--..., -..,,,,,,,,,, VU HlfU BINH (CHU BIEN) - f>am HIEU CHIEN ' IJEU ( BUYl1N 'l 1 0AN 'fllung HOC Cff Sff TAP. HAI. - HINH HOC. (Tdi bdn ldn

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1 --..., -..,,,,,,,,,, VU HlfU BINH (CHU BIEN) - f>am HIEU CHIEN ' IJEU ( BUYl1N 'l 1 0AN 'fllung HOC Cff Sff TAP. HAI. - HINH HOC. (Tdi bdn ldn tlui hai) NHA XUAT BAN GIA.O D()C VItT NAM

2 {[[W+bz0FkV43GmRt7u4DpvuYxd]]} PHAN CONG BIEN SOAN : Oam H1eu Chien : Cac chuyen de tct 1 den 7. VO HG'u Binh : Cac chuyen de 8, 9. Anh minh ho bia m(>t : Nha toan h9c c6 Hy Lc;1p Euclid {khoang trve1c CN) 2

3 l.!0 ndi d.iu ' 89 sach Tai li u chuyen Toan Trung ht;,c CO' sit gom 8 cuon, ta lop 6 den lop 9, moi lop 2 t p (t p m(>t: Of;li so, t p hai: Hinh h9c),. dli<;1c bien SO,;ln nham : - Cung cap cha cac em hqc sinh mc;>t tai li u c6 h thong va, kha day du di3 h9c t p chuyen sau ve mon Toan d cap Trung hqc Cd sc i. - Jf;lo nguon cho cac lop chuyen Toan.d Trng hqc ph6 thong va giup cac em hqc sinh lop 9 thi vaol(jp 10 chuyen Toan. - Giup cac can b(> chi d o mon Toan, cac thay co g1ao d;,:iy Toan c6 them tai li U d'3 chi di;!o va gilmg d;,:iy chuyen sau ve Toan tt;1i cac hot;1t d(>ng chuyen de, ngo;,:ii kh6a, cac cau, l1;1c be;> Toan hqc nham boi dl.tang cha hqc Sinh Ve kien th(tc, ki nang, tli duy Va nhung ph m Chat tot trong hqc t p, nghien CCfu.. va sang tf;jo, g6p phan vao Vi i;;. (fao. tf;)c va boi dllang nhan tai. Nh6m tac gia cua b9 sach la nhung thay co giao da va dang giang d1;1y d cac l(jp chuyen Toan cua cac trliong Ot;1i h9c, cac trllong Trung h9c cd sd c6 uy tfn, nhung chuyen gia Toan c6 kinh nghi m d T1;1p chi Toan h9c & tu5i tre va Nha xuat ban Giao dl,jc Vi tnam. Cuon sach Tai li u. chuyen toan THCS Toan 6, t'p hai: Hinh h9c gom 7 chuyen de Cd ban va 2 chuyen de nang cao. O moi chuyen de Cd ban, CUOn sach khong l p l;,:ii CaC n(>i dung da c6 trorig Sach giao khoa ma chi t6m t t cac kien thctc Cd ban, dong thoi b6 sung them m(>t so kien thctc can thiet Do d6, khi giang d y cac chuyen de, cac thay CO giao nen SU' dl,jng ket h<;1p.tal lif}u nay VOi Sach giao khoa. Cac vi dl,l minh h9a dl.t<;1c ch9n l9c nham giup h9c sinh biet each giai va trinh bay m(>t so d;,:ing toan nang cao thvong g p. Sau d6 la cac bai t p c6 hll(jng dan giai. 3

4 Cac chuyen de nang cao nham giup b9n d9c hi u sau va r9ng hon nho,ng n9i dung c6 lien quan den chu'dng trlnh, chung dt1<;1c trlnh bay d h9c sinh CO th tiep thu dl1(;1c. Chung toi hi _v9ng cuon sach nay se la m9t tai li u thiet thl/c va b6 fch giup cac em.hi u sau sac nhung kien thctc Toan da h9c, g6p phan vao vi c nang cao chat IU'c;mg dao t9o va boi dtjbng h9c sinh gioi Toan J cap Trung h9c Cd sci M c du cac tac gia da lam vi c nhi t tlnh, nghiem tuc va c6 trach nhi m v6i b9n d9c, tuy nhien, cuon sach van kh6 tranti khoi nhung thieu s6t. Chung toi rat mong nh{m du'c;lc y kien dong g6p cua b9n d9c. Cac g6p y xin gui ve Ban Toan - Tin, C6ng ty GP Dich Vt,J Xuat ban Giao dt,jc Ha N9i, Nha xuat bari Giao dt,jc Vi t Nam, 1878 Giang Vo, Ha N9i. Chu bien VU HUU BINH 4

