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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:

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
Boi
duong MSG Hinh hoc 10
B.MpTSODANGTOAN
i)ang
l.Dung
vecta.
•
SM'
dung
djnh
nghta.
• Cac
diem

can xdc
dinh
nen la
diem
ciioi
vecta.
<Bdi
18. Cho tarn
giac
ABC. Hay xac
dinh
cac diem D, E, F sao cho :
a) AD = 2AB ;
b) AE = ^AC ;
Giai.
c) AF = Ad + AE.
a) Ta CO AD = 2AB
<=>
AD
cung huong voi AB va AD =2AB
b)
Taco
AE = ACc:>AE
2
nguoc
huong voi AC va AE = ^AC.
c) Dung hinh binh hanh ADFE. Diem F la diem duoc xac djnh (nhu hinh ve).
^ang2.Xdc
dinh
diem

thod
man he
thuc
vecta
•
Neil
CO
the nen thu gon he
thuc
• Su
dung
mot
soky
thuat
sau de nit gon :
- Neu
CO
MA + MB thi got I Id trung
diem
AB de co MA + MB = 2MI
- Neu
CO
MA + MB + MC thi goi G Id
trong
tarn
tam
gidc
ABC de c6
MA
+ MB + MC-3MG '> i \ ? ,

• *
•
- Su
dung
hieu
cua hai
vecta
ciing
diem
ddu MA - MB = BA
^dj
2
9. Cho tam
giac
ABC. Xac djnh cac diem M, N thoa man ding thuc sau :
a) MB
+
2MC = 0;
a)
Taco
b)NA
+ NB + 2NC = 0.
Giai.
MB
+
2MC = 0 MB = -2MC
<=>
MB, MC
nguoc
huong va MB = 2MC

<=>
M nam tren doan BC va MB = 2MC.
b) Goi I la trung diem doan AB.
Khi
do NA + NB = 2NI.
Taco:
NA + NB + 2NC =
0.<i>2NI
+ 2NC-0
<=>
NI + NC = 0
<=>
N la trung diem IC.
Cty
TNHH
MTV DWH Khang Vic
i)ang
3.
Bieu
dim
vecta
theo
hai
vecta
khong
ciing
phuong
•
Su:
dung

qui tac ba
diem,
qui tac
hinh hinh
hanh
debieh
dot
vecta.
• 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 AM = MN = NB va AP = PQ = QB.
a) Tinh MP, QN
theo
vecto
BC .
b) Tinh cac
vecta
MQ, NP
theo
cac

vecto
AB, AC
Tu
ket qua do suy ra MQ + NP = BC.
Giai.
a) Ta CO
MP^AM^l^^p^l^^
BC AB 3 3
Chu
y rang MP va BC
1
cijng huong nen MP = - BC.
3
Ta cung
CO
QN = -BC va QN, BC
nguoc
huong.
3
Tudo QN = —BC.
3
1
1
b) De dang thay rang PQ = - AC va NM = —AB.
3 3
Tu
do ta CO
MQ
= MP + PQ = iBC + iAC = -(AC-AB) + -AC = AB + -AC
33 3^ '3 33

NP = NM + MP = IBC -
i
AB
= i(AC - AB) - -
AB
= AB +
i
A^^^
3 3 3^ '3 3 3
Nhu
the MQ + NP = AC - AB = BC.
^di21. Cho hinh binh hanh ABCD. Tren duong
cheo
BD lay cac diem G va H
sao cho DG = GH = HB.
a) Chung minh rang AB + AD = AG + AH .
. o
b) AH cat BC tai M, AG cat DC tai N. Chung minh rang AM + AN = - AC .
Giai.
a)
Taco
AG = AD + DG ;
AH=AB+BH;
Boi liuihig HSG Hinh hoc 10
suy
ra:
AG
+ AH = AD + AB +
DC
+ BH

Vi
DG
va
BH
la hai vecto doi nhau
nen
DG
+ BH = 0 .
Vay
AG
+ AH = AD + AB.
b) Vi
AD
+AB = AC nen
AG
+AH = AC .
Do
do tu giac
AHCG
la
hinh
binh
hanh, do do : ,
BH
= HG = 2HO (voi O la giao diem cua AC va
BD).
' ' '
Suy
ra H la trong tam cua
AABC,

ma AM la duong trung tuyen
nen
AM =
-AH.
' " '*
,'•>,•
\
2
Chung
minh
tuong tu ta
ciing
c6 N la trong tam cua
AADC,
ma AM la
3
duong
trung tuyen nen AN =
—
AG
.
Dodo:
AM + AN =
|(AG
+ AH) =
-AC.
^ang 4. Chung minh ha diem thang
hang.
•
De'chiing minh ba diem phdn biet A,B,C thang hang ta chung minh hai

vecto
AB,
AC
cilng phuong,
nghid
la AB =
kAC.
Trong nhieu trudmg hofp ta
bieu
dim cdc vecto
AB,AC
theo hai vecto
khong ciing phttong.
<Bdi
22.
Cho
tam
giac
ABC.
a) Xac
djnh
diem I sao cho lA + IB + 2iC = d. ' r
b) M va N la hai diem thay doi tren mat phang sao cho
MN
= MA + MB + 2MC.
Chung
minh
rang M, N, I thang hang.
Giai.
a)

GQI
J la trung diem doan AB. Khi do lA + IB = 21].
Do
do lA + IB + 2rc = 6 o 2
(ij
+
IC)
= d <=> I la trung diem doan JC.
b) Ap dung qui tac ba diem, ta c6
NW
= MA + MB + 2MC = 5^ +
IA
+
Ml
+
iB
+ 2(N«
'
= 4Mi + lA +
IB
+
2IC,
vi lA +
FB
+ 2rc = 0 nen
22
MN-4MI.
Tu
day
suy

ra
M,
N,
I
thang
hjiig.
Chtl
y. Kci ludn a cdu b) cd thcdien dqt each khdc nhxt
Id
: Dwang thang MN ludn
di qua mot
dic'nl
codinh.
Cty TNHH MTVDWH Khang Viet
^ai 23. Cho tam giac ABC N la diem tren canh AC sao cho NC = SNA. M la
diem
tren canh BC voi BM =
kBC.
Goi I la trung diem AM.
a)
Tinh
cac vecto
BI,
BN
theo cac vecto
BA,
BC.
b) Xac
dinh
k sao cho ba diem

B,
I, N thang hang.
Giai.
1
a) Ta
CO
AN =
-AC
=> AN =
-
AC
va BN =
kBC
'
4 4
Ta
CO
0;A
BI^^^^^^^BA.JJBC.
2
2 2
1
BN=BA+AN=BA+-AC
4
=
BA +
-(BC-BA)
= -BA +
iBC.
b) Ba diem B, I, N thang hang khi va chi khi BI, BN la hai vecto cung phuong.

Dieu
nay tuong duong
VOI
=
<=>k
= —.
^
^ ^ 3/4 1/4 3
'£>(ing
5. Chiing minh tinh chat hinh hoc, hai diem trimg nhau.
24. Cho tam giac
ABC
va diem
G.
Chung
minh
rang
a) Neu GA +
GB
+
GC
= 0 thi G la trong tam cua tam giac ABC ;
b) Neu
CO
diem O sao cho OA +
OB
+
OC
= 30G thi G la trong tam
ciia

tam
giac
ABC.
' , .
(Bai
tap Dai so'10 nang cao)
Giai.
a) Ggi G' la trong tam cua
AABC,
tac6:G'A + G'B +
G'C
= 0.
Dodo
GA
+
GB
+
GC-0
o
GG'
+
G'A
+
GG'
+
G'B
+
GG'
+
G'C

= 0
o
3GG"''
+
G'"A
+ G'B + G''C = d
o 3GG' = d
<?:i>
G = G'.
Vay GA + GB +
G;C
= d thi G la
trong
tam cua
AABC.
b) Taco:
30G
= 0A + 0B +
0C
=
0G
+
GA
+
0G
+
GB
+
0G
+

