V6 ANH DUNG (TONG CHU BIEN ) - TRAN f)UC HUYEN ( CHU BIEN)

NGUYEN DUY HIEU- NGUYEN THANH TU.l\N-NGUYEN LE THUY HOA-NGUYEN VAN MINH

( TRUdNG TRUNG HQC PH6 THONG CHUYEN LE H6NG PHONG TP.H6 CH{ MINH)

? ..1'

GIAI TOAN ' HINH HOC

•

(DUNG CHO HQC SINH LC5P CHUYEN)

r l u i hon !£in rlui·nitm)

• ' ,.., ? , ,..

NHA XUAT BAN GIAO Dl)C Vll;T NAM

    

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3

Lor Nor £JA.u

~ rong thCii gian vua qua, ·duc;tc Slf giup dB cua Nha xuat

ban Ciao dt,tc, TntCing Trung h9c ph6 thong chuyen Le

H6ng Phong da bien soan b(> sach "Giifi toan dcmh cho hQc sinh lap

chuyen" theo dtnh huang bam sat sach giao khoa, b6 sung cac chu

d~ nang cao theo trinh do trudng chuyen va cac n(>i dung thi di;li

h<?C . B(> sach da duc;tc dong dao h9C sinh va giao vien cac truCing

chuyen sU' dt,tng va tin c~y.

Trong qua trmh d6i mai giao dt,tc, dap ung yeu du mai cua sach

giao khoa chuyen ban, xay dt;tng phuong phap ki~m tra ket hc;ip giUa

tt;i lu~n Va tri\c nghi~m khach quan, Chung toi bien SOC;ln li;li b(> sach

Giai toan danh cho h9C sinh cac truCing chuyen va h9C sinh kha, gioi

Ci cac truCing trung h9c ph6 thong tren toim quoc. B(> sach "Gidi torin

lap 10" g6m ba quy~n :

- Giai toan D~i s6 10 ;

- Giai toan Hinh hQc 10 ;

- Giai toan Luqng giac 10.

N(>i dung quy~n "Gidi toan Hlnh h9c 10" bam sat theo cau true

cua sach giao khoa Hmh h9c 10 (Nang cao) va duc;tc trlnh bay theo

ba chuang nhu sau :

- Clutdug 1 : Vecto ;

- Clutdng 2 : Tich vo huang va ung dt,tng ;

- Chudng 3: Phuong phap tol;l d(> trong m~t ph~ng .

Trong m6i bai h9c, chung toi xay dt;tng h~ thong bai t~p rEm

luy~n dl,l'a theo cac van d~ Cl,l th~. Den ph§.n on t~p cuoi chuang,

4

chung toi gi6'i thi~u cac cau hoi tdc nghi~m khach quan va cac cau

hoi ttt lu~n v6'i dQ kh6 cao, yeu du b;;tn d9c phai t6ng hqp l;;ti cac

kie'n thuc da dttqc cung cap. Chung toi c6 cung cap dap an va

hti6'ng d~n gicli SCf lttqc cua ffiQt s6 bai 't~p tieu bieu. cu6i moi

chttefng nh~m giup cac b;;tn d9c on t~p, nang cao kie'n thuc, ren

luy~n ca.c ki nang giai toan, d~c bi~t la phftn tr~c nghi~m khach

quan.

Hi v9ng quyen sach .se giup fch cho cac b~n h9c sinh trong qua trlnh h9c t~p, ren luy~n nang cao b(> mon Toan 16'p 10 ; la tai li~u

ho trq cho giao vien Toan cac trttong trung h9c ph6 thong trong

cong tac dao t;;to h9c sinh gioi.

M9i y kie'n dong gop xin dttqc gui v~ dia chi sau :

• TniO'ng Trung hQc phd' thong chuyen Le Hdng Phong,

235 Nguy~n Van Cit, Qu~n 5, Thanh pho' H6 Chi Minh.

• Ban bien t~p Toan - Tin hQc, Nha xuat ban Giao dt,tc t~i

Thanh pho' H6 Chi Minh, 231 Nguy~n Van Cit, Qu~n 5,

TP. HCM.