5 {[[W+bz0FkV43GmRt7u4DpvuYxd]]} OO NTHANG Chuyen al 1. BI M. TIA D0-6.N THANG D11ng 1. E>IEM VA E>UCJNG THANG. I. KIEN THUC co BAN 1. Diem, ducmg thfing va m t phfing la ba hinh hlnh h9c kh6ng duqc dtnh nghia. Diem di.rqc d t ten bfuig chii' cai in hoa nhu A, B,..., M,... Ducmg thing duqc d t ten bang chii' cai in thucmg nhu a, b,..., m,... (h. l). 2. Nh n xet 1 (ve vt trfcua diem va ducmg thfing). Y ai mqt ducmg thfing.bit ki, c6 v6 s6 diem thuqc ducmg thfing do va c6 v6 s6 diem kh6ng thuqc ducmg thfing d6 (h.1). m A B Hinh 1 Diem A thuqc ducmg thfing m' duqc ki hi u la A e m. Tucrng 11! c6 B e mva Cem. Diem M kh6ng thuqc ducmg thing m duqc ki hi u la M t m. 3. Nh n xet 2 (ve Sl;f xac dinh ducmg thing). C6 mqt va chi mqt ducmg thing di qua hai diem phan bi t da cho. Khi m()t duang thing di qua hai diem A va B, ta co duang thing AB (ho c ducmg thing BA). Khi ba diem A, B va c cung thuqc mqt ducmg thing thl chung thdng hang. Ne'u ba diem A, B va M kh6ng cimg thuqc bit cu m()t dtrong thing nao thi chung khongthdng hang.(h.l). c 5

6 + Kien thuc bt> sung 4. Hai duoog thing phan bi t ho c c6 m(>t diem chung, ho c khong c6 diem chung nao. Th t v y. neu hai duoog thing c6 t6i 2 diem chung thi theo nhr;;,n xet 2, chung phai trimg nhau. V y hai duoog thing phan bi t c6 m(>t diem chung ho c khong c6 diem chung nao. 5. V 6i hai duoog thing bat kl thi gifra chung ho c c6 1 diem chung (hai duoog thing cat nhau), ho c khong c6 diem chung nao (hai duoog thing song song), ho c c6 VO s6 diem chung (hai duoog thing trimg nhau). II. VI Dl) MINH HO Vi d1;1 l.l. Ve 5 diem A, B, C, M va N trong d6 ba diem A, B, C thing hang, ba diem A, B,.t\1 khong thing hang v.a ba diem A, B, N thing hang. a) Giai thfch vi sao ve duqc nhu v y. b) Chung to b6n diem A, B, C va N cung thu(>c m()t duoog thing. (Gia sir d6 la duoog thing d). c) Trong cac each viet sau, each viet nao dung, each viet nao sai A E d ; B d ; M E d ; N d. d) Hai duoog thing AN va BC c6 la hai duoog thing phan bi t khong? Hai duoog thing AB va MN c6 la hai duoog thing trung nhau khong? ) C6 bao nhieu duoog thing di qua timg c p hai diem trong s6 5 diem da cho? Gidi. (h.2) d Hinh2 a) Ch9n hai diem Ava B. Ve duqc nhu v y. vi theo nh n xet 2, c6 m(>t va chi m<)t duoog thing di qua 2 diem A, B. Ta co duoog thing AB. Theo nh n xet 1, co diem C thu c duoog thing AB, c6 diem M khong thu c duoog thing AB va c6 diem N thu c duoog thing AB. b) Vi cac diem C va N cung thu(>c duoog thing AB nen b6n diem A, B, C va N thing hang. 6 c) Cach viet dung : A E d. Cach viet sai : B d ; M E d ; N d. Si:ra l<;ti cho dung : B E d ; M d ; N E d.

