35. Cho tam gidc ABC vk dudng thing d. Tim dilm M trtn dudng thing d sao
cho vecto M = MA + MB + 2MC cd dd ddi nhd nhdt.
36. Cho tii gidc ABCD. Vdi sd k tuy y, Id''y cac dilm M vk N sao cho
AM = kAB vk DN = kDC. Tim tdp hgp cdc trung dilm / cua doan thing
MN khi k thay ddi.
37. Cho tam gidc ABC vdi cdc canh AB = c,BC = a,CA = b.
a) Ggi CM Id dudng phdn gidc trong cua gdc C. Hay bilu thi vecto CM
theo cdc vecto CA vk CB.
b) Ggi / la tdm dudng trdn ndi tilp tam gidc ABC. Chiing minh ring
alA + bW + clc = 0.
38. Cho tam gidc ABC cd true tdm H va tdm dudng trdn ngoai tilp O. Chiing
minh ring
a)OA-i-Ofi + OC = 0 ^ ;
b) ^
-I- ^
+^
= 2113.
39. Cho ba ddy cung song song AA^, BB^, CC^ ciia dudng trdn (O). Chiing
minh ring true tdm cua ba tam giac ABC^, BCA^ vk CAB^ ndm tren mOt
dudng thing.''
40. Cho n diim Aj, A2,..., A„ va n sd k^, ^2. •••> k„ md ki + ^2 +••• + k„ = k^O.
a) Chiing minh ring cd duy nhdt mOt dilm G sao cho
k^GAi + k2GA2 + ... + k„GA„ = 0.
Dilm G nhu thi ggi Id tdm ti cu cua he diem Aj, gan vdi cdc he sdk^. Trong
trudng hgp cac he sd k-^ bdng nhau (vd do dd cd thi xem cdc k-^ diu bdng
thi G ggi la trgng tdm cua he diem A,b) Chiing minh ring nlu G Id tdm ti cu ndi d cdu a) thi vdi mgi dilm O bdt
ki, ta cd
OG = j (^jOAi + k20A2 + ... -I- k„OA^\.
41. Cho sdu dilm trong dd khdng cd ba dilm nao thing hdng. Ggi A Id mOt
tam gidc cd ba dinh ldy trong sdu dilm dd va A'' la tam gidc cd ba dinh Id
11
ba dilm cdn lai. Chiing minh ring vdi cdc cdch chgn A khdc nhau, cdc
dudng thing ndi trgng tdm hai tam gidc A vd A'' ludn di qua mdt dilm
cd dinh.
42. Cho ndm dilm trong dd khdng cd ba dilm ndo thing hang. Ggi A Id tam
gidc cd ba dinh ldy trong ndm dilm dd, hai dilm cdn lai xdc dinh mdt
doan thing 6. Chiing minh rang vdi cdc cdch chgn A khdc nhau, dudng
thing di qua trgng tdm tam giac A va trung dilm doan thing 0 ludn di qua
mdt dilm cd dinh.
§5. True toq dp va tie true toa do
I - CAC KIEN THQC
GO BAN
/. Dinh nghia ve true toq dd, toq do cua vecta vd cua diem tren mdt true.
Dd ddi dai sd cua vecta tren true.
2. Dinh nghia he true toq do, toq dd cua vecta vd cua diem ddi vdi he true
toq do. Mdi lien he giiia toq dd cua vecta vd toq do cdc diem ddu vd diim
cudi cua nd.
3. Bieu thdc toq dd cua cdc phep todn vecta: Phep cdng, phep trii vecta vd
phep nhdn vecta vdi sd.
4. Toq do cua trung diem doqn thdng vd toq do cua trgng tdm tam gidc.
II-D^BAI
43. Cho cac dilm A, B, C trtn true Ox nhu hinh 2.
C
O
A
B
Hinh 2
a) Tim toa dd cua cdc dilm A, B, C.
b) Tinh AB,BC,CA,~AB
12
+ CB,''BA-
''BC,
A5.M.
44. Tren true (O; /) cho hai dilm M vd iV cd toa dO ldn lugt la -5 vd 3. Tim
toa dd dilni P trtn true sao cho ^= = -—.