Tran tr9ng cam Cfn !

Thanh ph6 H6 Chi Minh, thang 07/2007 cAc TAC CIA

    

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5

~I~CTd

§1. CAC D'NH NGHTA

A. TOM TAT GIAO KHOA

I. KHAI NI~M VECTd

Vecta la IIIQt doq.n tluing c6 huang, nghla la trong hai die'm mut cua do~n th<ing da chi ro diem nao la diem dau, die'm nao la diem cuoi .

Ki hi~u AB chi vecto c6 :

+ Goc.(cliem d&uJ laA;

+ Ng9n (cli e'11z cuo'i) laB.

DuCing thcing AB la gia cua vecto AB.

D9 dai do~n thcing AB la d¢ dai cua vecto AB , ki hi~u /AB/.

Nhu v~y : /AB/ = AB .

Chieu di tu· goc A den ng<;m B la hrtang cua vecto AB.

D~c bi~t : Vecto c6 d9 dai bilng 1 g<;>i la vecto don vi. --+ - > --+ - >

B

A

Hinh 1.1

Luu y : Vecto con dli<;>'C ki hi~u la : a ; b ; c ; X ..... ne'u khong can chi

ro diem dau va die'm cuoi cua n6.

II. VECTd CUNG PHUONG

Hai vecto dli<;>'C g<;>i la Cltng plutang neu gia cua chung song song hay tning nhau .

Hai vecto cung phuong c6 the ding !utang hay ngrtr;~c !utang.

6 Chuang 1. VECTO ------·------------

B. - -• AB va CD ctmg

piHi\Jng t:LIIlg hu'(ing

- -• AB vit MN CLill g

A

Hinh 1.2 N

Nh~n xet : I Ba die'm A, B , c th.:'tng himg <=-> AB v;\ AC cung !JhLIO'ng.

Ill. VECTd BANG NHAU

) )

Hai vectO' a va 6 lHlng nl/011 n[;u chung C/lll ,!j /ut'ci'ng va cting d(J deli.

- >

Ki hi~u : a = b .

Cac tfnh chat :

i) AB == AB ;

ii J AB == CD => CD = AB

iii) AB = CD va CD = EF => AB = EF.

IV. VECTd 0 - t

Vecto· 0 la vectO' c6 goc va ngQn trung nhau. Ki hi¢u : 0 . ...

AB = 0 <=> A = B ; AA == BB = ... = 0 .

- )

VectO' 0 c6 c19 clai bAng 0 va c6 phu'O'ng bu't kl (cung phu·O'ng, cung hu'lmg

v6'i mQi vcdO'I.

§1 . CAC E>!NH NGHTA 7

v. xAc D!NH MOT DIEM BANG DANG THU'c vECTd

---+ Cho di~m 0 co d!nh va vecta v kh6ng d6i . T6n t~i duy nhat m(>t di~m

~ ---+

M sao cho : OM = v . (1)

Ta n6i : di~m M duqc xac d!nh bai diing thuc (1).

B. PHUONG PHAP GIAI TOAN

1) vi ov VI d1,1 1 : Cho hai diem phan bi$t Ava B. Hoi c6 bao nhieu doc;tn th~ng va

bao nhieu vecto kh'k nhau va khac vecto 6? Giai

Mi?t do~n thiing duy nhat AB ho~c BA ;

Hai vecta khac nhau va khac vecta 0 la AB va BA .

VI dl,l 2 : Cho ~ABC can tc;ti A. Gqi MIa trung diem cua BC va N Ia trung

diem cua AB. A

a) 8~ng thuc AB = AC dung hay sai ?

b) Cac vecto nao cung huang vai AC ?

c) Cac vecto nao ngu<;1c huang vai BC ?

d) Cac vecto nao bang nhau ? M

Giai Hinh 1.3

a) Hai vecta AB va AC khong cung phuang n€m chung kh6ng biing nhau.

b) MN la do~n noi trung di~m cua hai c~nh·BC va AB nen MN va AB ~ ~

song song nhau. Do d6 NM va AC la hai vecta cung huang.

c) Ba diem B, C, M thcing hang nen cac vecta nguqc huang vai BC la :

CB; CM; MB.

d) Ta c6 : BM = MC v'i hai vecta nay cung huang va cung di? dai .