7 d) Biet bon diem A, B, C va N cung thuc)c mc)t ducmg thing (cau b) nen hai ducmg thimg AN va BC la m t, do d6 chung trimg nhau va khong la hai ducmg thing phan bi t. Hai ducmg thing AB va MN chi c6 m t diem chung la N nen hai ducmg thing AB va MN la hai ducmg thing phan bi t va khong trung nhau. e) C6 ta'.t ca nam ducmg thing AB, MA, MB, MC, MN (ducmg thing AB d i di n cho cac ducmg thing c6 ten g9i khac nhau nhung v!n la m t: AB, AC, AN, BC, BN va CN). Vi d1: a) Cho 4 diem phan bi t. Cu qua hai diem ta ve duqc mqt ducmg thing. Hoi c6 ta'.t ca bao nhieu ducmg thing? b) Qua 5 diem ve duqc nhi u nha'.t bao nhieu ducmg thing? Gidi. a) So ducmg thing c6 the ve duqc la 1, 4 va 6 (h.3). Trucmg hqp 4 diem thing hang, ta ve duqc 1 ducmg thing (h.3a). Trucmg hqp chi c6 3 diem thing hang, ta ve duqc 4 ducmg thing (h.3b). Truang h<jp kh6ng c6 ba diem nao thing hang, ta ve duqc 6 duong thing (h.3c ). a) b) c) Hinh3 b), De c6 the ve duqc so ducmg thing nhieu nha'.t thl trong 5 diem da cho kh6ng duqc c6 3 diem nao thing hang. Cach thu nhat. Do kh6ng c6 ba diem nao thing hang nen Tu diem thu nha'.t c6 the ve duqc 4 ducmg thing vm 4 diem con l i. V ai diem th(i' hai CO the ve du(jc 3 duc111g thang Vffi 3 diem con l i (kh6ng tfnh duoog thing ve vm diem thu nha'.t). V m diem thu ba c6 the ve duqc 2 ducmg thing vm 2 diem con l i (kh6ng tfnh cac ducmg thing ve vm cac diem thu nha'.t va thu hai). 7

8 V m diem thu tu c6 tm ve duq'c 1 duang thing vm 1 diem con l i (khong tfnh cac duang thing ve vm cac diem thu nhat, thu hai va thu ba). C6 tat ca: = 10 (duang thing). Cach thu hai. Do khong c6 ba diem nao thing hang nen: Vm m6i diem c6 the ve duq'c 4 duang thing vm 4 diem con l i. C6 5 diem nen ve. duq'c 4.5 = 20 (duang thing). Nhung neu v y m6i duang thing duqc tfnh 2 Mn (nhu AB va BA chi la m<)t duang thing ma h;ti duq'c tinh lam hai). 4 5 V y co ' Hit, ca : = 10 ( d ucmg ' t h' rmg ). 2 Truang h<!fj ttfng quat. Qua n diem trong d6 khong c6 ba diem nao thing hang, ta ve duqc : (n - l).n duang thfing (phan chi tiet xem Vf dl;l 1.15). 2,,:, -,..? o,,ng 2. DIEM NAM GIUA HAI DIEM I. KIEN THU'C CO BA.N 1. Dau hitu 1 (Dau hi u DI - Ve thu tl:{ cua ba djem tren m<)t duang thing). V m ba diem thing hang, c6 m<)t va chi m<)t diem nam gifra hai diem con 1<;1.i. + Kii!n thuc b6 sung 2. Dau hitu 1 a (Dau hi u D 1 a). V m ba diem A, B, C thing hang ma diem B khong nam giua hai diem A va C, diem C khong nam giita hai diem A va B, thi diem A phai nam giiia hai diem B va C. 8 Th t v y. vi ba diem A, B, C cung thu(>c m(>t duang thing nen chung thing hang. Ta bie't trong 3 diem thing hang c6 m<)t va chi m<)t diem nam gifi'a 2 diem con l i (dau hi u DI), ma diem B khong nam gifi'a 2 diem A, C va diem C khong nam gifi'a 2 diem A, B. V y diem A phai nam gifra 2 diem B vi C. Dau hi u 1 a c6 the phat bieu m(>t each khac duai day (h.4 ).

