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^JSJBm. NGUYEN TRONG TUAN (CHU BIEN) ThS. DANG PHUC THANH - NGUYEN TAN SIENG (Gido vien cliuy§ii toan va nang khieu) hi,; 'Uit Bdi duong hoc sinh gidi nociffiT Danh cho hoc sinh kha, gidi va chuqen loan Biensoan thgochuongtrlnhnidi KH THIT VIEWTIfv'HeiN'H THUAW NHA XU^T BAN T6NG H0P THANH PH^ Hd CHi MINH JCgi noi ddu Cuon sach "Boi dtiditg hoc sinh gidi Hinh hoc 10" la mot tai lieu tham khao mon toan dung trong truong trung hoc pho thong, nh^m muc dich giijp cho hoc sinh ren luyen cac ky nang giai cac dang toan theo yeu cau Chuan Kien thuc - Ky nang cua chuang trinh, tren ca so do tai lieu con giiip cho cac em tiep can, giai cac de thi Dai hoc, Cao dang va cac bai toan nang cao danh cho cac hoc sinh kha - gioi. " • Noi dung cuon sach gom 3 chuang, tuan tu theo sach giao khoa Hinli hoc 10 hien hanh. Moi chuang gom cac chuyen de tuong ung vai cac bai hoc (§) duoc trinh bay theo cau true sau A. Tom tat li thui/et. ' ' I B. Mot so dang toan. C. Luyen tap. Cuoi nidi chuang la phan cac bai tap on cudi chuang bao gom cac bai tap tong hap kien thuc trong chuang. Trong hai phan B. Mot so dang toan va C. Luyen tap, cac bai tap dugc trinh bay theo tung dcing, tu dan gian den phiic tap, ngoai cac bai theo dang Chuan Kien thuc - Ky nang, chung toi da trinh bay them cac bai tap mai, dang kho, phiic tap phuc vu cho doi tugng hoc sinh kha - gioi nham phat huy kha nang tu duy linh boat, sang tao, doc lap a moi em. De hoc tot tai lieu nay, hoc sinh can nSm vung li thuyet, cac dang toan va cac bai tap c6 lai giai mau. Voi tinh than tu hoc mot each nghiem tue, khoa hoc, chiing toi hi vong rang tai lieu nay c6 the giiip cho cac em cai thien dugc nang luc hoc toan cua ban than va vuan len trinh do kha, gioi. Dii da CO gSng rat nhieu, nhung thieu sot la dieu kho tranh khoi, rat mong cac thay, c6 giao va cac em hoc sinh gop y de cuon sach nay dugc dieu chinh, bo sung, hoan thien han trong Ian tai ban. Cdctdcgid Wtd sdch Khang Viet xin trdn trgng giai thieu t&i Quy doc gid vd xin idng nghe moi y kien dong gop, de cuon sdch ngdy cang hay horn, bo ich hem. Thuxinguive: , Cty TNHH Mot Thanh Vien - Dich Vu Van Hoa Khang Viet. 71, Dinh Tien Hoang, P. Dakao. Quan 1, TP. HCM Tel: (08) 39115694 - 39111969 - 3911968 - 39105797 - Fax: (08) 39110880 Hoac Email: khangvietbookstore@yahoo.com.vn Cty TNHU MIV DVVH Khang Vi£> Chirofng 1 VECTO §1. KHAI NIEM VECTQ A. TOM TAT LI THUYET m 1. ^inhnghia ' • Vecta la mot doan thMng c6 huong, nghla la trong hai diem miit ciia dpan thang, da chi ro diem nao la diem dau, diem nao la diem cuoi. • Ki hieu vecto c6 M la diem dau va N la diem cuoi la MN . Nhieu khi nguoi ta dung ki hieu a de chi mot vecto AB nao do. • Vecta CO diem dau va diem cuoi triing nhau dugc ggi la vecta - khong, ki hi?u la 0. 2. Jiai vecta cung phuang, ciing hitang • Gia ciia vecto AB: Cho AB khac 0. Duong thJing AB dugc ggi la gia ciia AB. • Hai vecto cung phuang: Hai vecta dugc ggi la cung phuang neu chiing CO gia song song hoac trung nhau. • Neu hai vecta cung phuang thi hoac chiing cung huong, hoac chiing ngugc huang. Chu y. Vecta - khong AA c6 gia la mgi duang thJing qua A; 0 cimg phuang va cung huang voi mgi vecta. Tren hinh ve ta c6 cac vecto AB, CD, EG ciing phuong vai nhau, trong do AB, CD cimg huong, EG ngugc huang vai cac vecto AB, CD . 