GC
=
30G +
GA
+
GB
+
GC
=>
GA
+
GB
+
GC
= 0.
Vay
G la trong tam
ciia
AABC.
23
Boi
iditnig
use Uinli hoc 10
jVhdn
xet. Tir lai ^iai cua hai
loan
nay ta c6 cdc kct qua sau :
G la
tning
tam cua

tain
i^idc
ABC <=> GA + GB + GC = 0
> <»OA + OB + OC
^di 25. Cho tam
giac
ABC. Goi A' la diem doi xung cua A qua B, B' la diem
doi
xi'mg cua B qua C, C la diem doi xiVng cua C qua A. Chirng minh rang
hai tam
giac
ABC va A'B'C c6 cimg trong tam.
Giai.
Goi G la trong tam cua tam
giac
ABC. Khi do GA + GB + GC = 0.
Ta can phai chung minh
GA'
+ GB' + GC = 6 va tu do suy ra rang G cung
la trong tam ciia tam
giac
A'B'C. <
Taco
GA' = GA + AA\GF = GB +
BB',GC
= GC + CC\
Cong ba dang thuc tren
vetheo
ve, ta duoc: '*
GA'

+ GB' + GC = GA + GB + GC + AA' + BB' + CC = AA' + BB' + CC.
Talaico
AA'=
2AB,BB'=
2BC,CC'=
2CA. , . ,; , .
Tir
do GA' + GB' + GC = 2(AB + BC + CA) = 2.0 = 0. ; '1
Suy ra G la trong tam cua tam
giac
A'B'C.
Vay hai tam
giac
c6 cung trong tam. """I J
^ang 6. Tap hap
diem.
• AM = kv v&i A
codinh
va v
khong
doi thi tap hap cdc
diem
M Id
duong
thdng
qua A vd
cung
phuattg
vai gid
vecto

v.
•
I
MA 1=1 MB
I
vai A, B
codinh
thi tap hap cdc
diem
M Id
duaitg
trung
true
doanAB.
. • > .
•
I
AM 1=1 v
I
vai A
codjnh,
v ^ 0 vd c6 do ddi
khong
doi thi tap hap cdc
diem
M Id
duong
tron
tam A, hdn
kinh

bang
I
v I.
(Bdi
26. Cho tam
giac
ABC. Tim tap hop cac diem M sao cho
a) MA + kMB - kMC ;
b) MA + (l-k)MB + kMC = 0.
. Giai. "/V'
a) Gia su diem M
thoa
man dang thuc MA + kMB = kMC.
Khi
do ta c6: MA = k(MC-MB) = kBC
Suy ra hai
vecto
MA, BC cung phuong. Vi
vecto
BC c6' djnh nen tap hop
M
la duong thang qua A va
song song
voi BC.
b) Gia su M la diem
thoa
man MA + (1 - k)MB + kMC = 0. Ta c6
OA
Cty TNHH MTV DWH
Khaug

Vict
MA
+
(1
- k)MB + kMC = 0 » MA + k(MC - MB) + MB = 0
MA
+ MB + kBC = d » 2Mi
=-kBC
(I la trung diem AB)
Nhu
the hai
vecto
MI,
BC ciing phuong. Vay tap hop cac diem M la duong
thang qua I va
song song
voi BC.
<Bdi
27. Cho hinh binh hanh ABCD. Tim tap hqyp nhirng diem M sao cho
a)
|MA
+
MB|
= |MC + MD|; ,V. 'V. ; . .
b) |2MA-MB-MC| = |MC + 2MD|. MS-J^^/- :V
Giai.
a) Goi I va J Ian luot la trung diem cua AB, CD. Khi do M'/' • .
MA
+ MB = 2MI,MC + MD = 2MJ
,n<jb

,
Gia su M la diem
thoa
man dang thuc da cho, ta c6 /
MA
+ MB
MC + MD
<i=>
2MI
2Mj « MI = MJ
Vay tap hop cac diem M la duong trung true doan IJ.
b)
Taco
2MA-MB-MC = MA-MB + MA-MC = BA + CA .
Nhu
the
vecto
2MA - MB - MC la
vecta
khong doi. Do do dg dai cua no la
l2MA-MB-MCI=a khongdoi.
Goi K la diem
duoc
xac djnh boi KC + 2KD = 0 . Khi do K co djnh.
Gia su M la diem
thoa
man dang thuc da cho, ta c6 "
o a =
2MA-MB-MC
<::>l3MKI=a«MK

=
MC + 2MD
a
MK
+ KC + 2MK + 2KD
Vay tap hop cac diem M la duong tron tam K, ban
kinh

CLUYENTAP
• Dung
vecta
^
1-27.
Hinh
ben c6 4
vecto
a, b, c, d
Ian luot nam tren 4 duong thang
song
song.
Hay xac djnh cac so k, 1,
m,n
sao cho : d
a = kb,b = lc,c = md,a = nc.
Jiuang
dan
giai
a = b,b = c,c = d,a = -c.
25
Boi

tlumtg
USG Ilhih hoc 10
Cty TNHH MTV DWH KItaiig ViSt
1.28. Cho a
khac
0.
Cac khang
dinh
sau day dung hay
sai?
a) Hai
vecto
a
va-3a
ciinghuang;
b)
Vecta-3a
c6 do dai gap ba Ian do dai
vecto
a; ^ ,
3 -
c) Cac
vecto
-—a
va5a
cung phtrong ;
d) Cac
vecto
2
a

va-2 a CO do dai
khac
nhau.
''''j'"!^'' '''^^^j.
*
Jiuangddngidi
>'^> [ '""-^
^f'-^J^!
'
a) va d) sai; b) va c) dung.
1.29. Diem M tren doan thang AB sao cho 3MA = 2MB. Hay xac
dinh
so k de
dugc
dang thiic dung
Khido
-OA + -OB = OM + ON = OP
-OA+-OB
2
Vay:
• Tuong tu nhu tren ta c6
m = OQ =
VOM^
+
ON^
= 4- ^ =
i
77
4 16 4
•2 3a2

-0A+20B
2
V57
-a.
I'H) .DHA.
"ifii;^
r.';>:.f
t'>.] j
• Chmig minh he
thiic
vecto
1.31. Cho tu
giac
ABCD c6
I,
J la trung diem cua hai duong
cheo
AC, BD.
Chung minh rang
a) AB
+
CD = 2rj; b) AB + AD + CB + CD = 41].
Jiimng
dan
gidi
a) AM = kAB ;
b) MA = kAB ; c) MB = kAM ;
d) BM = kBA; e) AB =
kAM
;

Jiicang
ddn
gidi
g) BA =
kAM
.
a) AB
+
CD =
AI
+ IJ + JB + CI
+
IJ
+
JD
=
(AI + Ci) + (JB + JD) + 2Ij = 2IJ
b) AB + AD =
2AJ,CB
+ CD = 2CJ
a) AM = -AB.
5
b)MA
= -^AB. c)MB =
-AM.
b) BM = -BA.
5
e) AB =
-AM.
' 2

g) BA =
—AM.
6; 2
AB+AD+CB+CD
= 2 (AJ
+
Q) = -2 (JA + JC) = -4ji = 4IJ.
1.30. Cho tarn
giac
OAB vuong tai O voi AB = a va A = 60". Hay dung cac
vecto
sau day va
tinh
do dai cua chiing:
1^
3^^.
OA
+ 20B, - OA +-OB, -
OA
+ 20B.
2 2
Jiuang
ddn
gidi
Truoc
het ta c6 OA - -, OB = —
2 2
• Dung diem K
thoa
OK = 20B.