Tuang W ta cling c6 : CM = MB ; BN = NA va AN = NB .

8 Chu·ong 1. VECTO

Vf dl:J 3 : Cho tU giac ABCD. Chung minh rang :

ABCD Ia hlnh blnh hanh ~ AB = DC.

Giai Ar---------...8

( =>) ABCD la hl.nh bl.nh hanh, chung minh : ~-

AB =DC.

Ta c6 : ABCD la hl.nh bl.nh hanh nen D<-----c

Hinh 1.4 ~ -AB cung hu6'ng DC va AB = CD => AB =DC .

( <=) AB =DC , chung minh : ABCD la hl.nh bl.nh hanh. ~- ~-

Ta c6 : AB =DC nen AB , DC cung phuong va AB = CD.

Do d6 AB II CD va AB =CD => tU giac ABCD la hl.nh blnh hanh.

Vf dl:l 4 : Cho hlnh blnh hanh ABCD. Goi M va N lan IU'Qt Ia trung diem cua

AB va DC. AN va CM I an IU'Qt cat BD tc;1i Eva F.

Chung minh: DE = EF = FB.

Giai

Ta c6 AM = NC nen AMCN la hinh bl.nh hanh, do d6 AN II MC. Suy ra MF la duO'ng trung blnh cua ~ AEB va NE la duO'ng trung binh cua ~ DFC. Tu d6 ta c6 F la trung di~m cua EB va E la trung diem cua - ~-

DF. V~y DE = EF :::: FB .

2)BAIT~P

&11" D N C

Hinh 1.5

Bai 1. G9i C la trung diem eli a do;;tn th§.ng AB . Cac kh§.ng dinh sau day dung hay sai ?

a) AC va BC cung hu6'ng ; ~ ~

c) AB va BC nguqc hu6'ng ;

- ~

b) AC va AB cung huO'ng ;

d) IABI = !Bel ; n IABI = 21Bc1.

§2. TONG VA HI~U CUA HAl VECTCi

Bai 2. Cho lye giac deu ABCDEF co tam 0.

a) Tim cac vecto khac 0 va cung phuong v6'i OA.

h) Tim cac vecto bang vecto AB .

c) ILly ve d.c vecto bang vccto AB va c6 diem dau la 0, D, C.

9

Bai 3. Cho 5 diem A, B, C, D, E phfm bi~t. C6 bao nhieu vecto khac vecto­khong, c6 <licm <'hl.u va die'm cuoi la de oiem da cho.

Bai 4. Cho tam giac ABC c6 tnfe tam H va tam ctu·&ng trim ngoc_1i tiep la 0 .

G9i B' la oiem doi XLt'ng cua B qua 0. Chting minh: AI-l = B'C.

Bai 5. Cho W giac ABCD. G9i M, N, P, Q lan luqt la trung diem cua de -- --

cc_1nh .A.B, BC, CD, DA. ChLt'ng minh : NP = MQ va PQ = NM.

Bai 6. Cho hinh binh hanh ABCD. Dyng AM = BA, MN = DA, NP = DC,

PQ = BC. ChLt'ng minh : AQ = 0.

Bai 7. Cho lye giac d~u ABCDEF. H<:'ly ve cae vecto ui'mg vecto AB thoa :

a) Co diem tbu la 13, F, C.

b) C6 diem cuoi la F, D, C.

~ ' ~ ?