3. Jiai vecta bdng nhau • Dg dai ciia vecto AB : Dg dai ciia doan thang AB dugc ggi la do dai ciia vecta AB, ki hi^u la I AB I. Hai vecta a va b ggi la bang nhau neu chiing cung huang va cung dg dai. ta viet a = b. Boi liuchig IISG Hinh hoc 10 B. MOTSODANGTOAN i)ang 1. So sdnh cdc vecta Sit dung cdc dinh nghta vehai vecta cung phuong, cung huong, bang nhau ^di 1. Cho ba diem A, B, C phan biet. Cac menh de sau day diing hay sai ? a) Neu AB = AC thi AB = AC; b) Neu AB = AD thi B = D ; c) Neu AB = BC thi B la trung diem doan AC; d) AB = BA. Giai. , , , " • ' a) Sai vi hai vecto bang nhau khong nhCmg c6 do dai bang nhau ma con phai Cling huong. b) Dung. c) Dung vi neu AB = "BC thi AB,BC cung huong va AB = AC. Suy ra A,B,C th^ng hang theo thu tu do va AB = AC. Vay B la trung diem doan AC. d) Sai vi voi A, B phan biet thi cac vecto nay nguoc huong. <Bdi 2. Cho tam giac ABC. Goi M, N Ian luot la trung diem ciia AB va AC. Hay so sanh phuong, huong, dp dai cua cac cap vecto a) AB va AC ; c) AN va NC ; e) BC va NM ; b) BC va MN ; d) MA va MB; g) CA va NC. Giai a) Hai vecto AB va AC khong cung phuong; = 2 MN •Z b) Hai vecto BC va MN cung huong va BC c) Hai vecto AN va NC bang nhau; B d) Hai vecto MA va MB cung phuong, ngugc huong va ciing do dai; e) Hai vecto BC va NM ciing phuong, ngugc huong va BC NM -2 g) Hai vecto CA va NC cung phuong, ngugc huong va CA = 2 NC '£)ang2. Xdc dinh diem thod man he thuc vecta 'Bdi 3. Cho hai diem A va B phan bi^t. Hay tim diem M sao cho: a)AM = MB; ^ ^ b)AB = BM. Giai. a) M la trung diem doan AB. > b) M la diem doi xiing ciia A qua B. ,v ' 117 r,> ' Cty TNHH MTV DWH Khmig Vir. <Bdi 4.Cho tam giac ABC. Hay xac dinh cac diem D va E sao cho AD = BC,AE = CA. ji^ .gsiiOi Giai. ^'tv':ii.i ';"\ .^''''A: Ta CO ri'!'" " • AD = BC o D la dinh thu tu cua hinh binh hanh ABCD. • AE = CA <=> E la die'm tren tia CA sao cho CA = AE. (Hinh ve). Si ••• C. LUYEN TAP ^^'^^^ '] ' • 1.1. Cho hai diem A, B phan biet. Hay so sanh phuong, huong, do dai ciia hai vecto AB va BA. ' ' • \ d&n giai Hai vecto AB va BA ciing phuong, ngugc huong va cung do dai. 1.2. Cho hinh thang ABCD voi AB // CD va CD = 2AB. Goi M la trung diem CD, N la trung diem BC. Hay dien dau "x" vao cac cot ung voi cac tu "ci^mg phuong", "cung huong", "ngugc huong", "bang nhau" de dien ta moi quan he giira cac cap vecto vao bang sau : Cung phuong Ciing huong Ngugc huong Bang nhau AB va BC MN va BD AB va CD MA va BC MB va DA DMva BA BA va DC J-Iaang dan giai a) Vecto AB ngugc huong voi vecto DE . H J-!r>.r!)0 '•.I'lfiv .' b) Vecto BF ngugc huong voi vecto ED. j.^, | '\ , c) Vecto AE bang vecto FD . -la^i >:V'V' > . ' d) Vecto DE cijng huong voi vecto BA . a f ' v. e) Vecto FE cung huong voi vecto BC . v/ g) Vecto BD bang vecto DC. . . ; . - h) Vecto EC khong cung phuong vgi vecto BF. 4^^,, :,, Boi tluimg HSG Hinh hoc 10 1.3. Cho ba diem A, B, C. Co nhan xet gi ve ba diem do neu : a) AB=BC ; b) AB = AC ; c) AB = |AC| va AB, AC khong cung phuong. . . i • JiuangMngidi . a) Diem A nam tren duong trung tryc doan BC. r'j' * b) Diem B trung voi diem C. , c) A la trung diem doan BC. VJA 1.4. Cho ba diem A, B, C. M$nh de nao sau day diing ? a) AB = 0<=>A = B; b) AB = CD <=> A = C va B = D ;yj c) Neu AB =|AC thi B = C; d) Neu BA = CA thi B = C. -f''"'•• "'i '•'•'v^^!v'• Jilcang dan gidi a) dung b) sai c) sai d) dung. • v 1.5. Cho ba diem A, B, C phan biet. Ket luan gi ve ba diem A, B, C neu : a) Vecto AB cung phuong voi vecto BC; b) Vecto AB cung huong voi vecto AC ; c) Vecto AB bang vecto AC . Jiicang dan gidi a) Ba diem A, B, C thang hang. b) Ba diem A, B, C thang hang va B, C nam ve cung mot phia doi voi diem A. c) Hai diem B, C trung nhau. 1.6. Cho hinh binh hanh ABCD. Goi E, F Ian luot la trung diem cac canh AB, CD. Duong cheo BD cat AF tai G va cat CE tai H. Chung minh rang : a) DG = GH = HB ; b) AH = GC . Jiudng ddn gidi a) Ta CO DG = GH = HB = ^BD . 