0A + 20B = 0A
+
0N = 0P.
• ,
Do do: - ;
1.32. Cho tu
giac
ABCD. Goi M, N Ian luot la trung diem cua AB va CD.
a) Chung minh rang 2 MN = BC +
AD.
•
b) Goi I la trung diem doan
MN.
Chung minh rang iA + iB + IC + iD = 0.
Jiuang
ddn
gidi
Theo
qui tac ba diem, ta c6
1-
0A+20B
= OP
=
VOA2+ON2=.P-
+
3a2=^VT3.
V
4 2
• Dung cac diem M, N sao cho
OM

=
-OA,ON
= -OB.
2
26
60"
a
-I
N
Q
MN
= MA + AD + DN va MN = MB + BC + CN
Tudo
2MN
=
(MA
+
MB)
+
(DN
+
CN)
+
AD
+ BC
Chii
y rang: MA + MB = DN + CN = 6 •'
Tu
do suy ra dang thuc can chung minh .
b) Do M, N la trung diem cua AB, CD

nen lA + IB = 2iM va IC
+
ID = 2iN
Vay IA + iB + iC + iD = 2(lM + IN) = 0.
1-33. Cho doan thang AB. Goi M la diem
dxxqc
xac djnh boi M\=kIvB
(k^^-l).
Chung minh rang voi mpi diem O bat ki trong mat phang ta luon c6
Olvl=0^-kOB
1
+ k
27
Boi
duimg
HSG Hiiih hoc 10
Jiu&ng
dan
gidi
Voi
diem O bat
ki,
ta c6
MA
= kMB <=> OA - OM =
k
(OB - OM) (1 - k)OM = OA - kOB
Vi
k^l nen OM =
OA-kOB

1-k
1.34. Cho tam
giac
ABC. Goi M, N Ian
lirgt
la hai diem tren hai
canh
AB, AC
sao cho AM = 2MB va 2AN = 3NC. Goi I la trung diem doan MN. '
1
^ 3
a) Chune minh AI = —AB + —AC.
^ ^ 3 10
b)
GQ'I
K la trung diem
canh
BC.
Tinh IK
theo
cac
vecto
AB,
AC
.
Jiuang
ddn
gidi
a)
Truac

het, de y rang:
AM
=
-AB
va
AN
=
-AC.
3 5
Do I la trung diem
MN
nen
2 3 10
b) Vi K la trung diem BC nen AK =
AB + AC
1 1
Tudo IK =
AK-AI
= -AB + -AC.
6 5
1.35. Cho tam
giac
ABC. Xac dinh cac diem
I,
J, K, L sao cho
a) IA-iB + 2!C = 0 ; b) JA + jB + 3jC = 6 ;
c) 2KA + KB-KC = CA;
a) Ta
CO
d) LA + LB + 2LC = 0

Jiuang
dan
gidi
IA-IB
+ 2IC =
0<»BA
+ 2IC = 0
^ 1
1
<=>IC
=
—BA«CI
= -BA.
2 2
I
la diem dugc xac djnh nhu hinh ve.
b) Goi D la trung diem AB .
Khido
JA + JB = 2JD.
Tudo jA + jB + 3jC =
0<=>2jD
+ 3jC = 0. gf-
Nhu
the J la diem nam trong doan CD sao cho 2JD = 3JC (hinh ve).
c) Ta
CO
2KA + KB - KC = CA o 2KA + CB = CA o 2KA = CA - CB
1
2KA = BA o AK = - AB <=> K la trung diem doan AB.
Cty TNHH MTV DWH

Khang
Viet
J) Goi E la trung diem BC. Khi do LB + LC = 2LE
Ta
CO
2LA + LB + LC = 0 « 2LA + 2LE = 0 » LA + LE = 0 <=> L la trung diem
doan AE.
•••:;•?::—•
' ^.^
,
^^t^.,
<V;^v:\-!/:;''vir
,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
a) De tha'y I la diem doi xung ciia A qua B
va J la diem nam tren doan AC sao cho
3JA = 2JC.
Taco
ij =
A|-AI
= -2AB + -AC. (1)
5
Goi M la trung diem ciia BC. Khi do
AG
= -AM = -
3 3
AB + AC
=-AB+iAC
3 3
1
1
-5-
1
Vay IG =
AG-Ai
= -AB + -AC-2AB = —AB + -AC. (2)
3 3 3 3
b) (1) va (2) ^ Ta c6 5lj = 6IG = -lOAB + 2
AC,
tu do ta c6
I,

J ,G thang hang.
1.37, Cho tu
giac
ABCD. Goi Gj, G2 Ian luot la trong tam cac tam
giac
ABC
va BCD. Chung minh rang:
a)
G;G7 = ^2_GiA;
b) GiD + GiB + Gi"C + G2A + G^ + G^ = 0.
Jiuang
ddn
gidi
a)
Theo
gia thiet
G1A +
G7B
+
G7C
= G2B + G2C + G^ = 0
Tirdo
G7D = -G2B-G2C va G^ =-G^-G^ = BGi + CGJ
Do do G2D - GiA
=-G2B
- G^ + G]^ + G^ = 2GiG^
^ Ta
CO
aj) + GiB + QC =
3G^G2'

va G^ + G2B + G2C =
3G2G7
Tu
do suy ra dieu phai chung minh .
29
Boi ditaiig MSG Hinh 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
cac
vecto
a, b.
b) Goi I la trung diem AC. Chung minh rang D, E, I thing hang va D la trung
diem EI.
AP-AB-AC
c) Xac dinh diem N tren mat phang sao cho SNA + 2NB + 5NC = 0.
d) Tim tap hop nhirng diem P tren mat phang sao cho
k>0 la so cho
truoc.
Jiuang
ddn
gidi

a) AD = AB + BD = AB + -BC = AB + i(AC-AB) = ^a + ib
DE
= AE - AD = -
AB
- -
AB
-
i
AC
=-a - ib
2 4 4 4 4
=
k v6i
1
r 3
^
1
3- Ir
b) Taco DI =
AI-AD
=
-b a b
= —a + -b
2 4 4 4 4
So sanh ta
thay
DI = -DE => D, E, I thang hang va D la trung diem EI.
c) Goi J la trung dm BC. Ta c65NA + 5NC =
5(NA
+ NC) =

lONI.
. Dodo 5NA + 2NB + 5NC = 0c5 2NB + 10NI =
0c>NB
+ 5NI = 0
Vay
N la diem tren doan BI sao cho NB = 5NI.
d) Goi E la dinh cua hinh binh hanh
ABEC.
Khi do AB + AC = AE
=
ko
AM
-
AB-AC
-ko
AP-AE
EP
=
k o EP = k
Vay
tap hop diem P la duong
tron
tam E ban kinh k.
1.39. Cho tu
giac
ABCD
va diem M tuy y. Chiing minh rang cac
vecto
sau
khong phu

thuoc
vao M
a) a = MA-2MB + MC; b) b =-MA + 4MB -3S4C ;
c) c = MA + 3MB - 2MC - 2MD ; d) d = MA + MB + MC + 3DM.
'•'-'^
Jiuang
dan
gidi
Ta
chung minh cac
vecto
a, b, c, d bang mot
vecto
co'djnh.
a) a = MA-2MB + MC = MA-MB + MC-MB = BA + BC
b) b = -MA + 4Effi-3MC = 3(MB-MC) + MB-MA-3CB + AB
c) c = MA + 3MB-2MC-2MD =
MA-MC
+ 2(MB-iv^
=
CA + 2DB + CB.
d) Goi G la trpng tam tam
giac
ABC. Khi do MA + MB + MC = 3MG
Khi
do d = MA + MB + MC + 3DM = 3(MG-MD) = 3DG.
30
Cty
TNHIl
A;/ V

UVVii
Killing
1,40. Cho hinh binh hanh
ABCD
c6 tam O. Goi M, N Ian luat la trung diem
ciia
cac canh BC va CD. Dat AB = a, AD = b.
a)
Tinh
cac
vecto
BN, AM
theo
a, b.
b) Goi I la
giao
diem ciia BN va DM. Chung minh rang: IB + IC + ID = 0 .
3r
c) Ggi K la diem xac djnh boi AK = - h
•
Chung minh N la trung diem doan MK.
Jiuang
dan
gidi
- Ir
a) AM = a + -b, BN = a + b
b) I la
trong
tam cua
ABCD.

c) Ta CO AN = AD + DN = -a + b
' 2
MN=AN-AM
=
—a + b- a —b = —a + —b
2 2 2 2
- Ir
0"!
MK
=
AK-AM
=
-b-a b
= -a+b
2 2
De
thay
MK = 2MN . Tu do suy ra ket luan ciia bai
toan.
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 = 2AB.