§2. TONG VA Hlf;U CUA HAl VECTO

A. T()M TAT GIAO KHOA

I. TONG CUA HAl VECTd

- ) •· )

Cho hai vecto u va b . Lay m(lt die'm A bat kl, ta ve : -~ - )

AB = (/ ' tiep thco ve BC =· b . - ) - ) .. )

Vecto c :-:: AC duqc xac d!nh nhu tren duqc g9i la tO'ng cua a va b .

. ) · ·~) - )

Ki hi~u : u + b = c .

10 Chuang 1. VECTO ----------~ ----

B

A

- )

b

c

o )

Hinh 1.6

Nhti vf•.Y v6'i 3 dic'm A, B, C tuy y, ta c6 : iTB + BC = ACI -> . > - >

Cac tinh cha't : VO'i ba vectO" bat kl a . b , c , ta c6 :

)> - ) } - )

i) u + b = b + u ; (tinh giao hoanJ

~) -> - > ·> -·)

iiJ ( u + b)+ c = a + ( b +c); (tinh kc't hqp)

- )

iii) Cl + 0 = (/ ; (b;}L c!<\ng thuc tam gi<i.c)

Quy tiic 3 die'm

TCt dinh nghia cua phep c<)ng, ta suy ra : v6'i ba die'm bat kl A, B , C

ta c6: iAB + BC = i\C;I hay iAn + BC + C./\. = ol. Quy uic nay c6 the' to'ng quat cho n diem.

Quy tc'ic hinh binh hi:mh

'l'r~ ng hlnh binh hanh ABCD : [AB + AD = Acl <=> BA + DA = CA.

II. HI~U CUA HAl VECTd

> - > Cho vectO" (/ . VectO" c6 CIJI/g d(; drii va ng l/r/C lu/6'1/g v6'i (/ ctu·o·c gQi L't

-> > uccto cl6'i cua vectO" (/ . Ki hi~ u : ~ ([ .

- ) ~ > - > - > - > N6i m<)t cc.ich khac : "Neu a + b = 0 thl. ta n6i u la vectO" doi cua b hay 0 ~

b la vectO" doi cua Cl ".

§2. TC)NG VA HIEU CUA HAl VECTO 11

Cri.c tinh chat

i ) A, l3 : A TJ = - BJ\

ii) I Ll. trung c1ie"m AB ~ !A c:: - I U

iii 1 -- ( - .-\HJ - AB.

> > >

Cho h:ti \ ·cL·td o v~t b. Hi¢u dt:t u '.d {;, i-;i hi~·u u - 6 du"<.ic L1itth llgllla h('li :

l~:~~-=-; + ~ ?~ . ) - ) - ) ) . ) )

Suy ra : u - b = c ~ o = c + b .

Phep t.rt:t vectu c6 cac tinh cht!t nhu doi vlii phep tru c:..i. c so tht,Ic .

. > >

Clni y : Phcp toan tim tOng va h i¢u cua bai wcto u va b con gQi Ut >

phc p c(lng v<1 phep tru hai n :! cto" u Yit b .

Quy t:ic 3 diem :

V6i ba di c"m btlt ki /\, B, C, tad> :

~s~-~c; ~:_cji ) _

B. PHUONG PHAP Cl/\.1 TOAN

l van de· 1. CHUNG MINH MOT nANG THUC VECTO L______ ------------1) PHU'dNG PHAP

Ta co thi' s u· ch,mg m(lt trong ba phu·ung ph c:'tp- sau cL1.y :

• I3i e'n cloi vc nay ve ve' kia .

• Bie n doi Luong du·ong d<'ing thtk ccl.n chC!ng minh thanh m(lt drt ng thC!c ma ta cUt bie t lil dung .

• Bien doi m<)t dAng thC!c dung c6 stl n thimh drtng thCt<.: can chC!ng minh.

Lu'u y : Ne n ap dyng cac quy t<ic : ba di em, trung ai<Sm, hinh binh hanh

trong quci trinh bie n doi.

12 Chuong 1. VECTd

2) vf ov

Vf dl,l 1 : ChCing minh rang v6i bon diem A, B, C, 0 bat k1 taco :

AC + BD = AD + BC .