3 De tha'y cac vecto DG, GH, HB cung huong. Vay DG = GH = HB. b) TiV giac AEFC c6 AE = FC va AE // FC nen la hinh binh hanh. Do do AF // EC . Mat khac, AADG = ACBH => AG = CH. Nhu the AGCH la hinh binh hanh. Tu do AH = GC. Chu y:ABCD la hinh binh hanh <^A,B,C khong thang hang va AB = DC CUj TNHHMIV DVVII Khnug Vict 1.7. Cho tijf giac loi ABCD. Co ket luan gi ve tu giac ABCD neu AB = DC va AB AD Jilzang ddn gidi 1.7. Vi ABCD la tu giac loi va AB = DC nen ABCD la hinh binh hanh. Hon nira tu dieu kien c6n lai ta tha'y hinh binh hanh nay c6 hai canh lien tie'p bang nhau. Vay ABCD la hinh thoi. a+b §2. TONG CUA HAI VECTa A. TdMTATLITHUYET 1. ^inhnghla. ' • Cho hai vecto a va b. Tu mot diem A ba't ki dung cac vecto AB-a, BC = b . Khi do vecto AC duoc goi la tong ciia hai vecto a va b. Ki hieu AC = a + b. 2. Tinh chat • a + b = b + a; *a + 0 = a; * (a + b) + c = a + 3. Cac quy tdc • Quy tac ba diem : • Voi ba diem A, B, C tuy y ta luon c6 AB + BC = AC . • Quy tac hinh binh hanh : OABC la hinh binh hanh =;> OA + OC = OB. • Tinh chat trung diem : M la trung diem doan AB => MA + MB = 0 . • Tinh chat trong tam tarn giac : G la trong tam cua tam giac ABC => GA + GB + GC = 6 . B. MOTSODANGTOAN (b.c). t)ang 1. Chung minh dang thuc vecta Su dung qui tac ba diem Va qui tac hinh binh hanh. 5Sdt 5. Cho 4 diem A, B, C, D tuy y. Chung minh rang AB + CD = AD + CB. 7 Boi dudng HSG Hinh hoc 10 Giai Theo qui tac ba diem, ta c6: AB + CD = AD + DB + CB + BD Ap dung tinli chat giao hoan ciia phep cpng, ta c6 : AB + CD = AD + CB + DB + BD = AD + CB. <8di 6. Cho tii giac ABCD. Goi O la trung diem ciia AB. Chung minh rang OD + OC = AD + BC. Giai. Ap dung qui tac ba diem, ta c6 ^, . y •j-•!••;. <M OD=OA+AD OD + OC - OA + OB + AD + BC OC=OB+BC Chu y rang O la trung diem AB nen OA + OB = 0 Vithetaco OD + OC = AD + BC OV^JHl U 'bang2. Xdc dinh vd ttnh do dai cua vecto ,^. ~, ,5, ^, j; • • Dm vdo quy tac ba diem vd cdc ttnh chat cua tong hai vecto, ta rut gon kei qua phep todn tong cdc vecto. • Khi ttnh do dai vecto ta thuang xem do dai do la canh cua mgt tarn gidc ndo do. ^di 7. Cho tam giac ABC. Xac djnh cac vecto u = AB + CA, v = AB + CA + BC. Giai. Theo qui tac ba diem, ta c6 : ' '{{'•'' • u = CA + AB = CB, V = (AB + BC) + CA = AC + CA = 0. ^di 8. Cho hinh vuong ABCD tam O, canh a. Hay xac dinh va tinh do dai cac vecto: AD + AB, OA + OC, OB + BC, AB + AC. Giai. Truochettaco AC=BD=aV2, OA = OB = OC - OD = —. 2 Theo qui tac hinh binh hanh, ta c6 ,,,| AC =AC = a^. AD + AB = AC: AD + AB Vi O la trung diem AC nen OA + OC = 0 => OA + OC = 0 = 0. Cty TNHH MTV DWH Khang Vil-t • Theo qui tac hinh binh hanh: OB + BD = OD: • Dung hinh binh hanh BACE. Khi do OB + BD OD =OD = .72 AB + AC = AE: AB + AC AE = AE. Ta CO tam giac ADE vuong tai D biet AD = a, DE = 2a. Vay AE^ = AD^+DE^ =5a^ => AE = a>y5. Nhuvay AB + AC =aV5. ' ' ' ' 'bang 3. Chung minh tinh chat hinh hoc. (Bdi 9. Cho tu giac ABCD. Chung minh rang: ABCD la hinh binh hanh khi va chi khi voi moi diem M, ta c6 MA + MC = MB + MD. Giai : mynhmjf-unp Ap dung qui tac ba diem, ta c6 MA = MB + BA MC = MD + DC • MA + MC = MB + MD + BA + DC, vdi moi M (*) Ta c6: ABCD la hinh binh hanh <=> DC = AB o BA + DC = BA + AB = 0. Do do tir (*) ta suy dugc: ABCD la hinh binh hanh o MA + MC = MB + MD, voi moi diem M. bang 4. Tim tap hop diem thod man dang thuc ve tong cdc vecto ^di 10. Cho doan thang AB. Tim tap hop cac diem M thoa man dSng thuc : V ^ IMA + MBI=IABI. Giai. Ap dung qui tac ba diem voi O la trung diem AB, ta c6 ^' MA + MB=MO + OA + MO + OB = MO + MO + OA