Tiep
theo
dung hinh binh hanh
ACMD.
Khi
do AM = AC + 2AB . M la diem can xac dinh.
b) Ta CO
AN
= AB + BN=AB + kBC
- AB + k(AC - AB)= (1 - k)AB + kAC
A,
M, N thang hang <=> AM va AN ciing phuong
ol^
=
-<=>l-k
= 2kok = i.
2 1 3
1.42. a) Cho hai
vecto
khong ciing phuong a, b . Chung minh :
ma + nb = 0 <=> m = n = 0.
33
Boi
dumig
HSG
Hinh
hoc 10
b) Cho tam
giac
ABC c6 cac

canh
la a = BC, b = AC, c = AB va trong tarn G.
Chiing
minh rang neu a.GA + b.GB + c.GC = 0 thi tam
giac
ABC la tam
giac
deu. ,
^ \. v!
Jiic&ngddngidi
(.•
a) • Neu m = n = 0 thi ta c6 ma + n b = 0.
• Dao lai neu ma + n b = 0, ta c6 :
+ Neu m = 0 thi n b = 0 => n = 0. Ta c6 m = n = 0.
+ Neu m^tO thi ma + n b = 0
=> a- -—b, ta
CO
a va b cung phuong (trai gia thiet): Khong xay ra.
m
Do do ma + n b = 6 => m = n = 0.
Vay ma + n b = 0 <=> m = n = 0.
b) Vi ABC la mot tam
giac
c6 trong tam G nen ta c6
GA + GB + GC = 6 GA = -(GB + GC). •
Theo
gia thiet a.GA + b.GB + c.GC = 0 ta c6
-a(GB + GC) + b.GB + c.GC = 0 o
(b-a).GB
+ (c

-a).GC
= 6
Vi
GB va GC la hai
vecto
khong cung phuong nen
theo
cau a), ta c6 :
b-a = c- a =
0<=>a
= b = c.
Vay tam
giac
ABC la tam
giac
deu.
• Tap hgp diem
thoa
man thiic
vecta
,
1.43. Cho tam
giac
ABC. Tim tap hop nhung diem M sao cho
a) MA = MB + kMC; b) MA + kMB + MC = 0 ;
b) (l-k)MA + (l + k)MB + MC = 0; d) MA + MB + 2MC = kBC.
' ' "
Jiuang
ddn
gidi

a) Goi M la diem
thoa
man h^ thuc da cho, ta
CO
MA
= MB + kMC<:>MA-MB = kMC<=>BA = kMC ' •:
o BA, MC la hai
vecto
cung phuong A? ;
M
nam tren duong thang qua C va
song song
voi AB.
b)
G<?i
I ia trung diem AC. Khi do MA + MC = 2Mi. Tu do ta c6
_ , _ V
MA
+ kMB + MC =
0<=>
2Mi + kMB = 0 o MI = —MB
<=>
MI, MB
2 „,
cimg phuong hay M, B, I thang hang.
Vay tap hop cac diem M la duong thang BI.
•50
Cty
TNHH
MTV DWH

Khang
Vu
c) Gia su M la diem
thoa
man gia thiet bai toan, ta c6
(1 - k)MA + (1 + k)MB + MC = 0<::>MA + MB + MC + k(MB - MA) = 0
<:}.3MG
+ kBA = 0<::j>MG = —BA.
3 ^,
Voi
G la trpng tam tam
giac
ABC.
Vay tap hgp cac diem M la duong thang qua G va
song song
voi AB.
d) Gpi I la trung diem AB va J la trung diem CJ. Khi do
is^ + MB + 2MC - kBC o 2MI + 2MC = kBC o 4MJ
Vay tap hgp cac diem M la duong thang qua J va
song song
voi BC.
1.44. Cho hai diem A, B co'dinh. Tim tap hgp nhimg diem M sao cho
2^ ^ b) 2MA + 3MB = 4MA + MB
a)
c)
MA
+ 3MB =-MA-MB
3MA + MB-2MC
3MB-2MA-MC
Jlit&ng

ddn
gidi
1.44. a) Gpi I la diem
thoa
man IA + SIB = 0, khi do ta c6
MA
+ 3MB = Mi + IA + 3(MI + IB) = 4MI. ' "' ''^^ "
Ta CO
MA
+ 3MB
MA-MB
4MI
BA
oMI
= -BA.
6
V^y tap hgp cac diem M la duong tron tam I ban
kinh
BA/6.
c) 3MA + MB-2MC = 3MB-2MA-MC.
b) Ggi I va J la hai diem
thoa
man 2iA + 3IB = 0, 4jA - JB = 0
Cho M la diem
thoa
man yeu cau bai toan. Khi do ta c6
2MA + 3MB = 5MI, 4MA + MB = 5MJ
Suy ra
SMI
5MJoMI = MJ.

Vay tap hgp cac diem M la duong trung tryc doan IJ.
c)
Truoc
het chu y r^ng ' •. ^ "
3MB-2MA-MC
= 2(MB-MA) +
MB-MC
=2AB
+ CB = U
Ggi I la diem
thoa
man 3IA + IB - 2IC = 0. Khi do ta c6
3MA + MB-2MC = 2MI
Tu
do, gia thiet bai toan tro thanh
3MA + MB-2MC = 3MB-2MA-MC
2MI
u
«.MI =
V^y tap hgp cac diem M la duong tron tam I ban
kinh
bang
lui
33
Boi
ditmg HSG lihih hoc
10
§5.TRyC
TOA
Dp VA

HE
TRUC
TOA DO
A-TOMTATLI'THUYET
1.
True toa do
•
Toa do cua vecto
u
la so'
a
dugc xac
djnh
boi u = ai
. i
Toa
do
ciia
diem
M la so'm dugc xac
djnh
boi
OM
= mi.
Dp
dai dai
so
cua vecto AB tren mot true la mot so dupe ki
hieu
la. AB

va
dupe
xac
djnh
boi he thue AB =
AB.i
(T
la vecto don vi cua true).
*
H§ thuc
Sa-lo:
AB + BC = AC
voi
mpi
diem
A, B,
C
tren true Ox.
2.
Jie
true toa dd
•
Trong
he true toa dp Oxy
a
= (x;y)oa =
x.T+yj
M
0
y

A
r
0
M(x;y)»OM
= (x;y)
MN
=
(xN-XM;yN-yM)
•
Giasu
a
= (x;y),b =
(x';y').Khid6
-
_
|'x
=
x'
a=b<=>i
y
= y
a
+ b = (x + x';y+ y')
a-b = (x-x';y-y')
ka
=
(kx;ky)
vai keM
b
cimg

phuong vai a^O
<=>
x'
= kx,y' = ky vai k E R
•
Neu
I
la trung
diem
doan AB thi Xj =
^^A±^;
=
^^ +
76
•
Neu G la trpng tam tam giac
ABC
thi
_
XA
+
XB
+
B.
MOTSODANGTOAN
_XA
+
XB
+
XC

.
_yA + yB + yc
XG
^ ' yc - ^
^ang
l.Mot
sohai todn ve toa do cua diem tren mot true.
^di
28.
Tren
tryc x
'Ox
cho
3
diem
A, B,
C
Ian
lupt
eo toa dp la -3,
2, 5.
a)
Tinh
toa dp cac vecto AB, BC,
CA,
-3AB + 4BC + 2CA
b)
Tim toa dp
diem
M tren true sao cho

-
3AM
+
BM
+
MC
=
0.
Giai.
a) Toa dp
ciia
vecto AB la
2 -
(-3) = 5.
To?i
dp cua vecto BC la
5 - 2
= 3.
,
;
>
Toa
dp cua vecto CA la
-3 -5
= -8.
Toa
dp cua vecta -3AB + 4BC +
2CA
la -3.5 + 4.3 + 2(-8) = -19.
b)

Gpi
X la toa dp
ciia
M.
Khi
do toa dp
ciia
cac vecto AM, BM, MC Ian
lupt
la
X
+ 3, X
-
2,
5 -
x.
Theo
gia thiet
-
3AM
+
BM
+
MC
= 0 »-3(x + 3) +
X -
2 + 5
- X
= 0
<=>

X =-2 .
Vay
toa dp cua M la
X
=-2.
.
^dj
29.
Tren
mot true cho
4
diem
A, B, C, D.
Chung
minh
rang
AB.CD
+
AC.DB
+
AD.BC
=
0
(He thuc
Euler).
Giai,
Gpi
toa dp
eiia
4

diem
A, B, C, D Ian
lupt
la a, b,
e, d.
Khi
do vetrai cua dang
thiic
can
chung
minh
la
(b
-
a)(d
-
c) + (c
-
a)(b
-
d)
+
(d
-
a)(e
-
b)
" '
Thuc
hien

khai
trien
va
riit
gpn ta eo
dieu
phai
chung
minh.
(I
i)Qng
2. Tinh todn
ve
toa do diem
-
vecto tren mat phang.
•
Sic dung cac cong
thUc
ve toa do diem
-
vecto.
•
Su dung dieu kien cua hai vecto ciing phuong.
.
•,{!.
:ai;;.|
-f
',r
j'iil;

ia ,
:
A ; •
•••••
f-^ . >•
®di30.Choeaevecta
a
= (-3;l), b = (4;3), c-(-2;6). .,ay:r^,^tifM
a) Xac
djnh
toa dp cua eae veeta 2a
-
3b
, -
3a + 4b + 2e.
- I » >;
b)
Xac
dinh
hai so
x, y
sao cho
c
= xa + yb.
it
<[
^''L;
.
Giai.
a)