^
•
PN
2
45. Tren true (O ;7) cho ba dilm A, B, C cd toa dO ldn lugt la - 4, - 5, 3. Tun toa
va = .
dd dilm M tren true sao cho H^A + IdB + JiC = 0. Sau dd tfnh =
MB
MC
46. Cho a, b, c, d theo thii tu la toa dd cua cdc dilm A, B, C, D tren true Ox.
a) Chiing minh ring khi a + b^c + dt\n lu6n tim dugc dilm M sao cho
''MA.''MB=~MC
MD.
b) Khi AB vk CD cd ciing trung dilm thi dilm M d cdu a) cd xdc dinh khdng ?
Ap dung. Xdc dinh toa dd dilm M nlu bilt:
a = -i, b = 5, c = 3, d = -l.
Cdc bdi tap tic 47 den 52 duac x4t trong mat phdng toq dd Oxy
47. Cho cdc vecto a(l; 2), bi-3; I), c(-4; - 2).
a) T i m t o a d d c u a c a c vecto
- . - * - . * - .
_ 1 - > 1 _ _
u =2a -3b + c ; V = -a + —b - —c •,w = 3a + 2b+4c
vk xem vecto nao trong cdc vecto dd cung phuong vdi vecto /, cung
—•
phuang vdi vecto j .
—
*
b) Tim cdc sdm, n sao cho a =mb + nc.
48. Cho ba dilm A(2 ; 5), 5(1 ; 1), C(3 ; 3).
a) Tim toa dd dilm D sao cho AD = 3A5 - 2AC.
b) Tim toa dd dilm E sao cho ABCE Ik hinh binh hanh. Tim toa dd tdm
hinh binh hanh dd.
49. Bie''t Mixi; yi), Nix2; ^2), Pix^ ; ^3) la cdc trung dilm ba canh cua mdt tam
gidc. Tim toa dd cdc dinh cua tam giac.
50. Cho ba dilm A(0 ; -4), 5( -5 ; 6), C(3 ; 2).
a) Chiing minh ring ba dilm A,B,C khdng thing hang ;
b) Tim toa dd trgng tdm tam gidc ABC.
51. Cho tam gidc ABC cd A(-l ; 1), 5(5 ; -3), dinh C nam tren true Oy vk
trgng tdm G ndm tren true Ox. Hm toa dd dinh C.
13
52. Cho hai dilm phdn biet A(x^ ; >''^) vd 5(% ; yg). Ta ndi dilm M chia doan
thing AB theo ti sd k ne''u JiA = kJlB
^M
yM
_ ^A -
ik^l).
Chiing minh ring
^L
l-k
l-k
Bai tap on tap ctiuong i
53. Tam giac ABC la tam gidc gi ne''u nd thoa man mdt trong cdc dilu kien
sau ddy ?
a)
|A5
+ Acl =
|A5
- ACI.
b) Vecto AB + AC vudng gdc vdi vecto AB + CA.
54. Tii gidc ABCD Id hinh gi nlu thoa man mdt trong cdc dilu kien sau ddy ?
a) Jc-~BC
= ~DC.
b) D5 = m''DC + DA .
55. Cho G Id trgng tam tam gidc ABC. Tren canh AB Id''y hai dilm M vk N sao
cho AM = MN = NB.
a) Chiing td ring G ciing la trgng tdm tam giac MNC.
b) Dat GA = d, GB = b. Hay bilu thi cac vecto sau day qua a vd ^ :
GC,AC,GM,CN.
56. Cho tam gidc ABC. Hay xdc dinh cac dilm M, N, P sao cho :
a) MA + MB- 2MC = 0 ;
h)NA + m + 2NC = 0 ;
c)~PA-~PB + 2PC = 6.
57. Cho tam gidc ABC, vdi mdi sd k ta xdc dinh cac dilm A'', B'' sao cho
AX'' = k''BC, ~BB'' = kCA. Tim quy tich trgng tdm G'' ciia tam gidc A''B''C.