Dvmg vecto OC = MO, ta c6 MC = Md + OC = MO + MO (hinhve). M O Khido IMA + MBI=IABI o IMO+MOI-IABI»IMCI=IABI <=> MC = AB 2MO = 20B o MO = OB. M luon each diem O co djnh mot khoang khong doi OB nen tap hop cac diem M la duong tron tam O, ban kinh R = OB. , < 9 Boi dumig HSG Hinh hoc 10 C. LUYENTAP 1.8. Cac menh de sau day dung hay sai: a) Neu a = b+c thi b) Neu I la trung diem doan MN thi MI + NI = 0 ; c) Neu AB = CD thi AC = BD. Jittang ddn gidi a) Sai, ch3ng han trong truang hop b,c khong cung phuong. b) Diing. Neu I la trung diem doan MN thi MI = IN, do do: MI + NI = IN + NI = 0. c) Diing. Ap dung qui tac ba diem ta c6: AB = CD AC + CB = CB + BD <::> AC = BD. 1.9. Su dung qui tac ba diem de riit gon cac tong vecta sau day: a) u = AB + CD + BC + EA; b) v = AB + BC + CD + FE + DF. Jiu&ng ddn gidi a) U = AB + CD + BC + E^A = (EA + AB) + (BC + CD) = EB + BD = ED. b) V = AB + BC + CD + FE + DF = AC + CD + DF + FE = AD + DE = AE. 1.10. Cho tam giac ABC deu canh a. a) Xac djnh va tinh do dai cac vecto u = AB + AC , v = CA + BA. b) Goi M, N Ian lugt la trung diem ciia BC va AC. Xac dinh va tinh do dai vecto AM + BN . Jiuang ddn gidi a) Dung hinh binh hanh ABDC. Do tam giac ABC deu nen AD 1 BC. a>/3 Vay AD = 2AH = 2.— = aVS. b) Ta CO u = AB + AC = AD. Tu do aV3 AD = AD = aV3. AM + BN 1.11. Cho 5 diem A, B, C, D, E. Chung minh rang a) AB + CD + EA = CB + ED; b) CD + EA = CA + ED. Jiucfng ddn gidi a) AB + CD + EA = AB + CB + BD + EA = CB + EA + AB + BD - CB + ED. b) CD + EA = CA + AD + EA = CA + ED. Cty TNHH MVV }) VVU Khnitg Viet 1.12. Cho 4 diem A, B, C, D. Chung minh rang AB = CD khi va chi khi trung diem ciia hai doan thang AD va BC trung nhau. - •\ \'.i, JIu&ng ddn gidi ,r„ Goi I va J Ian luot la trung diem cua AD va BC. Khi do ,j /j . ^ I AI = iD, JB = CJ. I . Ta CO AB = CD AI + IJ + JB = Cj + Jl + ID » IJ = fl « IJ = 0 <=> I = J. 1.13. Cho hinh chCr nhat ABCD voi AB = 2a, AD = a . Tinh AB + AD AC + AD AC + BD AB + AD = 2a. Jiuang ddn gidi = aS, |AC + AD|=2a^/2, |AC + BD: 1.14. Cho tam giac ABC. Goi O la trung diem doan BC. Cac diem M, N theo thu tu do nam tren canh BC sao cho O la trung diem doan MN. Chung minh rang AB + AC = AM + AN. '' ' Jiuang ddn gidi Truoc he't de y rang O la trung diem ciia MN Goi D la diem doi xiing cua A qua O. Khi do cac tu giac ABDC va AMDN la hinh binh hanh. Do do theo qui tac hinh binh hanh AB + AC = AM + AN = AD. D, 1.15. Cho tam giac ABC. Goi D, E va F Ian lugt la trung diem ciia cac canh BC, CAva AB. Chung minh: AD + BE+ CF-d. : I.T Jiuang ddn gidi Cdch i;Ta c6 : AD + BE + CF =AB + BD + BC + CE + CA + AF = (AB + BC + CA) + (BD + CE + AF). Dira vao tinh chat duong trung binh trong AABC, ta CO BD = FE; CE = DF; AF = ED. /\ Do do AD + BE + CF = AA + FE + ED + DF = 0 + FF = 0 (dpcm) Cdch 2;Ta c6 AFDE, BDEF, CEFD la cac hinh binh hanh nen AD = AE + AF, BE = BF + BD , CF =CD + CE . 11 Boi dumtg HSG Hhih hoc 10 Cty TNHH MIV I > VVH Khang Viet Do do AD + BE + CF = AE + AF +BF + BD + CD + CE = AF+BF + BD+CD + CE+AE. s^ n Vi D, E va F Ian krot la trung diem ciia cac canh BC, CA va AB nen: AF + BF = 0,BD + CD = 0 va AE + CE = 0. 3"^^' Vay AD + BE + CF=0. 1.16. Cho tarn giac ABC. Tir cac diem A, B, C ta dung cac vecto bang nhau tiiy y AA = BB' = CC . Chung minh rang "^^"'^ '""^'^ '' a) BB' + CC" + BA + CA = BA' + CAT' ; b) AA' + BB" + CC' = BA' + CF + AC. Jiuang ddn gidi a) Theo qui tac hinh binh hanh, ta c6 BB' + CC' + BA + CA linn = BB' + BA + CC + CA = BA' + CA'. b) Taco BA' = BA + AA', CB' = CB + BB^,AC" = ATC + CC' Tudo'* "'"''••"'•^ . • ' BAVCB' + AC = M' + BF + CCVAC + CB + BA = AA' + BB' + C^ , i 1.17. Cho ngu giac deu ABCDE CO tarn O . m<\i 6b f/3 rinfiri rinM a) Chiing minh cac vecto u=OA + OB va v = OC + OE ciing phuong voi vecto OD. b) Chung minh rang OA + OB + OC + OD + OE-0. v.