidu di€n cira mOt hinh trong kh6ng gian li hinh chieu song song ctra hinh d6 trOn m6t mlt phing ho4c hinh ddng dang vdi hinh chidu d6. Hinh bidu di6n ctra duong trdn thudng ld duong elip hof,c dudng trdn. ll - Ciu tr6i tr,r kidm tra 1. Hdy nOu su kh6c biOt gifra hai dudng thing ch6o nhau vd hai duong thing song song. 2. NOu phuong ph6p chfng minh ba didm thing hdng. 3. NOu phuong ph6p chrlng minh ba duong thing ddng quy. 76 .i ..,. , i ., 4. NOu phuong phdp chfrng.minh duong thEng song song v6i m6t phdng. 5. NOu phuong ph6p chrlng minh hai mlt phing song song. ll! - Bdri tip 1. Trong c6c mOnh dd sau, mOnh dd nio dring ? a) Hai dudng thing ch6o nhau thi kh0ng c6 didm chung ; b) Hai duong th8ng khOng c6 didm chung thi ch6o nhau ; c) Hai drrong thing ch6o nhau thi kh6ng cing thu6c m6t mf,t phfrng ; d) Hai duong thing kh6ng song song thi ch6o nhau. 2. Trong c6c m0nh dd sau, mOnh dd nio dfng ? a) Hai dudng thing phan biOt cing song song vdi mQr m[r phing rhi song song v6i nhau ; b) Hai m[t ph8ng phan bi€t ctng song song vdi m6t dudng th&ng thi song song vdi nhau ; c) Hai mat ph8ng ph6n biOt kh6ng song song thi cit nhau ; d) Hai mlt phing phan biOt ctng song song vd mOt mf,r phtng thf ba thi song song vdi nhau ; e)"MQt duong thing c6t mot rrong hai duong thing song song thi c6t duong thing cdn lai ; 0 rv1Qt mlt ph8ng c6t mot trong hai duif'rg thing song song thi c6t duong thing cdn lai ; g) ivlQt duong thing c6t mot trong hai m[r phing song song thi c6t mlt phing cdn lai. 3. Trong c6c hinh sau, hinh nio li hinh bidu diOn cria m6t trl di0n ? B ,G,A5 A C A\\ pj1-t+ d) oN B /-'-\ CD e) Hinh 8l s) h) 77 4. Cho hai hinh binh hdnh ABCD vd ABEF nam trong hai m4t phflng kh6c nhau. Ldy cdc did,m M, N ldn lugt thuqc c6c dudng ch6o AC, BF sao cho MC : ZAM ; .NF = 2BN. Qua M, N, k6 c6c duong thing song song v6i AB c6t cdc canh AD, AF ldn lugt t4i M1, Nt.Chrlng minh rang : a) MN ll DE; b) MrNr ll mp(DEF) ; c) mp(MNNTM) ll mp(DEF). 5. Cho hinh lang tru tam gi6c ABC.A'B'C'. Ggi G, G'ldn luCI ld trong t6m cfia tam gi6c ABC vd A'B'C'. MQt mat phing (a) cit c6c canh Alq:, BB', CC', GG'ldn luot tai A1, 81, C1 vi G1. Chung minh rf,ng : a) GG' song song vi bang canh b€n cira hinh lf,ng trU : b) Gr li trgng tAm ctra tam giiic A1B1C1 i I - l. c) G1G' = :(AtA' + B1B' + Cp') : G1G = ;(A1A + B1B+ C1C). '3J 6. Cho hinh hQp ABCD.A'B'C'D'. Ve thidt dien ctra hinh hop t4o boi m4t phing di qua hai trung didm M, N cria cdc canh AB, AD vi t6m O cliuamat CDD'C'. 7. Cho hinh hop ABCD.A'B'C'D'. TrOn ba canh AB, DD', C'B'ldn lugt lfly ba didm M,N, P kh6ng trirng vfii cltcdinh sao ,no { = P'y- : !'f " AB D'D B'C' a) Chung minh rang mp(MNP) vd,mp(AB'D) song song vdi nhau. b ) X6c dinh thidt di0n cfra hinh h6p khi c5t boi mp(MNP). 