phần loại và phương pháp giải hình học 10

516.0076 PH121L NGUYEN PHU KHANH ^I^^IKWli^^j )AU THANH KY - NGUY§N MJNH NHIEN " _JYEN ANH TRUING - NGUYEN TAN SIENG D6 NGOG THUY (Nhom giao vien chuyen toan trifdng THPT) PHANLOAI & PHCONG PEIAP GlAl mmm WHIM (Tai Ban, SCfa ChOa Va Bo Sung) • Danh cho hoc sinh Idp 10 on tap va nang cao kien thufc • Bien soan theo noi dung sach giao khoa cua bp GD&DT AT BAN DAI HOC QUOC GIA HA NOI 1 w a\. 7«3UYElsrPFrtrKFwi\iFr DAU THANH KY - NGUYEN MINH NHIEN NGUYEN ANH TRUdNG - NGUYEN TAN SIENG D6 NGOC THUY (Nhom glao vien chuyen toan trudng THPT) if I! ri; : PEIANLQAI & FHlTdNG PHAP GIAI (Tai Ban, Sufa ChOa Va Bo Sung) Danh cho hoc sinh Idp 10 on tap va nang cao kien thiJc Bien soan theo npi dung sach giao khoa cua bp GD&DT mm, ^ ^^ ^Allj i/Sn riiA HA NOI Cac em hoc sinh than men!. "Phan loai va phuong phap giai Hinh hoc 10" la mot trong nhi>ng cuon thuoc bo sach "Phan loai va phuang phap giai lop 10,11,12 ", do nhom tac gia chuyen toan THPT bien soan. Vai each vie't khoa hoc va sinh dpng, cuon sach se giup ban doc tiep can voi mon Toan mot each tu nhien, khong ap luc, ban doc trd nen tu tin va nang dong hon; hieu ro ban chat, biet each phan tich de tim ra trong tarn cua van de va biet giai thich, lap luan cho tung bai toan. Su da dang cua he thong bai tap va tinh huong giup ban doc iuon hung thu khi giai toan. Tac gia chii trong bien soan nhung cau hoi ma, noi dung ca ban bam sat sach giao khoa va cau true de thi Dai hoe, dong thai phan bai tap thanh cac dang toan eo 161 giai chi tie't. Hien nay de thi Dai hoc khong kho, to hgp cua nhieu van de don gian, nhung chiVa nhieu cau hoi ma neu khong nam chac ly thuyet se lung tung trong viec tim loi giai bai toan. Vai mot bai toan, khong nen thoa man ngay voi mot loi giai minh vua tim dugc ma phai cogang tim nhieu each giai nhat cho bai toan dcS, moi mot each giai se c6 them phan kien thue moi on tap. Mon Toan la mot mon rat ua phong each tai tu, nhung phai la tai tu mot each sang tao va thong minh. Khi giai mot bai toan, thay vi dCing thoi gian de luc loi tri nha, thi ta can phai suy nghi phan tich de tim ra phuang phap giai quyet bai toan do. Doi voi Toan hoc, khong c6 trang sach nao la thua. Tung trang, tung dong deu phai hieu. Mon Toan doi hoi phai kien nhan va ben bi ngay tu nhung bai tap dan gian nhat, nhi>ng kien thue co ban nhat. Vi chinh nhiing kien thue co ban moi giup ban doe hieu dugc nhung kien thue nang cao sau nay. Ludwig Van Beethoven tung noi: "Gigt nuoe co the lam mon tang da, khong phai vi gigt nuoe co sue manh, ma do nuae chay lien tuc ngay dem. Chi CO sir phaii da'u khong met moi moi dem lai tai nang. Do do ta co the khSng djnh, khong nhich tung buae thi khong bao gio co the di xa ngan dam". Mac dii tac gia da danh nhieu tam huyet cho cuon sach, song su sai sot la dieu kho tranh khoi. Chung toi rat mong nhan dugc su phan bien va gop y quy bau ciia quy doe gia de nhung Ian tai ban sau cuon sach dugc hoan thi^n han. *' , r\ \ ^V'Q' ^^^y "^^^ nhom