*^,,*. ' ,u/^i;. Jiuang ddn gidi a) Truoc he't chii y rang do tinh chat doi xiing cua ngu giac deu nen OD 1 AB. Dung hinh binh hanh AOBF. Khi do li - OA + OB = OF Do OA = OB nen AOBF la hinh thoi. Nhu the OF 1 AB. Suy ra u = OF, OD cung phuong. Chiing minh tuong tu ta cung c6 hai vecto v=OC + OE, OD cung phuong. b) Nhan xet rang OA + OB + OC + OD + OE la mot vecto cung phuong vol OD. Chung minh tuong tu ta cung c6 OA + OB + OC + OD + OE cimg phuong voi OB (vi vai tro hai vecto OB, OD nhu nhau) Vay chi c6 the tong cac vecto tren la vecto - khong . 1.18. Cho hai vecto a va b khac vecto 0 . a) Chung minh rang la+bl<lal + lbl; dau " = " xay ra khi nao ? i; ii b) Ap dung : Tim tap hop cac diem M thoa man dieu kien I AM + MB 1=I AM I +1 MB I, voi A, B la hai diem cho truoc. ' . Jiu&ng ddn gidi Hit vsr i a) Tu mot diem A bat ki ta dung AB = a, dung BC = b . Khi do Ac = a + b. ' + Neu ABC la mot tam giac ta c6 AC < AB + BC, do do I a + b I < I a I + I b I + Trong truong hop A, B, C thSng hang thi AC < AB + BC khi AB va AC ngugc huong ; ' . a + b _ , AC = AB + BC khi AB va AC cijng huong. Vay la + bl<lal + lbl; dau " = "xay ra khi a va b cijng huong. ; . ' b) Dua vao ket qua cau a) ta thay rang IAM + MBI=IAMI + IMBI ^ I AB 1 = 1 AM I + I MB I, hay AB = AM + MB, dieu nay xay ra khi va chi khi AM va MB cung huong, hay M nam tren doan thang AB. Vay tap hop cac diem M la doan thang AB. ' " '* §3. HIEU CUA HAI VECTQ lil A.TOMTATLITHUYET , .vu.v~Jurt ' 1. Vecta doi cua mqt vecta ni i-,! i • Neu a + b = 0 thi tanoi a la vecto doi ciia vecto b vanguoclai. ' • Vecto doi cua vecto a (ki hieu -a) la mot vecto ngugc huong voi vecto a va CO cimg do dai voi vecto a. , <!• 4' , ^ , 'u • . id V • OA •-• i; {j-y-jv •Mv:uinm :>(./. Jiieu cua hai vecto: a) Djnh nghia: Hieu cua hai vecto a va b, ki hi|u a-b la tong ciia vecto a voi vecto doi cua vecta b. vj.u,» ji. . a-b = a + (-b) ^) Cach ve vecto a - b : Cho cac vecto a va b (nhu hinh ve). -v Tu diem O bat ki, ta ve OA = a, OB = b. 4 ,. Ta CO BA = a - b. ^) Quy tac ve hieu vecto: Voi ba diem M, N, O tiiy y thi ta c6: KS^J = - OM. 13 Boi duang HSG Hhih hoc 10 B. MpTSODANGTOAN •^ang 1. Chiifig minh dang thiic vecta SM' dung qui tac ba diem vd qui tac hinh hinh hanh. \ <^di n. Cho hinh binh hanh ABCD va M la diem tijy y. Chung minh rang MA-MB = MD-MC. Giai. ^ . : , : Do ABCD la hinh binh hanh nen BA = CD. (1) Chuyrang M = MA-MB va CD = MD-MC. (2) Tu (1) va (2) ta CO dang thiic can chiing minh. <Bdi 12. Cho 6 diem A, B, C, D, E, F tiay y. Chung minh dang thuc sau AE-FB + CD = AD-EB + CF. Giai. Taco AE-FB + CD = /S + DE-(FE + EB) + CF + FD = AD-EB + CF + DE-FE + FD = AD-EB + CF + DE + ED= AD-EB + CF. Vay AE-FB + CD = AD-EB + CF. ^q.ng2. Xdc dink vd tinh do ddi cua vecto • Rut gon kei qua cdc phep todn tong, hieu cdc vecta • Khi tinh dp ddi vecta ta thuang xem dg ddi do Id canh ciia mot tam giac ndo do. <^dil3. Cho tam giac ABC. a) Hay xac dinh cac vecta u = AB - AC, v = BA - BC - CA. b) Xac djnh diem M sao cho MA + MC - MB = 0. c) Xac dinh diem N sao cho AN = AB + AC - CB. a) Taco u = AB-AC = CB. v = BA-BC-CA = CA-CA-0. b) Taco MA + MC-MB = OoMA + BC = 0 oM4 = BC. D Do do M la dinh thii tu cua hinh binh hanh ABCM. c) Goi D la diem do'i xung ciia A qua trung diem cua BC. Khi do tu giac ACDB la hinh binh hanh. Nhu the AB + AC = AD. 14 Cty TNHH MTV DWH Khang Viet Vay AN = AB + AC-CB = AD + BC = AD + AM