+
Ta
CO
2a = (-6;
2),
3b = (12;
9).
-
' - ^ '
;
r
'
Tudo
2a-3b =
(-18;-7).
+
Ta
CO
-3a
= (9;-3), 4b = (16; 12), 2c = (-4; 12).
Nhu
the-3a + 4b + 2c =
(21;21).
t>)
Gia
su c = xa + yb.
-

30
'-3x + 4y =

-2
Khi
do ta
CO
X
+ 3y =
6
X
=
y
=
13
- 30- 16r , :
.
Vay c = —a + —b.
•
11
13 13
13
.jiiiiKku.,.
35
Boi
duong HSG Hinh hgc 10
^ang
3. Ba diem A, B, C thang hang, khong thang hang.
•
Tinh
toa do cac vecta AB, AC. '
"
' >

?
; r^r
Bang
tqa dg chting minh hai vecta
AB,AC
khong cung phuang.
Suy ra A, B, C khong thang hang.
^di
Sl.Cho ba diem
A(5;-2),
B(l;3),
C(2;-4).
Chung
minh
rang ba diem A, B, C khong thang hang.
Giai.
Taco
AB-(-4;5),
AC-(-3;-2)
-4 5
5>t-—=>AB,
AC khong cung phuong. . |. ^, - .,.t
w
,
Vay
ba diem A, B, C khong thSng hang. lA' i ' >
' •
^di
32. Cho
A(-l;2),

B(3;l).
a) AB cat true Oy tai N. Tim toa do diem N. ^ ' ' ^
b)
Tim tpa dp diem M tren (d): y = 1 de A, B, M thang hang.
Giai.
a) Vi N nam tren true tung nen N(0; y). Ta co :
AN
=
(l;y-2),AB
=
(4;-l)
-
AB
eat true Oy tai N
<=>
AN
ciing
phuong vai AB
1
y-2 7 ,„ ^,
<=>
- =
<=>
y = Vay N
4-1
^ 4
b)
Diem
M thuoe duong thang y = 1 nen ta c6 toa dp
M(x;

1). Ta c6 '
•
AM
= (x +1
;-l),
AB
=
(4;-l).
•. r
>
at
>
V;>0(:H
rt>'(.;i>-JKX
*
A,
M, B thang hang
<=>
AM cung phuong voi AB
'inifo
stX (
«
^^ =
—<»x
+ l = 4
<::>x
= 3. Vay
M(3;l).
t)ang
4. Xdc dinh tga do diem M thod dim ki^n cho truac.

•
Tim dac tncng hinh hoc cua diem
M.
* •: » - -
<
»•
*
»
Chuyen cac dac tntng hinh hgc sang toa do.
^di
33. Trpng mat phang Oxy cho ba diem
A(l;-2),
B(3;4),
C(0;5)
a)
Tinh
toa dp cac vecta
AB,
AC.
Suy ra A, B, C la ba
dinh
cua mpt tam giac.
Tim
toa dp trpng tam G cua tam giac do.
b)
Tim to? dp diem D sao cho
ABCD
la
hinh binh
hanh. '

36
Cty
TNHH
MTV DWH Khang Vi$t
Giai.
a) Taco AB =
(2;6),
AC = (-1;7).
Ta
thay cac vecta AB va AC khong cung phuang nen ba diem A, B, C
khong
thang hang. Do do A, B, C la ba
dinh
aia mpt tam giac.
Gpi
G la trpng tam tam giac
ABC.
Khi do ta
CO
'
XA + XB + xc _ 4
XG-
5 -3
yA
+
ys+yc
_7
3
3
b)

Gia sir
D(x;
y).
Vi
ABCD
la
hinh binh
hanh
•
Vay G
nen
AB
= DC.
Ta
CO
DC = (-x;5 -
y).
Do do
AB
- DC «
r2=-x
[6
= 5-:
•
1
'gitiaJH,,,
x
= -2
[y
=

-l
Vay
D(-2;-l).
Tam I cua
hinh binh
hanh la trung diem duong
cheo
AC. Do
/I
3^
do I
U'2
Bai
34. Cho tam giac ABC vai
B(l;-1),
C(6;-3).
Biet
trpng tam
ciia
tam giac nam
tren
true hoanh va trung diem M cua canh AB
each
deu hai tryc tpa dp. Xac
djnh
tpa dp diem A.
Gia
su A(a; b). Khi do trpng tam G
ciia
tam giac c6 tpa dp

^7 + a b-4^
.Bit
O!'.'
XA
+ XB +
XC.
yA + yB + yc
3 ' , 3
hay G
33;
Do
G nam tren true hoanh nen b - 4 = 0 hay b = 4.
VayA(a;4),
G
7 + a
;0
Trung
diem M
ciia
cgnh AB la M
.
Do M
each
deu hai true tpa dp
[
2 '2
nen x
M
yM
a + l 3

"a + l = 3
=
-0 <=>
2
2
a + l = -3
a = 2
a = -4.
Vay
A(2;
4)
hoae
A(-4;
4).
V7
i
Boi
dumig
HSG
Hinh
hoc 10
CLUYENTAP
• Tpa dp diem,
vecta
tren mpt true > t
i^'
1.45. Tren true x'Ox cho hai diem A va B c6 toa do Ian luat la 2 va 8.
a) Tim toa do diem M biet rang MB + 2MA = 0 .
b) Tim toa do diem N biet rang NB-3NA-0. .,',<n i. f Mo
c) Tim toa dp diem P doi

xiing
voi B qua A. *
A
'
• i *
' .
Jiuang
dan
gidi
a) Tpa dp Mia 4. b) Tpa dp N la-1. c) Tpa dp P la-4.
'^"^^^^^
AC
CB
1.46. Tren true x'Qx cho 3 diem A, B, C phan biet thoa man dieu ki^n =—.
2 3
30A 20B
Chung minh rang OC = + . n'w'
, i] I
Jlic&ng
dan
gidi
Ap
dung He thuc
Sa-la
ta c6 , t> .'Kl OA
AO
+ OC CO + OB
-30A + 30C
=-20C
+ 20B • ' { yrf .,

Hay 50C = 30A + 20B OC = - OA + - OB (dpcm). t.,
5 5
.••' <
Gpi
I la diem thoa man 3IA + IB - 2iC = 0.
• Tinh
toan
ve tpa dp diem -
vecta
tren
mat
phang
1.47. Cho hai
vecta
a =
(2;5),b
= (2; 3). Khi do
vecto
v = 2a - 3b cung phuong
vdi
vecta
nao sau day ?. ,.| ,^
a) ^ = (!;!) b) xI^ =
(-4;-2)
c)
ii^
=
|V2; ^)
d)
Ii^