9 M N -, Dau hi u lb (Dau hi u Dlb). Vm ba diem A, B, d C thing hang ma 2 diem A, B nam cung phfa d6i B A C vm diem C va 2 diem A, C nam cung phfa d6i vm ' Hinh 4 diem B, thl diem A nam giua 2 diem B va C. 4. Dau hi u 2 (Dau hi u D2). Nc!u diem A naffl giii'a hai di in B, C ma diem M nam giii'a hai diem A, B va diem N nam giii'a hai diem A; C thi diem A nam giii'a hai dieth M va N. Th<)t v<)y, bic!t diem M nam giii'a hai diem Ava B nen hai di'in M va B nam cimg phfa d6i vm diem A. Bic!t diem N nam giua hai diem A va C rien hai diem N va C nam cung phfa d6i vm diem A.. Li[li c6 diem A nam giii'a hai diem B va C nen hai diem B va C nam khac phfa d6i vm diem A. Suy ra hai diem M va N nam khac phfa d6i vm diem A hay diem A nam giii'a hai diem M va N. II. yi D{J MINH HO Vi d1; Cho 5 diem A, B, C, E va F theo thu ti! cilng thuoc mot ducmg thang d-(h.5). a) Hoi diem C nam giii'a hai diem nao? b) Hoi diem B nam giii'a hai diem nao? c) Hoi diem E khong nam giua hai diem nao? Gidi. a) Ditm C nam giii'a hai diem : Ava E ; A va F ; B va E ; B va F. b) Diem B nam giii'a hai diem: Ava C; Ava E; Ava F. c) Diem E khong nam giii'a hai diem: Ava B; Ava C; :S va C. L,ru j. Ch9n theo thu ti! tir diem ddu la A, r6i lrut luqt dc!n cac diem con li(li. Tir nay, khi n6i cho cac diem A, B; C, D va E theo thlt ti! cung thuoc mot ducmg thang thi ta c6 cac kc!t qua nhu tren. Vi du 1.4. Cho 5 diem A, B, C, Eva F. Bic!t ba diem A, B va C thing hang, ba diem B, C va E thing hang, ba ditm C, E va F thing hang. Giai thfch vi sao nam diem A, B, C, E va F thing hang? d. A B C E F Hinh 5 9

10 Gidi. Ta biet qua 2 diem ve duq'c m(>t va chi m(>t duang thing, ma ba diem A, B va C thing hang, nen ba diem A, B va C cung thu c m(>t duang thing d ching h<;1n. Luc nay duang thing d d<;1i di n cho cac duang thing AB, AC, BC (h.5). Ba diem B, C va E thing hang nen diem E thu(>c duang thing BC, suy ra diem E E d. Ba diem C, E va F thing hang nen diem F thu c duang thing CE, suy ra diem F E d. V y nam diem A, B, C, Eva F thu(>c d nen chung thing hang. D11ng 3. TIA. HAI TIA 801 NHAU I. KIEN THU'C CO BAN 1. Hinh g6m diem O va m(>t phlln duang thing bi chia ra bai diem O duq'c g9i la m(>t tia g6c O (h.6). x A 0 y Tia Ox con g9i la m t ni'ra duang thing g6c 0. Hinh 6 Tren hinh 6, tia Ox con duq'c gqi la tia OA. Tia Ox khong bi gi6i h<;lll ve phfa x, tia Oy khong bj gi6i h<;1n ve phfa y. 2. Hai tia chung g6c Ox va Oy t<;10 thanh duang thing xy duqc gqi la hai tia d6i nhau. Moi diim tren duung thdng la goc chung cua hai tia dot nhau. 3. Hai tia OA va Ox la hai tia trung nhau, cfing v y hai tia AO va Ay trung nhau. + Kien th(rc bo sung 4. Cho hai tia chung g6c Ox va Oy, ta c6 : ho c d6 la hai tia d6i nhau, ho c la hai tia trung nhau, ho c la hai tia khong d6i nhau, khong trung nhau. 5. Dau hi u 3 (Da'.u hi u D3). Ve thu tlj cua ba diem tren m(>t duang thing. Cho ba diem A, B, C thing hang, neu hai tia AB va AC d6i nhau thi diem A n!m gifra hai diem B va C. 6. Dau hi u (Ve S\J xac dinh tia). Neu diem A n m gifra hai diem B va C thl hai tia AB va AC d6i nhau, hai tia BA va BC trung nhau, hai tia CA va CB trung nhau. II. Vi DlJ MINH HOA.. Vi d1;1 l.s. Cho diem O thu c duang thing xx'. La'.y hai diem A, B thu(>c tia Ox sao cho diem A nam gifra hai diem O va B (h.7). Tren h'inh ve c6 bao nhieu tia? C6 bao nhieu c p tia d6i nhau? IO