bien soan y\n Phu Khanh Cty TNHH MTV UVVHKmHg VTgt SAH4 Hinh l.I §1. CACDINHNGHIA A. TOM TAT LY THUYET 1. Dinh nghia vecto: Vccta la doan thang co huong, nghia la trong hai diem mut cua doan thang da chi ro diem nao la diem dau, diem nao la diem cuoi. Vecto CO diem dau la A, diem cuoi la B ta ki hieu: AB Vecto eon dugc ki hieu la: a, b, x, y, Veeta - khong la vecto co diem dau trung diem cuoi. Ki hieu la 0 2. Hai vecto cung phuomg, citng huong. - Duong thang di qua diem dau va diem cuoi ciia vecto ggi la gid cua vecto - Hai vecto co gia song song hoac triing nhau ggi la hai vecto ciing phwang - Hai vecto ciing phuang thi hoac eung huong hoac nguge huong. c D Hinh 1.2 H Vi du: 6 hinh ve tren tren (hinh 2) thi hai vecto AB va CD ciing huong eon EF va HG nguge huong. Dac biet: vecto - khong cung huong voi mgi vec to. 3. Hai vecto bSng nhau A B - Dg dai doan thSng AB ggi la do dai vecto AB,kihieu AB Vgy AB =AB. L 1 c - Hai vecta hhng nhau neu chung ciing huong va eiing do dai. Hinh 1.3 D Vi du: (hinh 1.3) Cho hinh binh hanh ABCD khi do AB = CD B. CAC DANG TOAN VA PHlTaNG PHAP GIAI. DANG TOAN 1: XAC DINH MOT VECTO; PHUONG, HUONG CUA ' VECTO; DQ DAI CUA VECTO 1. PHUONG PHAPGlAl. • Xac djnh mot vecto va xac dinh su cung phuang, ciing huong ciia hai vecto theo djnh nghla • Dua vao cac tinh chat hinh hoc cua cac hinh da cho biet de tinh dp dai cua , mpt vecto 2. CAC VI Dg. • I Vi du 1: Cho tu giac ABCD. Co bao nhieu vecto khac vecto-khong c6 diem dau va diem cuoi la dinh cua tu giac. Lai gidi Hai diem phan biet, chang han A, B ta xac djnh dupe hai vecto khac vecto- khong la AB, BA . Ma tu bon dinh A, B, C, D ciia tu giac ta c6 6 cap diem phan biet do do c6 12 vecto thoa man yeu cau bai toan. Vi du 2: Chung minh rang ba diem A,B,C phan bi^t thang hang khi va chi khi AB, AC ciing phuong. Lcri giai Neu A,B,C thang hang suy ra gia cua AB, AC deu la dudng thang di qua ba diem A,B,C nen AB, AC cung phuong. i Ngup-c lai neu AB, AC ciing phuong khi do duong thSng AB va AC song song hoac triing nhau. Nhung hai duong thang nay ciing di qua diem A nen hai duong thang AB va AC triing nhau hay ba diem A, B, C thing hang. Vi du 3: Cho tam giac ABC. Goi M,N,P Ian lupt la trung diem cua BC,CA,AB. a) Xac djnh cac vecto khac vecto - khong ciing phuong v6i MN c6 diem dau va diem cuoi lay trong diem da cho. b) Xac dinh cac vecto khac vecto - khong ciing huong vdi AB c6 diem dau va diem cuoi lay trong diem da cho. c) Ve cac vecto bang vecto NP ma c6 diem dau A, B . Lcri gidi (Hinh 1.4) a) Cac vecto khac vecto khong ciing phuong voi MN la NM, AB, BA, AP, PA, BP, PB. b) Cac vecto khac vecto - khong cung huong voi AB la AP, PB, NM . c) Tren tia CB lay diem B' sao cho BB' = NP ^ ' ^ Khi do ta CO BB' la vecto c6 diem dau la B va bang vecto NP. ' Qua A dung duong thang song song voi duong thang NP. Tren duomg thang do lay diem A' sao cho AA" ciing huong vo-iNP va AA" = NP. Hinh 1.1" Khi do ta CO AA' la vecto c6 diem dau la A va bang vecto NP. Vi du 4: Cho hinh vuong ABCD tam