=
(2VS;Vs)
' '
Jiiccmg
dan
giai
Taco
2a =
(4;10),3b
=
(6;9)=^v
=
2a-3b
=
(-2;l)
s
^ ^ 2 1
Tir
do ta thay v cung phuong voi
vecto
U3 vi
—j=
=
—z—.
V2
;;«ifj;t
1.48. Trong mat phang tpa dp, moi menh de sau day dung hay sai ?
a) Hai
vecto
doi nhau khi va chi khi chiing c6 hoanh dp doi nhau va tung dp

doi
nhau. 1
b) Neu
vecta
a 6 cung huang vdi
vecta
I thi no cd hoanh dp duong.
Cty TNHU MIV
DVVH
Khang
Vic
Neu
vecta
a 0 ngupc hudng vdi
vecto
j thi no cd tung dp am.
J) Neu
vecta
a khong cung phuong vdi ca hai
vecto
i va j thi ca hai toa do
cua no deu
khac
0.
e) Hai
vecto
a = (-3; V3) va b = (V3; -1) la hai
vecta
cung hudng.
Jiuang

d&n
gidi
a) Dung
b)
Diing
VI
i = (1;0) va neu a^O cung huong vdi i thi a = ki = (k;0) vdi k>0.
c) Dung (tuong tu cau b)
d) Dung. Ta luon vie't dupe a = (x;y) = xi + yj .
^
i''' *
Do gia thiet nen ca x va y deu
khac
0. v
jn
/ -
j,
„ «1 . ^,
e) Sai. Ta cd a =
-V3b=>a,b
ngupc hudng. <\i 'i uv h im '
1.49. Cho cac
vecto
u =
—i
+ 3i,v = 2i + xi
2
iSi-/M
1 • U
a) Xaedjnh

xsao
cho hai
vecta
u,v ciing phuong.
b) Xae
dinh
x sao cho
vecto
2u -3v cung phuong vdi
vecto
i .
•
y,,^:^
•:r
• '
•
: Ji^g d^ri
gidi
,
,
^t^j
ejjp
yV-,f
a) Hai
vecto
u, v cung phuong khi va chi khi =
—
<z> x = -12
2
b) Ta cd 2u = -T +

6],
3'v = 61 + 3x]
=>
2u - 3v = -7T + (6 - 3x)]
Vecto
2u - 3v ciing phuong vdi
vecto
i khi 6 - 3x = 0 <::> x = 2.
1.50. Cho cac
vecto
a = (-3; 4), b =
(-1;3),
c -
(2;0).
?
a) Tim toa dp cac
vecto
2a + b, - a + 3b - 2c. .'^'^ "
b) Xae
dinh
k,
1
sao cho ka + lb + e = 6.
c) Xae djnh cac so k, 1 sao cho
vecta
c ciing phuong vdi
vecto
v = ka + lb vi
I
V1= 5 .

Jiuang
dan
gidi
. \
a) Tim toa dp cac
vecto
2a + b = (- 8;]!), - a + 3b -2c =
(-2;5).'
"'
b)
Tacd
ka =
(-3k;4k),lb
=
(-l;31):^ka
+ lb =
(-3k-l;4k
+ 3l) . ^
^ . -3k-l + 2 = 0
Tacd: ka + lb + c = 0<=>-^ ^, „ o<i
4k + 31 = 0
5
1
= -^
5
Bdi
duang
HSG Hinh hpc 10
c) Ta thay hai
vecto

cung phuang khi 4k +
31
= 0<:>l =
-4k
Khi
do dp dai
vecta
v la j-3k
-1|
=
5 k
-3k
+
-k
3
_5|k
Theo
gia thiet = 5ok = ±3
Vay ta c6 hai dap so (k;l) =
(3;-4)
hoac
(k;l) =
(-3;4).
^f ^
1.51. Trong cac m^nh de sau, m^nh de nao
diing
?
a) Neu OH = 1, OK = 2 voi H, K Ian lugt la hinh chieu ciia A len Ox, Oy thi
diem
A c6 toa dp la

A(l;
2). j, - *•> > , - -v, .r (•H / n/ ? r.,
••^-vi^
'
h) Neu A(x;y) va CB=xi-yj vai y^O thi A va B doi
xiing
nhau qua true Oy.
c) Neu hai diem A va B c6 hoanh dp doi nhau, tung dp doi nhau thi trung
diem
doan AB la go'c toa dp O. i
d) Neu A(a; 1) va B(-a; 3) thi trung diem doan AB nam tren true hoanh.
JIudng
dan
gidi
a) Sai (phai sua lai la OH = 1, OK = 2,
b) Sai (doi
xiing
qua Ox), • ;'
^
- ^, '^^wVl,
c)
Diing,
d) Sai (nSm tren Oy).
^>no-nui:'fnh
v-,u
•
v:k>ll i,
• Xac dinh tpa dp diem
thoa
man dieu kifn cho

tmoc
1.52. Trong mat phang toa dp cho ba diem A(-2; 5), B(2; 2), C(10 ; 4).
a) Chung minh rang A, B, C la ba
dinh
ciia mpt tam
giac.
Tim toa dp trpng tam
cua tam
giac
do.
b) Tim diem D tren true hoanh sao cho A, B, D thang hang. • >5
<
fif
i
c) Tim diem E tren true tung sao cho ABCE la hinh thang vai hai day la AB, CE.
a)
Taco
AB-(4;-3),AC =
(12;-l)
Hai
vecta
AB,AC khong cimg phuang nen ba diem A, B, C khong thang
hang. Vay A,B,C la ba
dinh
cua mpt tam
giac.
^10 n\
Gpi
G la trpng tam tam
giac

ABC. De c6 G
3 3 j
b) Lay D(x;0) la diem nam tren true hoanh. Khi do AD = (x + 2;-5)
Ba diem A, B, D thang hang
<=>
AB, AD cung phuang
X + 2 -"i 14
o = —<»3x + 6 =
20ox
= —. Vgy D
I
3
4n
Cty TNHH MTV DWH
Khang
Viet
Lay E(0; y) la diem nam tren true tung. Khi do CE = (-10; y - 4)
ABCE la hinh thang
<=>
AB,CE ciing phuang
,d£ =
^c>4y-16
= 30<=>y = —.Vay E
2 )
153. Trong mat phang
t<?a
dp cho ba diem A(2;-l),
B(x;2)
va C(-3;y).
a) Xae

dinh
x, y sao cho B la trung diem eiia doan AC.
b) Xac
dinh
x, y sao cho tam
giac
ABC nhan go'c toa dp O lam trpng tam.
c) Tim h? thiic lien h? giija x, y sao cho A, B, C la ba diem thang hang.
Jiudng
ddn
gidi
a) B la trung diem doan AC khi va chi khi
XA+XC
<=>s
X
=
2-3
2 =
y-1
X
= II )
2
[y
= 5
b) Vi tam
giac
ABC nhan goc O lam trpng tam nen
r2 + x-3 = 0 fx = l
-l
+ 2 + y = 0 [y =

-l.
••••
1 t ,
c) AB = (x-2;3),AC = (-5;y + l)
A,
B, C thang hang
<=>
AB,
AC cung phuang (x - 2)(y +1) +15 = 0.
1.54. Cho A(2;l),B(2;-l),C(x;-3),D(-2;y). ,
a) Tim x, y de ABCD la hinh binh hanh .
b) Tim
X,
y sao cho ba diem A, B, C tao thanh mpt tam
giac
nhan D lam trpng tam.
Huong
ddn
gidi
a)
Taco
AB = (0;-2),AC =
(x-2;-4),
DC = (x +
2;-3-y).
ABCD la hinh binh hanh <:>AB = DC vaA,B,C khong thang hang
'0.(-4)
+
2(x-2)7^0
x = -2

y
=
-l.
x + 2 = 0
-3-y = -2
Dl
thay la ba diem A, B, C khong thang hang. Do do :
^BC nhan D lam trpng tam khi va chi khi
; = XD
ZAJ-YB
+ yc
<=>
=
yD
2 + 2 + x = -6 rx = -10
l-l-3
= 3y ^1y =
-l.
iiiiMiii
41
Boi
dumg HSG Hinh hoc 10
1.55. Cho
giac
ABC biet A(3; 7), trong tam G -;3 , diem B
riclm
tren tia Oy vo
V
3 )
(

diem C nam tren tia Ox. Tim toa do B va C. ,
'
! JJu&ngddngidi ;
>
Goi C(xc ; 0) va B(0 ;
yB)
Ian
lirgt
nam tren cac truoOx va Oy sao cho AA\]Q
nhan
G lam trong tam. Ta
CO:
i , , , f
'
^
f3
+
0
+
xc=7^jXc=4
*
, , ,
|7 +
yB+0
= 9
[yB=2
I . ' ,
Vay
B(0 ; 2) va C(4 ; 0).
1.56. Trong mat

phSng
toa do cho ba diem
M(2;-3),
N(-1 ;2), P(3;-2).
a) Xac djnh toa do diem Q sao cho MP + MN - 2MQ = 0.
b)
Xac djnh toa do ba dinh cua tam
giac
ABC sao cho M, N, P Ian luot la trung
diem cua BC, CA, AB.
Jlacmg ddn
gial
a)
Taco
MN = (-3;5),MP = (1;1) => MN + MP = ( 2;6)
Giasir Q(x;y).Khid6 MQ = (x - 2;y + 3) => 2MQ = (2x - 4;2y + 6)
Cty
TNHH
MTV
DWH
Khang Vict
§6.
BAI TAP ON
CUOI
CHl/QNG
I
J 57. Cho luc
giac
deu ABCDEF c6 tam O. Chung minh rang
a) OA + OB + OC + OD + OE + OF = 0;

b)
Vol mpi diem M tuy y thi MA + MD = MB + ME = MC +
MF.,

Jiuang
ddn
gidi
a) ABCDEF la luc
giac
deu tam O. Khi do cac cap diem A va D, B va E, C va F
doi
xung qua tam O. Do do: OA + OD = OB + OE = OC + OF = 0 .
b)
Dedangc6
MA>MD
= MB + ME = MC + MF = 2K^
1.58. Cho tam
giac
ABC. Hai diem M va N Ian lugt di dong tren cac duong
thang AB va AC sao cho AM = mAB, AN = nAC . Tim dieu kien cua m va
n
sao cho vecto v = BN + CM cung phuong voi vecto BC . , . ^
Jiuang
ddn
gidi
Truoc het ta c6 BC =
-
AB
+ AC. :
Taco