Vay N la dinh thu tu cua hinh binh hanh ADNM. Cac diem M, N dugc xac dinh nhu tren hinh ve. « » ^di 14. Cho hinh thoi ABCD c6 tam O, AB = a va ABC = 60". Xac dinh va tinh dp dai cac vecta: AB + AD va AB - AD . Giai. Theo quy tac hinh binh hanh, ta c6 : " ' AB + AD = AC . Do do : 1 AB + AD 1 = AC = a (vi AABC deu). Theo quy tac ve hieu cua hai vecta, ta c6 AB-AD = DB. , , • Vi ABCD la hinh thoi CO AB = a va ABC = 60° nen BD = 2BO = a x/3 (vi AABC la tam giac deu). Vay I AB-AD I =DB = a>y3. ©ang- 3. Chitng minh tinh chat hinh hoc ^di 15. Cho tu giac ABCD thoa man dong thoi cac dieu kien sau day : ii) i) AB = DC; ii) AB + AD Chung minh rang tu giac ABCD la hinh chir nhat. Giai. Tir dieu kien i), ta thay rang ABCD la hinh binh hanh. Taco AB + AD = AC va AB-AD = DB AB-AD Do do tir dieu kien ii), ta dug-c AC DB => AC = BD. Vi hinh binh hanh ABCD c6 hai duang cheo bang nhau nen do la hinh chu nhat. ^at 16. Cho hai vecta a va b khong cimg phuong. Chung minh rang : Ne'u I a - b I = I a + b I thi vecta a va b c6 gia vuong goc vai nhau. Giai, Vai hai vecta a va b khong cung phuang, tu diem O bat ki, ta ve OA = a, OB = b. Dyng hinh binh hanh OACB, khi do BA = a-bvaOC = a+b. Dodo la - b I = la + b loAB = OC OACB la hinh chij nhat. B Boi duang HSG Hinh hoc 10 Suy ra OA J_ OB. Vay neu I a - b I = I a + b I thi a va b c6 gia vuong goc voi nhau. • ? '£>ang 4. Tim tap hap diem thoa man dang thiic vehieu cac vecto Bai 17, Cho tam giac ABC. Tim tap hap cac diem M thoa man dang thiic: IBA + MBI=iCM-CBI. , . , . , n • • ' Giai ,< j ^ , '/^ -tip Ta c6: IBA + MB 1=1 CM - CB 1^1 BA - BM 1=1 BMI • "' <=> I MA I=IBMI <::> MA = MB. Vay tap hop cac diem M la duang trung tryc doan thang AB. C.LUYENTAP 1,19, Chon khang dinh diing trong cac khang dinh sau a) Hai vecto doi nhau c6 dO dai bang nhau. b) Neu hai vecto AB va AC doi nhau thi A = C. | c) Hai vecto doi nhau thi cung phuong. d) Vecto doi ciia vecto a - b la vecto a + b. , ' ) ^ ( . < Jiuong dan gidi a) Dung vi hai vecto doi nhau c6 do dai bang nhau. j., , b) Sai vi khi do A la trung diem BC. c) Dung vi gia ciia chiing trimg nhau. d) Sai vi vecto doi ciia a-bla-a + b. 1,20, Cho tam giac ABC va diem M tiiy y. Cac khang dinh sau day dung hay sai? a)MA-MB = BA > b) BA-CM = AB-MC c) MA-BA = MC-BC d) MA + MB = CA + CB. J / iiiij^ Jiuong ddn gidi i a) Khang djnh diing. b) Khang dinh sai vi ' BA - CM = AB - MC o BA + BA = CM + CM » BA = CM (sai). c) Khang dinh diing vi CA = MA - MC = BA - BC MA - BA = MC - BC. d) Khang dinh sai. j « , • • 1,21. Cho 4 diem A, B, C, D. Chiing minh rang a) AB + CD = AD-BC; ; b) AB-CD = AC-M Cty TNI in Ml V DWH Khmig Vii Jiuang ddn gidi a) AB + CD = AD + DB + CD = AD + CB = AD - BC. : " ' =' ' b) AB - CD = AC + CB - CD = AC + DB = AC - BID. 1.22. Cho tii glac ABCD. Dung ben ngoai tii giac cac hinh binh hanh ABEF BCGH, CDIJ, DAKL. a) Chiing minh rang KF + EH + Gj + It = 0. b) Chiing minh rang EL - FH = FK - GJ. - i , . > Jiicang ddn gidi a) Sii dung ti'nh chat ciia hieu hai vecto ta c6: KF = AF - AK, EH = BH - BE, GJ = CJ - CG, IL = DL - DI Tir do KF +EH + Gj + IL = AF-Ak>M-BE+ CJ-CG + DL-Di = (AF - BE) + (BH - CG) + (C| - Dl) + (DL - AK) = 0 ' b) Ta CO EL = EF + FK + KL = BA + FK + AD, HI = HB + BA + AD + DI. => EL - HI = FK - HB - Di = FK + CG - CJ = FK + JG = FK - Gj. 1.23. Cho tam giac ABC. Tim tat ca nhung diem M thoa man cac truong hop sau day a) MA-MB = CA + BC; b) MA-MB + MC = 0 ; . , c) |BA - BM = CB - CM| ; d) |BA - BM = BA - BC Jiuang ddn gidi a) Gia sii diem M thoa man MA - MB = CA + BC o BA = BC + CA o BA = BA (luon diing) Vay M la diem bat ki tren mat phang