O canh a . Goi M la trung diem cua AB, N la diem doi xung voi C qua D. Hay tinh do dai ciia vecto sau: MD, MN . ^ Lai gidi (hinh 1.5) Ap dung djnh ly Pitago trong tam giac vuong MAD ta c6 DM^ = AM^ + AD^ = Suy ra MD = MD- a' 2 5a' + a = — Qua N ke duong thing song song vol AD cat AB tai P. | Khi do tu giac ADNP la hinh vuong va| a 3a PM = PA + AM - a + - = —. -4 M Ap dung djnh ly Pitago trong tam giac vuong NPM ta c6 MN^ =NP2+PM^ =a^ + ax/TS 3a^ 2 . 13a^ .MN = iVl3 Suy ra MN = MN = • 3. BAI TAP LUY£N TAP. *^ ""^ Bai 1.1: Cho ngii giac ABCDE . Co bao nhieu vecto khac vecto-khong c6 diem dau va diem cuoi la dinh ciia ngii giac. ^ Huang dan gidi I Hai diem phan biet, chang han A, B ta xac dinh dupe hai vecto khac vecto- khong la AB, BA. Ma tir nam dinh A, B, C, D, E ciia ngu giac ta c6 10 ca I diem phan biet do do c6 20 vecto thoa man yeu cau bai toan. P Bai 1.2: Cho ba diem A, B, C phan bi^t thang hang. a) Khi nao thl hai vecto AB va AC cung huong ? ^. b) Khi nao thi hai vecto AB va AC ngiroc huong ? Huang dan gidi ^ a) A nam ngoai doan BC b) A nam trong doan BC Bai 1.3: Cho bon diem A, B, C, D phan biet. a) Neu AB = BC thi c6 nhan xet gi ve ba diem A, B, C f>r{> b) Neu AB = DC thi co nhan xet gi ve bon diem A, B, C, D Huang dan gidi , (^fy j a) B la trung diem cua AC b) A, B, C, D thang hang hoac ABCD la hinh binh hanh, hinh thoi, hinh vuong, hinh chij' nhat Bai 1.4: Cho hinh thoi ABCD c6 tam O. Hay cho biet khang djnh nao sau day dung ? ^ b) AB = DC c) OA = -OC a) AB = BC d) OB = OA e) AB BC BD f) 2 OA Huang dan gidi a) Sai b) Dung c) Dung d) Sai e) Dung f) Sai Bai 1.5: Cho luc giac deu ABCDEF tam O. Hay tim cac vecto khac vecto-khong CO diem dau, diem cuoi la dinh ciia luc giac va tam O sao cho a) Bang voi AB b) Ngugc huong voi OC Huang dan gidi a) Fd,OC,ED b) c6,OF,BA,DE Bai 1.6: Cho hinh vuong ABCD canh a , tam O va M la trung diem AB. Tinh dp dai cua cac vecto AB, AC, OA, OM, OA + OB. Huang dan gidi Ta CO AB = AB = a; OA =OA=-AC= 2 AC = AC = \/ABVBC^ = aV2 OM = OM = - Ggi E la diem sao cho tu giac OBEA la hinh binh hanh khi do no cung la hinh vuong Ta CO OA + OB = OE => OA + OB = OE = AB = a Hinh 1.40 Bai 1.7: Cho tam giac ABC deu canh a va G la trong tam. Goi I la trung diem cua AG. Tinh do dai cua cac vecto AB, AG, BI. 1 Huang dan gidi S. Ta CO AB = AB = a Goi M la trung diem cua BC Ta CO AG -AM = -\// 3 3 = AG = -AM = -7AB2-BM2 =^ BI —j_(aW^aV2T = BI = N/BM'+MI2 =^^ + ^ = — Bai 1.8: Cho truoc hai diem A,B phan biet. Tim tap hop cac diem M thoa man Huang dan gidi o MA = MB => Tap hop diem M la duong trung true cua doan MA MB MA MB thang AB DANG TOAN 2: CHUNG MINH HAI VECTO BANG NHAU. : 1. PHLTONG PHAPGlAl. ' • De chiing minh hai vecto bang nhau ta chung minh chung c6 ciing do dai va cung huong hoac dua vao nhan xet neu tii giac ABCD la hinh binh hanh thi AB = DC va AD = BC 2. CAC VI DU. .y ;,rt 7. , Vid«l:Ch^ giac ABCD. Goi M, N, P, Q Ian lugt la trung diem AB, BC, CD, DA. Chung minh rSng MN=QP • ' • Lai gidi (hinh 1.6) Do M, N Ian