Giiii

Ta c6: --~ -~-- -- - --AC + BD =-= AD + DC + BC + CD = AD + BC + (DC + CDJ = AD + BC.

Vf dl,l 2 : Cho h1nh b1nh himh ABCD voi tam 0 . Cac kh~ng d!nh sau day

dung hay sai ?

b) AB + BD = BC;

-- - --c) OA + 013 = OC + OD; d) BD + AC = AD+ BC .

Giiii

a) Taco : AB + AD = AC (quy tEi.c hl.nh binh h<1nh)

I~_, i~l

= AB + ADI = lAC: = AC .

Vi IBDI = BD => m~nh eM sai.

b) Taco : AB + BD = AD (quy ti'ic ba diem). Hinh 1.7

Vi IJC = AD => m~nh d~ dung.

c) Ta c6 : OA + 013 = OC + OD <=> Oi\ - OD = OC - OB ~ DA = BC

=> m¢nh de sai.

d) Ta c6 : BD + AC = AD + BC <:::> BD - BC = AD - AC <:::> CD = CD

=> m~nh de dung.

Vf dl,l 3 : Cho tam giac ABC. Dl,ing cac h1nh

b1nh hanh ABIJ, BOPQ, CARS nam phia

ngoai tam giac do. Chu·ng minh rang:

- ~- ~ RJ + 10 + PS = 0 .

Q

Hin/J 1.8

s

    

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§2 . TONG VA Hlt;U CUA HAl VECTO 13

Giai

Ta c6 : ABIJ la hinh binh hanh n E'm : AJ = BI

BCPQ la hinh binh hanh nen : BQ = CP ;

~ --CARS la hinh binh hanh nen : CS = AR .

Ta l~i c6 :

RJ + IQ + PS = AJ - AR + BQ - BI + CS - Cf

~~----~- ->

= AJ - BI + BQ - CP + CS - AR = 0 .

VI dl:J 4 : Cho tam giac ABC va G Ia trqng tam tam giac. Chung minh ding : ~ ~ ~ -

GA + GB + GC = 0. •

A

Giai

Ve hlnh binh hanh BGCD, M la trung di~m BC.

Theo quy t iic hinh binh hanh ta c6 : ~ ~-

GB + GC = GD , rna GD = 2GM (tinh

chat duO'ng cheo), GA = 2GM (tinh

s ~------\'-'.:....__----=~c cha t tr9ng t am)

~ GD = GA ~ GA = - GD . 0 -- ~-

Hinh 1.9 Vf}y GA + GB + GC = 0.

3) BAI TAP

B~li 1. Cho hinh binh h anh ABCD voi t am 0. Hay dien vao ch6 trong (. .. ) dC' duqc diing thU'c dung. --

a) AB + AD = .. . ; b ) AB +CD = ... - ~

~ ~

c) AB + OA = ... ; d) OA + OC = . .. ~ ~ ~

e) OA + OB + OC + OD = .. . .

14 Chuang 1. VECTO

Bai 2. Cho hinh binh hanh ABCD v6"i tam 0. Cac khting dinh du·o-i day dung hay sai ?

a) OA - OB = AB; bl CO-OB = BA; ~ -

c) AB - AD = AC; dl AB- AD= BD ; -- ~ ~

e) CD - CO = BD - BO .

Bai 3. Chung minh ding neu AB = CD thi AC = BD .

Bai 4. Cho hinh blnh h~mh ABCD va m9t die'm M tuy y. ~- ~-

Chu·ng minh di.ng : MA .,.. MC = AlB + MD.

Bai 5. Cho hlnh binh hanh ABCD tam 0. Chu·ng minh di.ng : --

a) CO - OB = BA ; b l AB - BC = DB ; - - -

c) DA - DB = OD - OC d) DA - DB + DC = 0 .

Bai 6. Cho hinh binh hanh ABCD. Chu·ng minh ding : DA - DB + DC = 0.

Bai 7. Cho 6 die'm A , B, C , D, E , F. Chu·ng minh rAng:

b ) AB + CD = AD + CB - - --

c) AB - CD = AC - BD.