14
58. Trong mat phing toa dd Oxy, cho hai dilm A(4 ; 0), 5(2 ; - 2). Dudng
thing AB cdt true Oy tai dilm M. Trong ba dilm A, 5, M, dilm ndo ndm
giiia hai dilm cdn lai.
Cac bai tap trie nghiem chi/dng I
1.
Cho tam gidc diu ABC cd canh a. Dd dai cua tdng hai vecto AB vk AC
bdng bao nhieu ?
(A)2fl;
2.
(B)a;
iC) a43 ;
(D)
^ •
Cho tam giac vudng cdn ABC cd AB = AC = a. Dd ddi cua tdng hai vecto
AB vk AC bing bao nhieu ?
iA) a42 ;
(B) ^ 2
;
,
(C) 2 a ;
,
(D)fl.
Cho tam gidc ABC vudng tai A va A5 = 3, AC = 4. Vecto CB+ JB cd dd
ddi bing bao nhieu ?
(A) 2 ;
(B) 2VI3 ;
(C) 4 ;
(D) Vl3.
Cho tam giac diu ABC cd canh bdng a, H la trung dilm cua canh BC.
Vecto CA-Hc
iA)-; a
5.
cd dd dai bing bao nhieu ?
,^. 2aV 3
(C) - ^ ;
3a
(B) — ;
,T^X
(D)
a4l
2 "
Ggi G la trgng tdm tam gidc vudng ABC vdi canh huyin BC =12. Tdng hai
vecto GB + GC cd dd dai bang bao nhieu ?
(A) 2 ;
6.
(B) 2V3 ;
(C) 8 ;
Cho bdn dilm A, 5, C, D. Ggi / vd / ldn lugt Id trung dilm cua cdc doan
thing AB vk CD. Trong cdc dang thiic dudi ddy, ding thiic nao sai ?
(A) 277 = AB + CD ;
7.
(D) 4.
(B) 277 = AC + 5D ;
(C) 2lj = AD +''BC ;
(D) 277 -l- CA + D5 = 6.
Cho sdu dilm A, 5, C, D, E, F. Trong cdc ding thiic dudi ddy, ding thiic
ndo sai ?
(A) ''M>+ ~BE+^ = JE+
(C) AD + ^
''BD+
''CF
; (B) JD + ''BE+CF^JE
+ ''BF + CE ;
+CF = AF + BD + CE ; iD) AD+ ''BE+CF = AF+ M:+ CD.
15
8.
Cho tam gidc ABC vk diim I sao cho IA = 2IB. Bilu thi vecto CI theo hai
vecto CA vk CB nhu sau :
—.
pM— OJTR
(A) CI =
^
;
(C)C7 = ^ ± ^ ;
9.
>
>
•
(B) C / = - C A - K 2 C 5 ;
(D)C7 =
^
^
.
Cho tam giac ABC vk I Id dilm sao cho 1A + 21B = 0. Bilu thi vecto C?
theo hai vecto CA vk CB nhu sau :
(K)a=i~i^:
(B)a = -C/1 + 2C5;
(C)a = ^ ± 2 « ;
(D)a = ^ ± | ^ .
10. Cho tam gidc ABC vdi trgng tdm G. Ddt CA = a, C5 = S. Bilu thi vecto
AG theo hai vecto a vd ^ nhu sau :
(A)AG = 2 3 _ l i ;
(B):^ =
^
(C)Ag = ^
(D)AG =
^
;
11. Cho G Id trgng tdm tam gidc ABC. Ddt ^
;
.
= d, CB = b. Bilu thi vecto
CG theo hai vecto a vd 6 nhu sau :
•^
3
—•
(C) CG = ^
^ ^ ^
3
3 •
12. Trong he toa dd Oxy cho cdc dilm A(l ; - 2 ) , 5(0 ; 3); C ( - 3 ; 4), D ( - 1 ; 8).
Ba dilm nao trong bdn dilm da cho Id ba dilm thing hdng ?
(A)A,5,C;
16
;
(D) CG =
(B)5,C,D;
(C)A,5,D;
(D)A,C,D.