luat la trung diem cua AB va BC nen MN la duong trung binh cua tam giac ABC Suy ra MN//AC va MN = iAC (1). Tuong tu QP la duong trung binh cua tam giac ADC suy ra QP//AC va QP = ^AC (2). Tir (1) va (2) suy ra MN//QP va MN = QP do do tur giac MNPQ la hinh binh hanh Vaytac6MN=QP Vi du 2: Cho tarn giac ABC c6 trong tarn G . Goi I la trung diem ciia BC . Dung diem B' saocho B'B = AG. ^ ,>^,.,i i,, a) Chung minh rang BI = IC % -; • '''-^ b) Goi J la trung diem ciia BB'. Chung minh rang BJ = IG. ^ Loi^ax (hinh 1.7) a) Vi I la trung diem cua BC nen BI = CI va BI ciing Huang voi IC do do hai vecto BI, IC bang nhau hay BI = IC . b) Taco B'B = AG suy ra B"B = AG va BB7/AG. Do do BJ, IG cung huong (1). Vi G la trong tam tam giac ABC nen * IG = ^AG, J la trung diem BB' suy ra BJ=1BB- ^ Vivay BJ = IG (2) I Tu (1) va (2) ta CO BJ = IG. Hinh 1.7 Vi du 3: Cho hinh binh hanh ABCD. Tren cac doan thSng DC, AB theo thu hr lay cac diem M, N sao cho DM = BN. Goi P la giao diem ciia AM, DB va Q la giao diem cua CN, DB. Chung minh rang AM = NC va DP = QB. LOT ^«» (hinh 1.8) Ta CO DM = BN => AN = MC, mat khac AN song song voi MC do do tu giac ANCM la hinh binh hanh v SuyraAM = NC. Xet tam giac ADMP va ABNQ ta c6 DM = NB (gia thiet), _ PDM = QBN (so le trong) m Mat khac DPM = APB (do! dinh) va APQ = NQB (hai goc dong vj) suy ra DMP = BNQ. Do do ADMP = ABNQ (c.g.c) suy ra DP = QB . De thay DP, QB ciing huong vi vay DP = QB. 8 3. BAI TAP LUYCN TAP. Bai 1.9: Cho tu giac ABCD . Goi M, N, P, Q Ian lugt la trung diem AB, BC, CD, DA. Chung minh rang MQ = NP . m-ofi Huong dan gidi • V^**^ Do M, Q Ian lugft la trung diem ciia AB va AD nen A MQ la duong trung binh ciia tam giac ABD suy ra MQ//BD va MQ = ^BD (1). Tuong tu NP la duang trung binh ciia tam giac CBD suyra NP//BD va NP = iBD (2). Tir (1) va (2) suy ra MQ//NP va NP = MQ do do tir giac MNPQ la hinh binh hanh Vay taco MQ=NP . Bai 1.10: Cho hinh binh hanh ABCD. Goi M, N Ian lugt la trung diem cua DC, AB; P la giao diem ciia AM, DB va Q la giao diem ciia CN, DB . Chung minh rang DM = NB va DP PQ = QB . Huang dan gidi ' Ta CO tu giac DMBN la hinh binh hanh vi DM = NB = | AB, DM / /NB. Suy ra DM = NB. Xet tam giac CDQ c6 M la trung diem ciia DC va MP//QC do do P la trung diem ciia DQ. Tuong tu xet tam giac ABP suy ra dugc Q la trung diem ciia PB Vi vay DP = PQ = QB tu do suy ra DP = PQ = QB Bai 1.11: Cho hinh thang ABCD c6 hai day la AB va CD voi AB - 2CD. Tu C ve Ci = DA. Chung minh rang ^ b) Ai = iB = DC V Huang dan gidi a) AD = IC va DI = CB a) Ta CO CI = DA suy ra AICD la hinh binh hanh =>AD = IC Ta CO DC = AI ma AB - 2CD do do AI = i AB => I la trung diem AB Ta CO DC = IB va DC / /IB => tir giac BCD! la hinh binh hanh Suy ra DI = CB b) I la trung diem cua AB=>AI = IB va tu giac BCDI la hinh binh hanh => IB = DC suy ra AI = IB = DC Bai 1.12: Cho tam giac ABC c6 true tarn H va O tarn la duong tron ngoai tiep . Goi B' la diem doi xung B qua O. Chung minh: AH =B'C . Huang dan gidi Taco B'CIBC, AH 1 BC ^ B'C //AH , B'AIBA, CH 1 AB => B'A //CH Suy ra AHCB' la hinh binh hanh do do AH =B'C . §2 TONG VA HIEUHAIVECTO A. TOM TAT LY THUYET 1. Tonghaivecta a) Dinh nghla: Cho hai vecto a; b. Tir diem A tuy y ve AB = a roi tu B ve BC = b khi do vecto AC duoc goi la tong cua hai vecto a; b. Ki hieu AC = a + b (Hinh 1.9) b) Tinh chat: + Giac hoan : a + b = b + a + Ket hop : (a + b) + c = a + (b + c) + Tinh chat vecto - khong: a + 0 = a, Va 2. Hieu hai vecta Hinh 1.9 ? a) Vecto doi cua mot vecto. Vecta doi ciia vecto a la vecto ngugc huang va cung do dai vol vecto a Ki hieu -a Nhu vay a + (-a) = 0, Va va AB = -BA b) Djnh nghta hieu hai vecto: Hieu ciia hai vecta a va b la tong ciia vecto a va vecto doi ciia vecto b. Ki hi^u la a-b = a + (-b) 3. Cac quy tic: Quy tac ba diem: Cho A, B ,C tuy y, ta c6 : AB + BC = AC Quy tic hinh binh hanh: Neu ABCD la hinh binh hanh thi AB + AD = AC Quy tac ve hieu vecto: Cho O, A, B tiiy y ta c6: OB -OA = AB 10 Chu ij: Ta c6 the mo rpng quy tic ba diem cho n diem A,,A2, ,An thi A1A2 + A2A3" + + A„_,An = A, A„ B. CAC DANG TOAN VA PHl/QNG PHAP GIAI. DANG TOAN 1: XAC DINH DO DAI TONG, HIEU CUA CAC VECTO. 1. PHLTONG PHAP GIAI. De xac dinh do dai tong hieu ciia cac vecto • Truoc tien sir dung djnh nghla ve tong, hieu hai vecto va cac tinh chat, quy tac de xac djnh phep toan vecto do. • Dua vao h'nh chat cua hinh, sir dung djnh li Pitago, he thuc lugng trong tam giac vuong de xac djnh do dai vecto do. 2. CAC ViDU. Vidu 1: Cho tam giac ABC vuong tai A c6 ABC = 30" va BC = aVS . Tinh do dai ciia cac vecto AB + BC, AC - BC va AB + AC . Lai gidi (hinh 1.10) Theo quy tac ba diem ta c6 . AB + BC = AC AC Ma sin ABC ^ BC AC = BC.sin ABC = ax/5.sin30° aVS Do do AB + BC = AC = AC • AC-BC=AC+CB=AB Ta CO AC^ + AB^ = BC^ ^ AB = >/BC^ - AC^ = a>/T5 Vi vay AC-BC AB =AB = Goi D la diem sao cho tir giac ABDC la hinh binh hanh. Khi do theo quy tic hinh binh hanh ta c6 AB + AC = AD Vi tam giac ABC vuong 6 A nen tir giac ABDC la hinh chu nhat suy ra AD = BC = a>/5 Vay AB + AC AD = AD = a>/5 11 Vtdu 2: Cho hinh vuong ABCD c6 tarn la O va canh a. M la mpt diem bat ky. a) Tinh AB +AD OA - BO CD - DA b) Chung minh rang u = MA + MB - MC - MD khong phu thuoc vi tri diem M . Tinh do dai vecto u Lai gidi {hinh 1.11) a) + Theo quy tac hinh binh hanh ta c6 AB + AD = AC Suy ra AB + AD AC = AC. Ap dung dinh li Pitago ta c6 AC^ = AB^ + BC^ = 2a^ ^ AC = >/2a Vay AB + AD = ayfl + Vi O la tam ciia hinh vuong nen OA = CO suy ra OA - CB - CO - BO - CB Vay OA - BO CB = a Hinh 1.11 + Do ABCD la hinh vuong nen CD = BA suy ra CD - DA = BA + AD = BE) • h Ma |BD| = BD = 7AB^ + AD^ = suy ra |CD - DA| = a^f2 b) Theo quy tSc phep tru ta c6: u = (MA - MC) + (MB - MD) = CA + DB Suy ra u khong phu thuoc vj tri diem M . Qua A ke duong thang song song v6i DB cat BC tai C . Khi do tu giac ADBC la hinh binh hanh (vi co cap canh doi song song) suy ra DB = AC' Do do u - CA + AC' = CC' Vi vay Icci = BC + BC' = a + a-2a 3. BAI TAP LUY^N TAP. ^ Bai 1.13: Cho tam giac ABC deu canh a. Tinh do dai cua