8. Cho hai tiaAxvi By nim trOn hai dudng thing ch6o nhau. MQt didm M chay trOn Ax vh m6t didm N chay tren By sao cho AM = kBN (k > 0 cho tru6c). a) Chung minh rang MN song song vdi m6t mit phing cd dinh. b) Tim t4p hqp c6c didm l thu6c doan MN sao cho IM = klN. ,| L. Cho trl di1n ABCD. Goi M, N ldn luot ld trung didm cira cl,c canh AD vit BC ;G th trong tam tam gi6c BCD. Khi dy, giao didm cira dudng thhng MG vd mp(ABC) ld : (A) Didm C : 78 (B) Giao didm cria duong thing MG vd duong th&ng AN ; (C) Didm N ; (D) Giao didm cfia duong thing MG vitduong thhng BC. Cho tr1 di€n ABCD vh ba didm E, F, G ldn luot nam tr€n c6c cqnh AB, AC, AD md kh0ng trtng vdi cdc dinh. Thidt diOn cira hinh trl di}n ABCD khi c6t b&i mp(EFG) li : (A) MOt doan thing ; (C) MOt tit gi6c; (B) MOt tam gi6c; (D) MOt ng0 gi6c. Cho tr1 di}n ABCD vd ba didm 1,.I, K ldn luot nam tr0n ba cpnh AB, BC, CD mi kh6ng trtng v6i cdc dinh. Thidt diOn cfia hinh trl dien ABCD khi cit b&i mp(I/K) 1I : (A) MQt tarn gi6c ; (c) MQr hinh thang ; (B) MQt tfi gi6c i (D) MQt ngfl gi6c. Cho hinh ch6p S.ABCD. Gqi AC a BD = I, AB a CD = l, AD n BC = K, D&ng thrlc nio sai trong c6c ding thrlc sau day ? (A) (SAC) n (SBD) =.11 ; (C) (SAD) n (SBC) = ^SK ; (B) (SAB) n (SCD) = S,I i (D) (SAC) n (SAD) = AB. 5. Cho hinh ch6p S.ABCD. Mgt mit phing kh6ng di qua dinh nio cira hinh ch6p c6t cdc canh SA, SB, ,SC, ,S, ldn luCI tai A', B', C', D'. Goi O li giao didm cira AC vdBD. Tim mOnh dd ddng trong cdc mOnh dd sau dAy. (A) C6c duong thing A'C', B'D',SO doi m6t ch6o nhau ; (B) Cdc duong th&ng A'C', B'D',SO d6ng ph8ng ; (C) C6c duong thing A'C', B'D', SO ddng quy ; (D) Hai dudng th&ng A'C' vi B'D' cht nhau cdn hai dudng thing A'C'vd SO ch6o nhau. Cho ttl diQn ABCD. Goi G vI E ldn luot ld trgng t0m ctra tam gi6c ABD vit 6. ABC. MOnh dd nio dudi dAy dting ? (A) Duong thing GE song song vdi dudng thhng CD ; (B) Duong thing GE cit dudng thhng CD ; (C) Hai duong thing GE vd CD ch6o nhau ; (D) Duong thing GE cat duong thhng AD. Cho trl didn ABCD. Goi M, K ldn lugt li trung didm cira BC vi AC, N le 7. didm trOn canh BD sao cho BN = zND, Gqi F li giao didm ctra AD vir mp(MNI}. Trong c6c mOnh dd sau dAy, m€nh dd ndo dring ? (A)Ar=FD; (C) AF = 3FD ; (B) Ar = 2FD ; (D) FD = 2AF. 8. Cho trl diQn ddu ABCD c6 canh bing a. Ggi G li trong t6m tam gi6c ABC. CAt ft diQn boi mp(GCD) thi dien tich cira thi6t dien li : (x\ o2& , 2 rcs o20 , 6 rn>ff; @# g. Cho hinh ch6p S.ABCD c6 ddy lh hinh binh hinh. Goi 1,,I ldn luot lh trung didm cfra AB vi CB. Khi dy, giao tuydn cira hai mat phing (SAB) vn (SCD) li duong th8ng song song v6i : (A). Duong thhng AD ; (C) Duong thhng BI ; (B) Ducrng thhng BJ ; (D) Duong thfrng /.,r. 10. Cho hinh ch6p S.ABCD c6 ddy lh mQt hinh binh hinh. Goi A', B', C', D'ldn lucn li trung didm cira c6c canh S,A, SB, SC vi SD. Tim m€nh dd dring trong ciic mOnh d0 sau dAy : I (A) A'B' ll mp(SAD); (B) A'C' ll mp(SBD); (C) mp(AC'D) llmp(ABC) ; , (D) A'C' ll BD. 11. Cho tr1 diOn dil,ABCD c6 canh bang a, di6m M ffOn canh AB sao cho AM = m (O < m < a).Khi d6, diOn tfch thidt diOn cira hinh trl diQn khi c6t b&i m4t phing qua M vd song song vdi mp(ACD) li : @#, c>g:ff, 61 @-ry)zJz . (D) @ * d2Jl 12. Cho hinh ch6p S.ABCD c6 diy ld mQt h-inh binh hinh. MQt m4t phfing (P) song song v1i AC vi SB ldn luot cfut cdc canh SA, AB, BC, SC, SD, BD t4i M, N , E, F , I ,,I. Trong c6c mOnh dd sau d6y, mOnh dd ndo dting ? (A) Bdn dudng thing MN , EF, I.I, SB dOi m6t song song ; (B) Bdn dudng th&ng MN, EF,lJ, SB ddng quy ; (C) Bdn duong thing MN, EI;,1J, SB ddng phing ; (D) Ce ba monh dd trcn ddu sai. 80