BN =
AN-AB
= -AB + nAC,
(1)
CM
=
AM-AC
=
mAB-AC
f-2
= 2x-4 [x=l
t>'
Ta
CO
MP + MN - 2MQ = 0oMP + MN =
2MQ«^
^ ^c^^
[6
= 2y + 6 [y =
^)
Nhu
the V = BM + CN = (m - 1)AB + (n - 1)AC . (2)
'V.tJ
/
Vay
Q(];0).
b)
Gia su
A(x;y).
Vi N la trung diem CA nen

:>
[Xr-
=2XK]
-x^ =-2-x
yc=2yN-YA=4-y
M
la trung diem BC
fxg
=2XM
-XC
=6 +
X
•J
,0
Tu
(1) va (2), ta c6: BC va v ciing phuong khi va chi khi
m-1 n-1
-1
1
•
<=>
m + n = 2.
•
C(-2-x;4-y).
lyB = 2yM-yc = -io + y
B(6
+ x;-10 + y).
P la trung diem AB
Xp
=

XA
+ Xr
y,.
=
yA
+yB
[6
+
2x
= 6
x
= 0
1.59. Cho hinh binh
hanh
ABCD
tam O. Dat AC = a, BD = b
a)
Tinh
cac vecto
AB,BC,
AD
theo
a va b.
b)
Hai diem M va N Ian lugt tren hai duong thang AB va AD. Gia su
AM
=
xAB,
AN = yAD. Tim dieu
kif

n
cua x va y sao cho ba diem M, N, B
thSnghang.
' 0,-'CTT 'ir!
4/A'b.
Jiit&ng
ddn
gidi
' ''^
a) Taco: AB-OB-OA
2y-10 = -4 [y = 3.
Vay
A(0;3),B(6;-7),C(-2;1).
= BD+iAC=-a b
2
2 2 2
BC=OC-OB=iAC+iBD=ia+ib
^)
Taco
AM = xAB = x
—a—b
.2 2 .
x
- X r
=-a b
2
2
AN
= yAD = yBC = y —a +—b
U

2
43
Boi
duang
HSG Hinh hgc 10
Tirdo MN =
AN-AM
= ^^i + ^^b
2 2
- X- Xr 2-X- X;
MC
=
AC-AM
= a—a + -b = a + -b
2 2 2 2
Chii
y rang cac
vecta
AM,AB va
AN, AD
cixng huang. Do do x,y >0. Ba
diem M, N, C thang hang neu va chi neu MN va MC la hai
vecto
cung
• phuong. Nhu the ta phai c6
^.yZ^-Z±^.2z2i
=
o«xy-x2-(2y-xy
+
2x-x2)

=
0
' '
22 22 ^ \ y
.'i*
Hay
xy-x^-2y
+
xy-2x
+ x^
=Ooxy
= x + y
Vay
dieu kien de M, N, C thang hang la xy = x + y. ' '
1.60.
Cho tam
giac
ABC . Tren cac canh BC, CA, AB Ian lugt lay cac diem M,
N,
P sao cho BM + CN
+
AP = 0. Chung minh rang cac tam
giac
ABC va
MNP CO
cung
trong
tam. • ,, .,,»,f >/r e/-v ^
<
O ui .cisi

:x>ij-ti
Gpi
G la
trong
tam tam
giac
ABC.
Ta c6 GA +
GB
+
GC
= 0. Khi do
= (GA
+
GB+GC)
+
(AM+BN
+
CP)
=
O
GM
+
GN
+ GP =
GA
+
AM + GB + BN
+
GC +

CP= " 'f
, ^ \< I HI
Tu
do ta
CO
dieu phai chung minh. ' " " '* ' ' * ;
1.61.
Chung minh rMng dieu ki^n can va du de hai hinh binh hanh
ABCD
va
A'B'C'D'cocungtamla
AA'
+
BF'
+
CC'
+
DD'
= 0. , .,,
, , ,.,
JiuangMngidi
,,; ^ ^.^.^^.^
Gpi
I, r Ian lugt la tam ciia hai hinh binh hanh.
;
A
- ^1
(\
>:
: |/?

A
Khid6iA
+ iB +
IC
+ iD = r^ + r^ + rC + rD' = 6
sofirl^rftijli
Taco AA' = Ai + ir + rA';
BB"
= BI + IT + rB" ' >
CC
=
CI
+ ir
+
rC ;
DD*'
=
DI
+ ir
+ rD'
,j.
Cpng
4 dang
thuc
tren ve
theo
ve, ta
dugc
AA'
+

BB^
+
CC
+
DD'
= 4ir •
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
AM, AN
theo
hai
vecto
AB = a, AC = b .
b) Gpi I la trung diem AM, J la diem tren AN sao cho AJ =
kAC.
Xac djnh k
debadi&nB, I,
Jthanghang.

t ; I d r ' *^ ; fc"' H'
Jiucng
ddn
gidi
a) Dithay BM =
JBC,BN
= JBC
Tvr
do
AM
= AB + BM = AB + -BC = AB + -(AC-AB) = iAB + -AC = ii + -b.
A
4^ '4 4 44
4
5:
4
^ = AB + BN = AB + -BC =AB +
-(AC-AB)
4 , 4 . A
=_iAB+-AC=-ia+-b
4 4 4 4
=
AB +-(AC - AB) =-i
AB +
-
AC
4 ^ '4 4
=
—a + -b
4 4

4"" 4
^b) Ta
CO
t - 1
B
M 9
AMI
'iA^Mfx-'ilh
Ai
= -AM = -a + -b, AJ = kAN = a + —b, ;^«'i:;rfn-^,r? a
2 8 8 4 4
Bi
=
AI-AB
= ia + -b-a = —a + -b '
8 8 8 8
.,V:v^,:.,
BJ = AJ-AB = —a
+—b-i
= ^^a + —b
4 4 4 4 A"
0 Ba diem B, I, J thang hang khi va chi khi
BI, BJ
cung phuong
. -k-4 -7 5k 3 2(k
+
4) 10k „, „, , 3
hay
:
— = —

:-<=>-^
- =
<=>
3k + 12 = 35k o k =
^ 4 8 4 8 7 3 8
1.63.
Cho hinh binh hanh
ABCD.
Hai diem M va J Ian lugt
thay
doi tren cac
canh DC va BC voi DM =
yDC,BJ
= xBC. Ggi I la trung diem AD. AM cat
BI
taiN.
a) Cho y = -,tinhtiso —,
•^5 BI
b) Xac dinh x va y sao cho tam cua hinh binh hanh
ABCD
ciing la
trgng
tam
cua tam
giac
MIJ.
Jiu6ng
ddn
gidi
1