Bai 8. Cho sau die'm A , B, C, D, E , F. Chung minh rAng:

------- - -AD + ~ +cr= M + M +m = M + W +~.

Bai 9. Chu·ng minh rAng : AB = CD khi va ch:i khi trung die'm cua hai

doc.tn thting AD va BC trung nhau.

"' .... ' ? ,..? ....

Van dtJ2. TINH DO DAI CUA VECTO TONG, VECTO HIE;U

1) PHlidNG PHAP : bien doi vecto Long, vecto hi¢u da cho thanh m(>t vectc>" -7 - >

duy nhat u . Tinh d<) dai cua vecto u . Tu d6 suy ra d<) dai cua vecto tong, vecta hi~ u .

§2 . TONG vA HI~U cuA HAl VECTCi 15

2) Vf Dl) : Cho tam gi;k ABC vuong tai A biet AC =a va AB = 2a. Tinh d<)

clai cua Vl'Cto tong : AB + AC va vccto hi0u AB - i\C.

Theo quy u\c hlnh blnh hanh thl 1\B + 1\C = 1\D vc.ii AD b c1u·ung c:hco

hlnh blnh hanh ABCD. l\Ia goc A vuong nen ABCD L\ hlnh chu nh(lt

.::::> AD = BC. Ap d~mg c1inh li Pi-ta-go trong tam gi;lc vu6ng ABC ta c6 :

BC2 = AB2 + AC2 = 4a 2 + a 2 = 5u 1

.

V;~y : lAB + A.CI = IADI = AD ~ BC =- (/ ?;

Theo quy Ui.c ve hi~u vecto La c6 :

AB - AC = CB .

V<~y: \AB - A.Cj =jeRI= BC = uJ5. D

Hinh 1. 10

3)BAITAP

Bai 10. Cho tam giac deu ABC c;:mh b:mg a. 'l'inh d(> dai cua cac vectu - -- -- - -AB -+ BC va AB - BC .

Bai 11. Cho tam giac ABC vu6ng L<~i A hi[;L AB =a v;\ g6c B = (-)Ou. Tinh d<, --· - -

dai cac vecto AB + AC va J\B - AC.

Bai 12. Cho hlnh vuong ABCD qnh a. Tinh c19 clai cc-l.c vecto :

a) AC - AB; b) AB + AD; c) AB -+ BC.

Bai 13. Cho I ~ + bl = 0. So sanh <19 diti' phLIUng va hu'6ng cua hai vee tO' (/

va b .

Bai 14. Tu giac ABCD Ia hinh gl ne·u AB = DC va lAB\ = \BC\ ?

Bai 15. Cho a, b la hai vectCi kh<i.c 0. Khi n;\o xay ra cti.c ct\ng thCic

cluoi day ?

16 Chuang 1. VECTO -------- - -------

I ? ~ i Van d(J' 3. XAC O!NH Oll~~TH~-~ O~NG THUC VECTO CHO TRUde __j

1) PHlJdNG PHAP : De· xac d~nh die'm M thoa d!'ing thCic vectO' cho tni6'c, ta c6 cac phu'O'ng phap sau :

- > ''' Bien doi dAag thl!c vectO' da cho ve d~ng : AM = u , trong c16 A lc\ c1i ~·m

- >

co d~nh va u la vecto· co d~nh.

-->

== = Lay A lam die m goc, dyng vectO' bi'ing u thi di e·m ng<;m chinh lit c1i e'm

M can dt,/ng.

2) vi Dl) : Cho tam giac ABC. I lay X<ic cljnh di e'm M tho<:l m a n c1i eu ki ¢n :

- - -Ll1A - M /3 + MC = 0.

Giii.i /SJM B C

-----Ta et'> : MA. - MB + MC = 0

<=> MA. - MB = CM

<=:> BA = CM . Hinh t. 11

V(ty M c1u·qc xac dinh bO'i h~ thCic BA = CM => lvl la dinh thu· tu trong hinh binh hanh ABCM.