cac vecta sau AB-AC, AB + AC. ^ ^ Huong dan gidi '^A' Theo quy tac phep tru ta c6 AB-AC = CB: AB-AC =BC = a Gpi A' la dinh ciia hinh binh hanh ABAC va O la tam hinh binh hanh do. Hinh 1.45 Khi do ta CO AB + AC = AA'. Ta CO AO = yf^. AB'-OB'=Ja'-^ Suy ra • •••• ' / AB + AC = AA' = 2AO = aS Bai 1.14: Cho hinh vuong ABCD c6 tam la O va canh a. M la mot diem bat ky. a) Tinh AB + 0D|, AB-OC + OD b) Tinh do dai vecto M A - MB - MC + MD Huang dan gidi a) Ta CO OD = BO => AB + OD = AB + BO = AO AB + OD = AO = AC ayfl B' Ta CO OC = AO suy ra 1 1 V = 6 B = 0 Hinh 1.46 => AB-OC + OD b) Ap dung quy tac phep tru ta c6 MA - MB - MC + MD = (MA - MB) ^ (MC - MD) = BA - DC LayB' la diem dol xung cua B qua A do -DC = AB' BA - DC = BA + Ais^BB' Suy ra MA-MB-MC + MD BB' = BB' = 2a Bai 1.15: Cho hinh thoi ABCD canh a va BCD = 60". Goi O la tam hinh thoi. Tinh AB + AD OB-DC Ta CO AB + AD OB-DC AC CO Huang dan gidi = 2acos30" =aS, 0 a>/3 = a cos 30 = OB+AC-OA Bai 1.16: Cho bon diem A, B, C, O phan biet c6 do dai ba vecto OA, OB, OC cung bang a va OA + OB + OC = 0 a) Tinh cac goc AOB, BOC, COA b) Tinh Huang dan gidi a) Tu gia thie't suy ra ba diem A, B, C tao thanh tam giac deu nhan O lam trongtamdodo AOB = BOC=COA = 120" , b) Gpi I la trung diem BC. Theo cau a) AABC deu nen AI = — a 2 OB + AC-OA" =a>/3 Bai 1.17: Cho goc Oxy. Tren Ox, Oy lay hai diem A, B . Tim dieu ki?n cua A,B sao cho OA + OB nam tren phan giac ciia goc Oxy. ! Huang dan gidi Dung hinh binh hanh OACB. Khi do: OA+OB = OC Vay OC nam tren phan giac goc xOy <=> OACB la hinh thoi <=> OA = OB. DANG TOAN 2: CHUNG MINHDANG THUC VECTO. 1. PHLTONG PHAPGIAI. • De chung minh dang thuc vecta ta c6 cac each bien doi: venay thanh vekia, bien doi tuong duong, bien doi hai veciing bang mot dai lirong trung gian. Trong qua trinh bien doi ta can su dung linh boat ba quy t3c ti'nh vecto. Lmi I/: Khi bien doi can phai liitmi<^ dicli, chang han bien doi vc'phai, ta can xem vetrai c6 dai luong nao de tu do lien tuang den kie'n thuc da c6 de lam sao xuat hien cac dai lugng 6 vetrai. Va ta thuang bien doi ve'phuc tap ve vedan gian hon. 2. CAC Vi DU. Vtdu 1: Cho nam diem A,B,C,D,E . Chung minh r5ng .aMB +CD + EA = CB + ED b) AC + CD-EC = AE-DB + CB Lai gidi a) Bien doi vetrai ta c6 ^ VT = (AC + CB) + Cb + (Eb + DA) = (CB + Eb) + (AC + Cb) + DA =(CB + Eb) + AD + DA = CB + ED = VP (DPCM) : . b) Dang thuc tuong duong voi (AC-AE) + (CD-CB)-EC + DB = d<:>EC + BD-EC + DB = 0 BP + DB = 6 (dung) (DPCM). Vidu 2: Cho hinh binh hanh ABCD tam O. M la mot diem bat ki trong m|it phang. Chung minh rang . a) BA + DA + AC = d c) MA + MC = MB + MD . b) OA + OB + OC + OD^d LaigidHHinh 1.12) a) Ta CO BA + DA + AC = -AB - AD + AC = -(AB + AD) + AC ^ Theo quy tac hinh binh hanh ta c6 AB + AD = AC suy ra . ^ _. v^,? n BA + DA + AC = -AC + AC = 0 b) Vi ABCD la hinh binh hanh nen ta c6: OA = Cd=>OA + OC = OA +Ad = d M.n,,,,,^ , Tuong tu: OB + OD = 0 => OA + OB + OC + OD = 0 . c) Cach 1: Vi ABCD la hinh binh hanh nen AB = DC => BA + DC = BA + AB = 0 MA + MC = MB + BA + MD + DC = MB + Mb + BA + DC = MB + Mb Cach 2: Dang thuc tuong duang voi Ac; MA - MB = MD - MC » BA - CD (dung do ABCD la hinh binh hanh) Vi du 3: Cho tam giac ABC. Goi M, N, P Ian lugt la trung diem cua BC, CA, AB . Chung minh rang . . a) BM + CN +AP = d b) AP + AN-AC + BM = d c) OA + OB + OC = OM + OIV + OP voi O la diem bat ki. ' Lai gidi {H\nh 1.13) a) Vi PN,MN la duang trung binh cua tam giac ABC nen PN//BM, MN//BP suy ra tu giac BMNP la hinh binh hanh BM = PN N la trung diem cua AC => CN = NA Do do theo quy tac ba diem ta co BM + CN + AP = (PN + NA) + AP = PA + AP = d b) Vi tu giac APMN la hinh binh hanh nen theo quy tMc hinh binh hanh ta c6 AP + AN = AM , ket hop voi quy t^c phep tru => AP + AN - AC + BM = AM - AC + BM = CM + BM Ma CM + BM = d do M la trung diem ciia BC. Vay AP + AN-AC + BM-d. c) Theo quy tac ba diem ta c6 OA + OB + OC = (OP + PA) + (OM + MB) + (ON + NC) = (OM + ON + OP) + PA + MB + NC , = (OM+ON+OP) - (BM + CN + AP) • Theocau a) ta c6 BM + CN + AP = d suy ra OA + OB + OC = OM + ^+^. 3. BAI TAP LUY^N TAP. Bai 1.18: Cho bon diem A, B,C, D. Chung minh ring a) DA-CA = pB-CB b) AC + DA + BD = AD-CD+BA „ ' • Huang dan gidi a) Ap dung quy tac phep tru ta c6 _ DA-CA = DB-CB<»DA-DB = CA-dB c>BA = BA (dung) V b) Ap dung quy tac ba diem ta c6 AC + DA + BD = AD - CD + BA » (DA + AC) + BD = (BA + AD) - CD o DC + BD = 150 - CD (diing) Bai 1.19: Cho cac diem A, B, C, D, E, F . Chung minh rang AD + BE + CF = AE + BF + CD Hu&ng dan gidi Cdch 1: Dang thuc can chung minh tuang duong voi ( AE) - AE) + (BE - BF) + (CF - CD) = d o ED + FE + DF = 6 o EF + FF = d (dung) Cdch 2: VT = AD + BE + CF = (AE + ED) + (BF + FE) + (CD + DF) = AE + BF + CC> + EC> + FE + DF • = AE + BF + CC) = VP Bai 1.20: Cho hinh binh hanh ABCD tam O. M la mot diem bat ki trong mat phang. Chung minh r^ng a) AB + OD + OC = AC b) BA + BC + 0B = 0D c) BA + BC + OB = MO-MB Huang dan gidi a) Ta CO OD = BO do do AB + OE) + OC = AB + B6 + OC = Ad + OC = AC b) Theo quy tac hinh binh hanh ta c6 BA + BC + OB = BD + OB = Oli + BD = OD Hinh 1.47 c) Theo cau b) ta c6 BA + BC + OB = OD Theo quy t3c trir ta c6 MO - MB = BO Ma ob = BO suy ra BA + BC + OB = MO - MB Bai 1.21: Cho tam giac ABC. Goi M, N, P Ian lugt la trung diem ciia BC, CA, AB . Chung minh rang a) NA + PB + MC = 0 b) MC + BP + NC = BC Huang dan gidi ^ a) Vi PB = AP, MC = PN nen NA + PB + MC - NA + AP + PN = NP + PN = 0 b) Vi MC = BM va ket hgp voi quy tac ba diem, quy tac hinh binh hanh ta c6 MC + BP + N(: = BM + BP + NC: = BN + NC = BC u , Hinh 1.48 Bai 1.22: Cho hai hinh binh hanh ABCD va AB'C'D' c6 chung dinh A. Chung minh rang B'B + CC' + D'D = d Huang dan gidi , . Theo quy tac trir va quy tac hinh binh hanh ta CO ^ , ,.