b) Gia sii diem M thoa man MA-MB + MC = 6«>BA + MC = d OCM = "BA Vay M la dinh thii tu ciia hinh binh hanh ABCM. ^) Gia sii M la diem thoa man MA BA-BM = CB-CM MB « MA = MB <=> M nam tren duang trung true doan AB. Gia sii M thoa man BA - BM = IBA - BC MA CA <=>MA = CA <=> M nam tren dirniiv lirin hi in A, ]}j\f\ 17 Boi tiuaiig use, lliitli hoc 10 1.24. Cho hinh chu nhat ABCD c6 AB = 2a, AD = a. Hay xac dinh va tinh do dai cua cac vecto sau day : Jf i a) AC + DA; b) AD + AC ; Jiicang ddn giai c) BC + BA - BD. AC + DA = DC IAC + DA = DC = 2a. Dung hinh binh hanh DACE. Khido AD + AC = AE=> AD + AC =|XE| AD + AC = N/ABVBE^ = 2aV2. BC+BA-BD=BC+DA=BC-BC=0 BC+BA-BD = 0. 1.25. Cho hinh thang vuong ABCD c6 hai day AB = a, CD = 2a, duong cao AD = a. Hay xac djnh cac vecto sau va tinh do dai ciia chiing: CD - BA, AC - BD, DA - AB - CD, AB - EA, AC - DA. , Jiuang ddn gidi • CD - BA = CD + AB = CD + DE = CE CD-BA CE - CE = a. AC-BD = BF-BD = DF => AC-BD = DF =DF = 3a. DA-AB-CD = DA + BA-CD = EA-CD = CB-CD = DB DA-AB-CD = DB =aV2. AB-EA=AB+AE=AC => AB-EA = AC =asl5. 1 N D c F AC - DA = AG = 2aV2. • AC-DA-AC + AD = AG 1.26. Duong tron noi tiep ciia tarn giac ABC tiep xiic voi cac canh BC, CA, AB Ian lugt tai M, N va P. Cho AM + BN + CP = 0. Chung minh rang tarn giac ABC la tarn giac deu. Jiuang ddn gidi AM + BN + CP = 0r:^AB + m + BC + CN + CA + AP = 6 hay (AB + BC + C\ + (AP + BM + CN) = 0 <^ AP + BM + CN = 0, vi AB + BC + CA = 0. 18 CUj TNHII Ml/V DVVII KItiuig Vi Dung mpt tarn giac DEF c6 DE = AP, EF = BM, FD = CN, ta c6 AABC dong dang ADEF =^— = — = — . (1) ^ ^ DE EF FD ^ ^ ^ : Datx = AP = DE, y = BM = EF, z = CN = FD.Tac6 '[ ; (1) «^=:yii=£i2i = ^(^Liy±^=2. (2) X y z x + y + z .•,{;,. , • . (2) ^ l + ^ = l + - = l + - = 2 x = y = z. vyA;r, >/,;:; ; X y z VayAABCdeu. <*k'if:/ ^ ' §4. PHEP NHAN VECTQ VQI MOT SO A. TOMTATLITHUYET 1. 'f)inh nghla r 7 Tich cua vecto a voi so thuc k la mot vecto, ki hieu la ka, duoc xac din nhusau : (,,,,,, 1) Ne'u k > 0 thi vecto ka cung huong voi vecto a ; Neu k < 0 thi vecto ka nguoc huong voi vecto a 2) Do dai ciia vecto ka bang I k I. I a I. 2. 'Tinh chat • Voi moi vecto a, b va mpi so thuc k, 1 ta c6 : i.r-' 1) k(la) = (kl)a; ^ 2) (k + \)a = ka + la ; 3) k(a + b) = ka + kb; k(a-b) = ka-kb; 4) ka = d <=> k = 0 hoac a = 0. • I la trung diem doan AB o MA + MB = 2MI, voi mpi diem M. . • Ne'u G la trpng tarn tarn giac ABC thi vol mpi diem M ta luon c6 : MA + MB + MC = 3MG. 3. ^ieu kien de hai vecta ciing phuang • b Cling phuong a (a ;t 0) o 3ke M: b =ka. • Ba diem phan biet A, B, C thang hang o 3 ke K : AB = kAC. 4. S jeu thi mdt vecta qua hai vecta khong cung phuang • ' Cho hai vecto khong cijng phuong a va b. Khi do mpi vecto c deu c6 the bieu thj dupe mot each duy nhat qua hai vecto a va b, nghla la eo duy nhat cap so'm va n sao cho c = ma +nb. 1 [...]... MSG Hinh hqc 10 Cty TNHIl A ; / V UVVii 1.38 Cho tarn giac A B C G(?i D la diem tren canh B C sao cho DC=3DB, E Ici 1,40 Killing Cho hinh binh hanh A B C D c6 tam O Goi M , N Ian luat la trung diem ciia cac canh B C va C D Dat A B = a, A D = b ,v diem tren tia doi ciia tia B A sao cho AB = 2BE Dat A B = a, A C = b a) Tinh cac vecto A D , D E theo cac vecto a, b a) Tinh cac vecto B N , A M theo a, b... = 0 c 5 2NB + 10NI = 0 c > N B + 5NI = 0 Vay N la diem tren doan BI sao cho N B = 5NI d) Goi E la dinh cua hinh binh hanh A B E C K h i do A B + A C = A E AM - A B - A C - k o A P - A E = k o EP = k o EP = k De thay M K = 2 M N T u do suy ra ket luan ciia bai toan 1.41 Cho tam giac A B C a) Xac djnh diem M sao cho A M = A C + 2AB b) Cho N la