3) sAr rAP

Bai 16. Cho tam giac ABC. H <:1y xa c dinh di em lH thoa man c1ieu ki$n sau : --- - · AlA - MI3 + lllC = IJC .

Bid 17. Cho tam giac ABC. Hay tim cac di e'm M thoa m <) t trong ca c c1i eu ki ~n sau day :

a) MA - MB = BA c) fMA - CA l = fAC - ABf .

§3. TiCH CUA M(>T SO VOl MQT VECTd 17

, ? - ~ , -

§3. TICH CUA MOT SO VOl MOT VECTO

A. TOM TAT GIAO KHOA

I. PHEP NHAN MQT SO Vdl VECTO

--+ --+ --+ Cho vecto a :to 0 va m9t so thy'c ll :to 0. Tich cua so k vO"i vecto· a , ki

--+ --+ hi~u h. a , chi m9t vecto cung phuang vO"i a va thoa cac tinh chat :

--+ + Cung huO"ng vO"i a neu h > 0 ;

--+ + Nguqc huO"ng vO"i a neu h < 0 ;

--+ ~

+ C6 d9 dai : h. a = /ll/ a

--+ --+ --+ Quy uO"c : k. 0 = 0. a = 0 .

--+ -~

V O"i hai vecto a va b , vO"i m<;>i so thy'c h, l cho truO"c, ta c6 :

-~ ~ -~ ~

i) ll . (a + b ) = ll a + h b

--+ -~ -ii) (h + l ) a = h. a + l. a

--+ ~

iii) h. (l a ) = (ld). a

-~ ~ --+ - ) iv) 1. a = a ; ( - 1). a = - a

Quy tcic trung di~m

I la trung di~m cua AB ~ ~ ~

<=> !A+ IB = 0 ~~ ~

<=> MA + MB =2M!.

(M la m9t diem ba t ld)

2.GIAI TOAN HiNH HOC 10·A

( tinh ph an bo)

( tinh ph an bo)

(tinh ket hqp)

Quy tcic tr<_>ng tam

G la trc;ng tam cua tam giac ABC ~ ~ ~ ~

<=> GA + GB + GC = 0 ~~ ~ ~

<=> MA + MB + MC = 2MG.

(M la m(>t diem ba t ki)

18 Chuong 1. VECTO

II. QUAN H~ GIUA HAl VECTd CUNG PHU'dNG

-~ --+ --} ·-~

Dieu ki~n can va du de' hai vecta a va b cung phuang (v6'i a -:t 0 ) la

ton tq.i so tht,tc h sao cho b = h.~ .

,... .... ,...? , • .? 2. '

Ill. DIEU Klf;:N DE BA DIEM THANG HANG

Ba di~m phan bi~t A , B , C th:ing hang <:::::> ton tai so h -:t 0 : AB = k. A C.

IV. BIEU TH! MOT VECTd THEO HAl VECTd KHONG CUNG PHU'dNG

- } - )

D~nh li : Cho hai vecta khong cung phu·ang a va b . Khi do m9i vecta --+ --+ 0

X deu phan tich dliQ'C IDQt each duy nhat qua ha i vectO' Cl va b , nghia --+ - ) -~

la c6 duy nhat c<'[tp so m va n sao cho : x = m a + n b .

B. PHUONG PHAP GIAI TOAN

, ,.. 2 I'

Van dtJ 1. CHUNG MINH MOT DANG THUC VECTO

1) vi ov VI d1,1 1 : Cho hlnh blnh hanh ABCD.

~- -

Chung minh ding: AB + AC + AD = 2AC .

Giai

Ta c6:

AB + AC + AD = (AB + AD)+ AC = AC + AC = 2AC

(quy ti:ic hl.nh bl.nh h <'mh).

A~----;8

D c Hinh 1.12

VI d1,1 2 : Cho tam giac ABC. Lan luQt lay cac diem M, N, P tren cac do<;~n AB,

BC va CA sao cho :

AM = _:1_ AB ; BN = _:1_ BC ; CP = _:1_ CA . 3 3 3

Chung minh : AN + BP + CM = O.