<^ B'B + CC' + ITD = (AB - AB') + (AC' - AC) + (AD - AD' j^pl =(AB +AD)-AC-(AB' +AD') +AC = 0 Bai 1.23: Cho ngu giac deu ABCDE tam O. '"^ Chung minh rang OA + OB + OC + OE + OF = 0 Huang dan gidi Dat u = OA + OB + OC + OE + OF * Vi ngu giac deu nen vecto OA + OB + OC + OE cung phuong voi OF nen u cung phuong voi OF . • . / < Tuong tu u cung phuong voi OE suy ra u = 0. ^ • Bai 1.24: Cho hinh binh hanh ABCD . Dung AM = BA, MN = DA, NP = DC, PQ = BC. ••• Chung minh rang: AQ =0 . A; Humg dan gidi * j ' Theo quy tac ba diem ta c6 AQ = AM + MN + NP + PQ = BA + DA + DC + BC Mat khac BA + BC = BD, DA + DC = DB suy ra AQ = BD + DB = 0 [...]... sau TIT a) u + 2v vai u = 3 i - 4 j va v = — i ' 2 b) k = 2a + b va 1 = -a + 2b + 5c b) Taco 2a"" = (6;4) b = (-l;5) suy ra k = ( 6 - l ; 4 +5) = (5;9); -a =(-3;-2), 2b =(-2 ;10) va 5c =( -10; -25) suy ra U ( - 3 - 2 - 1 0 ; - 2 + 10- 25) = (-15;-17) Vidu 2: Cho a = (1;2), b = (-3;4); c = (-1;3) Tim toa dp ciia vecto u biet 1 X = — 3 ^=2 ^1 3> ^3'2, 3 BAI T A P LUYfiN T A P Bai 1.77.Cho cac vecto a = (2;0),... Cho 4 diem A, B, C, D; I la trung diem AB va J thuoc CD thoa man A D + BC = 2 f i Chung minh J la trung diem ciia CD ^^ ^^ Huang dan gidi A D + BC = 2ij giac A B C D Gia sit ton tai d i e m O sao cho O A = OB = OC = O D m'a + n ' b + p ' c = 0 va O A + OB + O C + O... o t k h o n g cung p h u o n g va thoa m a n d i e u kien: + a-b + , dau bSng xay ra k h i a, b cijng h u o n g , dau bang xay ra k h i a, b nguoc h u o n g 47 fnan loai va phuarng phap giai Hinh hoc 10 • Dua bai toan ban dau ve bai toan tim cue tri cua M I voi M thay doi Huang dan giai + Neu M la diem thay doi tren duang thang A khi do M I dat gia tri nho po a + P + y O nen ton tai duy nhat diem I... 34 triing voi G' la trong tam AA'B'C Cdch 2: Goi G la t r o n g tam AABC (tuc ta c6 G A + GB + GC = 0 ) ta d i chung m i n h G A ' + GB' + G C ' = 0 PJiaii loiii vc'i phutriig phiip ^it'ii IIluli hoc 10 2 CAC VI Dg * Vt du 1: Chung minh rang AB = CD khi va chi khi trung diem cua hai doan thang AD va BC trimg nhau Goi I, J Ian lugt la trung diem cua AD va BC suy ra A I = ID, CJ = JB Do do AB = CD »... (1) va (2) suy ra DE = - D I Vay ba diem D, E, I thJing hang 3 Hinh 1.35 r 3 BAI T A P L U Y ^ N T A P Vi du 4: Hai diem M, N chuyen dpng tren hai do^n th^ng co'dinh BC va BD (M;:tB, N^B)saocho 2 +3 = 10 ^ BM BN Chung minh rang duong thang MN luon di qua mot diem codjnh Lcri gidi De thay luon ton tai diem I thupc MN sao cho 2 — I M + 3 — I N = 0 ( l ) BM BN ^ ' GQi H la diem thoa man 2HC + 3HD = 6... CC, song song voi nhau Nen ba vecto AA,,BBi,CC,^ co cimg phuong Do do hai vecto HjHj va HjHg cung phuong hay ba diem Hi,H2,H3 thang hang 3JA + 2JC = 0 o 3rA + 2ic = Sij Suy ra 2(1 A + IB + IC) = SIJ . "Phan loai va phuong phap giai Hinh hoc 10& quot; la mot trong nhi>ng cuon thuoc bo sach "Phan loai va phuang phap giai lop 10, 11,12 ", do nhom tac gia chuyen toan. : PEIANLQAI & FHlTdNG PHAP GIAI (Tai Ban, Sufa ChOa Va Bo Sung) Danh cho hoc sinh Idp 10 on tap va nang cao kien thiJc Bien soan theo npi dung sach giao khoa cua bp GD&DT mm,. PHANLOAI & PHCONG PEIAP GlAl mmm WHIM (Tai Ban, SCfa ChOa Va Bo Sung) • Danh cho hoc sinh Idp 10 on tap va nang cao kien thufc • Bien soan theo noi dung sach giao khoa cua bp GD&DT AT