diem thoa man B N = k B C Xac dinh k sao cho A, M , N... phSng toa do cho ba d i e m M ( 2 ; - 3 ) , N ( - 1 ; 2 ) , P(3;-2) Cho tam giac A B C H a i d i e m M va N Ian l u g t d i d o n g tren cac d u o n g thang A B va A C sao cho A M = m A B , A N = n A C T i m d i e u k i e n cua m va a) Xac d j n h toa do d i e m Q sao cho M P + M N - 2 M Q = 0 b) Xac d j n h toa do ba d i n h cua tam giac A B C sao cho M , N , P Ian l u o t la trung n sao cho vecto... sofirl^rftijli ' > DD*' = DI + i r + rD' Cpng 4 dang thuc tren ve theo ve, ta dugc A A ' + BB^ + C C + D D ' = 4ir ,j • Tu do suy ra dieu phai chung minh 1.62 Cho tam giac ABC M la diem tren canh BC sao cho BM = SMC, N la diem doi xung ciia M qua C a) Tinh cac vecto A M , A N theo hai vecto AB = a, A C = b b) Gpi I la trung diem AM, J la diem tren A N sao cho AJ = k A C Xac djnh k debadi&nB, I, Jthanghang t... •••:;•?::—• ' ^.^ , ^^t^., < V ; ^ v : \ - ! / : ; ' ' v i r ,v j_36 Cho tam giac ABC c6 trong tam G Goi I, J la hai diem xac dinh bai IA = 2IB, 3JA + 2JC = d a) Tinh IJ, IG theo hai vecto AB va AC - '^ b) Chung minh rang I, J, G la ba diem thang hang ' > Jiuang ddn gidi b) GQ'I K la trung diem canh B C Tinh IK theo cac vecto A B , A C 2 3 10 b) V i K la trung diem BC nen Khang Viet AB + AC 1 1 T u d...Boi duong MSG Hinh hoc 10 Cty TNHH MTV DWH Khang Vic B.MpTSODANGTOAN i)ang l.Dung i)ang 3 Bieu dim vecta theo hai vecta khong ciing phuong vecta • • • Cac diem can xdc dinh nen la diem ciioi vecta c) AF = A d + AE b) AE = ^AC ; a) Tinh MP, Q N theo vecto BC b) Tinh cac vecta MQ, NP theo cac vecto AB, AC Giai a) Ta CO Tu ket qua do suy ra M Q + NP = BC... B C va Q N , BC nguoc huong 3 • Neil CO the nen thu gon he thuc • Nen chon hai vecta ca so (khong cung phuong) (Bdi 20 Cho tam giac ABC Cac diem M , N tren canh AB va P, Q tren canh AC sao cho A M = M N = NB va AP = PQ = QB l l • Xac dinh tpa dp diem thoa man dieu kifn cho tmoc i, [y = - l •• •• c) AB = ( x - 2 ; 3 ) , A C = (-5;y + l ) 1 t , A, B, C thang hang < > AB, AC cung phuang (x - 2)(y +1) +15 = 0 = 1.52 Trong mat phang toa dp cho ba diem A ( - 2 ; 5), B(2; 2), C (10 ; 4) 1.54 Cho A(2;l),B(2;-l),C(x;-3),D(-2;y) a) Chung minh rang A, B, C la ba dinh ciia mpt tam giac... l 41 Boi dumg HSG Hinh hoc 10 Cty TNHH MTV DWH 1.55 Cho giac A B C biet A(3; 7), t r o n g tam G - ; 3 , d i e m B riclm tren tia O y vo V3 ) ( d i e m C n a m tren tia Ox T i m toa do B va C , ' ! > JJu&ngddngidi G o i C(xc ; 0) va B(0 ; yB) Ian l i r g t n a m tren cac t r u o O x va O y sao cho AA\]Q i ' ^ f3 + 0 + x c = 7 ^ j X c = 4 |7 + yB+0 = 9 *, , , , [yB=2 J 57 Cho luc giac deu A B C D E... Cty TNHH MTVDWH Boi liuihig HSG Hinh hoc 10 ^ai 23 Cho tam giac A B C N la diem tren canh A C sao cho N C = S N A M la suy ra: A G + A H = A D + A B + D C + B H diem tren canh B C voi B M = kBC Goi I la trung diem A M V i D G va B H la hai vecto doi nhau a) Tinh cac vecto B I , B N theo cac vecto B A , B C nen D G + B H = 0 Vay b) Xac dinh k sao cho ba diem B, I , N thang hang AG + A H = AD + AB . hoc sinh gidi Hinh hoc 10& quot; la mot tai lieu tham khao mon toan dung trong truong trung hoc pho thong, nh^m muc dich giijp cho hoc sinh ren luyen cac ky nang giai cac dang toan theo. hqc 10 1.38. Cho tarn giac ABC. G(?i D la diem tren canh BC sao cho DC=3DB, E Ici ,v diem tren tia doi ciia tia BA sao cho AB = 2BE. Dat AB = a, AC = b. a) Tinh cac vecto AD, DE theo . 1.41. Cho tam giac ABC. a) Xac djnh diem M sao cho AM = AC + 2AB. b) Cho N la diem thoa man BN = kBC . Xac dinh k sao cho A, M, N thang hang. Jiuang ddn gidi a) Dung diem D sao cho AD