2.GIAI TOAN HiNH HOC 10·8

§3 . TicH cuA MOT s6 v61 MOT VECTcJ 19

Giai A

Ta c6:

- · 1 --· --- - · 1 --BN = - BC <=> AN - AB = - BC (1 J

3 3

CP = }:_CA <=> BP - BC = l_CA · (2) 3 3 ,

AM = I_ AB <=> CM - C'A = I_ J\.B. (3J 3 3

c B

Hin!J 1. 13

C(lng theo ve ( 1), (2), (3) ta du·qc :

--· - - - - 1 - --/\N 1 BP + CM - (AB + BC + CA) = - (AB + BC + CAl.

3 ~ -~ - - - - - -

VI AB + BC + CA = 0 => AN + BP + CM = 0.

a) ChCtng minh rang : A1A2 + 8182 -1 C1C2 = 3G1G2 .

b) Suy ra m¢t dieu ki$n can va du de hai tam giac tren c6 cung trQng tam.

Gi:ii

a) Ta c6 : · - - --

A.LGL + B1G1 + C1G1 = 0 (vi G1 Ia trQng t{un tam giac A1B1C1);

-- - -~ - -G2A2 + G2B2 + G2C2 = 0 (vl G2 la trQng tam tam giac A2B2C2 ).

Tn l~.li c6 :

A1A2 + B1B2 + C1C2 =

= (A1G1 + G1C2 + G2A2 )+(B1G1 + G1C2 + G2B2 )+(C1G1 + G1G2 + G2C2. )

= 3C1G2• + ( A1G1 + B1G1 + C1G1 ) +( G2A2 + G2B-;_ + G2C2 )

--) ·)

3G1G2 + 0 + 0 = 3G1G2 .

b) Hai tam gi<ic AlBlCl Vtl A'!.B'2C2 co cung trQng tam - - - -~----- -

<=> 0 1 = C2 <=> C 1G2 = 0 <=> A 1A2 + B1B2 + C1C2 = 0 . (do cau a).

20 Chudng 1. VECTO

Bai 1. G9i AM la trung tuye'n cua tam giac ABC va D la trung die'm cua do~n AM. Chung minh riing :

a) 2DA + DB + DC = 0 ; b) 20A + OB + OC = 40D, v6'i 0 la die'm tuy y. ·

Bai 2. G9i M vaN Ian luqt la trung diem cac do~n th<ing AB va CD. Chung minh rAng:

2MN = AC + BD = AD + BC .

Bai 3. Cho tam giac d~u ABC c6 0 la tr9ng tam va M la m(>t die'm tuy y trong tam giac. G9i D, E, F Ian luqt la chan duong vuong g6c h~ tLt M den BC, CA, AB. ChU'ng minh rAng:

~ ~ ~ 3~

MD+ ME+ MF =-MO. 2

Bai 4. Cho tam giac ABC. G9i D , E, F Ian luqt la trung die'm cua BC, CA, AB . Chung minh :

--- ~ AD + BE + CF = 0 .

Bai 5. Cho 4 diem A, B, C, D bat ki. Chung minh rAng: -- -- --- -AB + CD = AD + CB va AB - CD = AC - BD .

Bili 6. G9i E, F Ian luqt la trung die'm cua AB, CD va 0 la trung die'm cua EF. Hay chung minh cac d<ing thuc sau :

- 1(- -) a) EF = '2 AC + BD ;

---- -~ b) OA + OB + OC + OD = 0; -- ~- ~ c) MA + MB + MC +MD = 4MO (v6'i M la m9t diem bat ki).

Bai 7. Cho lye giac ABCDEF. G9i M, N, P, Q, R, S Ian luqt la trung die'm cua cac c~nh AB, BC, CD, DE, EF. Chung minh rAng: hai tam giac

MPR va NQS c6 cung tr9ng tam.