K
a) D^t AB = a,AD-b.Giasu — = z:^BN = zBI.
BI
' ^ ' .
'•
^'
Ti'nh
dugc
AM = ya + b,
AN
= (1 - z)a + —
b
45
Boi
diam^ liSC
Uiiili
hoc 10
Cac
vecto
AM, AN ciing phuong. Suy ra
, , 3 , 10
Tu
do vai V = — ta CO z = —
^ 5 13
1-z z/2
<=>z(y
+ 2) = 2
b) Goi O la tarn ciia
hinh
binh hanh. De dang tinh

dugc
1
_ 1 , /
OJ
= -a +
1
X
—
2)
y-2
b,
OM =
Neu
O la trong tarn tarn
giac
MIJ thi
1^
a + -h, OI = a
2 . . 2.
" 21
a + xb =
0<=>
x = 0
1
y=2'
OM
+ OI + OJ-0c:>
1.64. Cho tu>
giac
ABCD

. \>\k. .:l.: ,,Af''. k
a) Xacdinhdiemlsaocho IA + iB + iC + !D = 0.
b)
Chiing
minh OA + OB + OC + OD = 40I voi O la diem bat
ki.
•
c) Xac dinh diem M
deM
MA + MB + MC + MD
I
c6 gia trj be nhat.
d) Tim tap hop nhirng diem N sao cho AB cung phuong vai
vecto
v=NA+NB+NC+ND.
Jiwmg
dan gidi
a) Goi M, N Ian lugt la trung diem cua AB va CD.
Taco rA + iB =
2iM,iC
+ ID = 2iN
Nhuthe:
IA + IB +
IC
+ ID = 0^2IM + 2IN = 0
<=>
Ila trung diem doan MN.
b) Ap dung quy tac ve
hif
u

hai vecta, ta c6: *"
•'.
iA+iB+iC+ID
= 0
«
OA-OI
+
OB-OI
+
OC-OI
+
OD-OI
= 0
<=> OA +
OB
+
OC
+
OD
= 40I voi mpi O.
c) Ap dung ket qua cau b) ta c6: \ ;
IMA
+ MB + MC +
MDI
=
4IMII
=4MI.
DodolMA
+ MB + MC +
MDI

dat gia trj be nhat o MI = 0.
VayMs
Ithi
IMA + MB + MC +
MDI
dat gia tri be nhat.
d) Ap dung ke't qua cau b), ta c6: v = NA + r^ + NC + ND = 4NI.
Nhu
vay, v ciing phuang vai AB o NI ciing phuang voi AB <=> NI song
song vai AB.
Nhu
the tap hgp nhiing diem N sao cho v ciing phuang vai AB la duang
thang qua I va song song vai AB. ,w:
ft,,
fNlUl
MVV
DVVn
Khaug Viet
J
65- 8'^^ ABC. Cho d, d' la hai duong thang phan biet thay do!
nhung
luon song song vol BC. Biet d cat hai canh AB, AC Ian lugt tai M, N ;
d'
cat hai canh AB, AC Ian lugt tai P, Q.
Chung
minh
vecto
v = MQ + PN c6 huang khong doi.
Jiicang ddn giai r r
, AM AN , , AP AQ , ,. ^ , , ,

Taco-—
= -— = k va— = — =
1
vol 0<k, 1<
vi*y;v.;,
>
AB
AC AB AC
AM
=
kAB,AN
=
kAC,
' '
A
-K
AP
= 1AB,AQ = 1AC
O
l-l M If ft
Taco MQ =
AQ-AM
=
lAC-kAB
va
PN =
AN-AP
=
kAC-lAB
'

Nhu the '-iU
.••:\iV'X
i'^'^ v
.j ::,','
v
= MQ + PN =
lAC-kAB
+
kAC-lAB
' : ^ ''
=
(k + 1)AC -(k + 1)AB = (k + 1)(AC - AB) = (k +
I)BC.
'
-;4 , trung diem canh CD la
P(2;2).
Hay xac dinh tga do cac dinh A,
Vay
V luon cung huang vai BC.
1.66, Cho
hinh
binh hanh
ABCD
c6 trung diem AB la M(l;5), trung diem BC
laN
B,
C, D.
Jiuang ddn gidi
Cho
A(x; y). Vi M la trung diem AB nen

JXB
=2XM -XA =2-X
.yB=2yM-yA=10-y
Vi
N la trung diem BC nen
\XQ
= 2XN - xg = 7 + X
::^B(2-x;10-y)
yc=2yN-yA
=y-2
^C(7
+ x;-2 + y)
Gpi
I la
giao
diem hai duong
cheo
hinh
binh hanh.
I
la trung diem AC nen
I
cung la trung diem MP nen I
_ + xc _ 7 + 2x
(3 7_\
[I'l]
Boi
duang
HSG
Hinh

hoc 10
Tu
do ta CO
f7 + 2x ^ 3
2 ~2
^ 2
Vay toa do cac diem la A
x = -2
y
= 2
,C 5;- ,D
V 2y
1.67. Cho cac diem M(-2;3), N(0;4),
P(-4;l).
^ ;
a) Chiing minh M, N, P la ba dinh mpt tarn
giac.
, j
b) Gia sit M, N, P
thoa
man MA = -MB, NB = -2NC, PC = 2PA . ,
^
.
Hay xac dinh tpa dp cac diem A, B, C. ,. j , j ,
Jiucfngdangiai
vj
^
^^^^^ ^ ^ ,^
Q,,,,;;
^

a) Chung minh hai
vecto
MN, MP khong cung phuang.
/'I
b) A
3.8
V
'4'5y
r 13 38'
•4'
5 J
13.11
U
' 5 j
1.68. Cho diem M(2;3).
a) Tim hai diem A, B Ian luot nam tren true Ox,Oy sao cho M la trung diem
doan AB.
b) Tim hai diem C, D Ian lugt nim tren tryc Ox,Oy sao cho M, C, D thing
hangvaMC = 2MD. "• u j rr^i^ ,v'' '
Jiu&ng
dan
gidi
a) Gia su A(a ;0), B(0; b). Vi M la trung diem cua dogin AB nen
XA+XB ^ h< 'H" .^nvrUAtJ W
ry
a = 4 • . i : " ,
YA+yB
b = 6. ic- , „
it
•'•1/':

XM=-
Vay hai diem can rim la A (4;
0),B(0;
6). nki :/
b)
Giasu
C(c;0),D(0;d)
Xet truong hgp MC = 2MD
Taco
MC =
(c-2;-3),MD
=
(-2;d-3)
c-2 = -4
-3 =
2(d-3)'
Nhuthe MC = 2MDo-^
c = -2
2
Vay hai diem can tim la
C(-2;0),D
0;-
\/
• Xet truong hpp MC = -2MD
f
9
Thuc
hien nhu tren ta
dxxgc
hai diem C(6; 0), D 0; -

V
2
1,69. Trong mp Oxy cho hinh binh hanh ABCD vai
A(l;3),
D(5;2),
canh
BC
-^-;-lJ
•
Biet
tam aia hinh binh hanh nam tren tr\ic hoanh. Hay
qua diem M
tim
toa do cac diem B va C. ' - ,
Jiu&ng
ddn
gidi
Gpi
I la tam hinh binh hanh. Khi do
l(a;0).
Gpi N la diem doi xung cua M
( 3 ^
qua I. Ta c6 N
thuoc
canh
AD va N 2a ;1
Ta
tinh
dug-c
cac

vecto
AD =
(4;-l),
AN
=
Vi
hai
vecto
AN, AD cung phuang nen 4(-2) =
^21 ^
^ 5^
2a
2)
/ IS 21
(-1) =^ a = —
4
Tu
do ta tim
dugc
tam I —;0
V 4
-V
^9.
,B
-3
,B
—;-2
I2
J
,B

I2 ' J
Do
tinh
do'i xung nen de dang
tinh
dugc
C
1.70.
Cho tam
giac
ABC. Xac dinh vi tri diem M tren duong tron ngoai tiep tam
giac
ABC de
|MA
+ MB +
MC|
C6 gia trj be nhat.
Jiurnig
dan
giai
G(?i G la trong tam cua AABC, ta c6 : MA + MB + MC = 3 MG.
Suyra
IMA + MB + MC I =3IMG I =3MG.
Ta
CO
: I MA + MB + MC I
CO
gia trj be nhat
<=>
MG

CO
gia tri be nhat.
M
chinh la
giao
diem ciia duong thang OG
voi
duong tron (O) ngoai tiep cua AABC.
Th^t vay, voi N bat
ki,
N ?t M, ta c6 :
OM=ON<OG+GN
<=>OG + GM<OG + GN,do do GM<GN
E5iern
M
dugc
xac djnh nhu tren hinh ve.
wA