C. CAU HOI VA BAITAP
b) Af i+ fiC = AC.
GIAI Ta cd : NAPM la hinh binh hanh
suy raiVA = MF(h. 1.31).
MP = (-2 ; 3).
Suy ra \x. - 2 = - 2 A A
1^.-2 = 3 [y^=5.
vay toa đ cua A la (0; 5).
Tuong tu, tir MC = FiV, MB = iVF ta tfnh dugc B(-2 ; 1), C(4 ; -1).
Vi du 4. Cho hinh binh hanh AfiCD cd Ă-1 ; 3), fi(2 ; 4), C(0 ; 1). Tim toa do dinh D.
GlAl GiasirD= {x^;y^).
Tacd AD = BC, AD = (.x^+1; >'^-3), BC = (-2 ; -3) (h.1.32). Do đ,
jx^+l = - 2 ^ ^ =-3
Vay toa do dinh Dia (-3 ;0).
Hinh 1.32
VAN dg 1
Tim toa do cua cac vecto u + v, u — v,kụ 1. Phuang phdp
Tfnh theo cac cdng thiic toa do cua u + v, u-v ,ku . 2. Cdc vi du
Vi du 1. Cho u = (3 ; -2), v^ = (7 ; 4).
Tính toa do cua cac vecto u + v, u-v ,2u, 3u-4v ,-{3u-4v).
GIAI
u + v = (10 ; 2), M-v = (-4 ; - 6 ) , 2M = (6 ; -4) 3M = ( 9 ; - 6 ) , 4 v =(28; 16). 3M = ( 9 ; - 6 ) , 4 v =(28; 16).
Vay 3M - 4 V = (-19 ;-22) va-(3M - 4 v ) = (19 ; 22).
Vi du 2. Tim x de cac cap vecto sau cung phuong :
a) a = (2 ; 3), b = (4 ; x) b) u = (0 ; 5), ỉ = (X; 7)
GlJd . A x a) — = — => J: = 6. 2 3 b);c = 0. c)—= — ^x^ = 3^x=±S. - 2 2x
Chiing minh ba diem thang hang, hai dudng thang song song bang toa do
1. Phuang phdp
Sir dung cac dilu kien cin va dii sau :
• Ba diim phan biet A, B, C thing hang <;:> AB = kAC ; • Hai vecto a,b^Q cimg phuong o Cd s d ^ d l a =kb.
2. Cdc vi du
Vi du 1. Cho ba diim Ă-1 ; 1), fi(1 ; 3), C(-2 ; 0). ChCrng minh ba diim A, fi, C thing hang.
GlAl
AB = (2 ; 2), AC = (-1 ; -1).
vay JB = -2AC. Do đ ba diim A,B,C thing hang.
Vi du 2. Cho Ă3; 4), fi(2 ; 5). Tim x 6e diim C(-7 ; x) thudc dudng thing Afị GlAl
Diim C thudc dudng thing AB khi va chi khi : ba diim A, B, C thing hang
^^6 = k'AB.
Ta cd AB = (-1 ; 1), AC = (-10 ; ;c - 4).
AC = kAB <^=^ = ^^=^x-A=l0^x=lẠ
Vi du 3. Cho bdn diim Ă0 ; 1), fi(1 ; 3), 0(2 ; 7), D(0 ; 3). Chumg minh hai
dudng thing Afi va CD song song. GiAi
AB = (1 ; 2), CD = (-2 ; -4). Vay CD = -2AB. Do đ hai dudng thing AB va CD song song hoac triing nhaụ
Ta cd AC = (2 ; 6), ma AB = (1 ; 2). Vay hai vecto Jc va AB khdng ciing phuong. Do đ diim C khdng thudc dudng thing AB. Vay AB II CD.
VAN dE y
Tinh toa do trung diem cua mot doan thang, toa do cua trong tam mot tam giac
1. Phuang phdp
Sit dung cac cdng thtic sau :
• Toa đ trUng diim ciia mdt doan thing bing trung binh cdng cac toa do tuong ung ciia hai diu miit.
• Toa đ ciia trgng tam tam giac bang trung binh cdng cac toa đ tuong iing ciia ba dinh.
2. Cdc vi du
Vi du 1. Cho tam giac AfiC vdi Ă3 ; 2), fi(-11 ; 0), C(5 ; 4). Tim toa do trgng tam G cua tam giac.
GIAI
Theo cdng thiic toa do ciia trgng tam tam giac ta cd 3-11 + 5
XQ- = -1'3'G =
2 + 0 + 4 = 2. = 2.
Vi du 2. Cho tam giac AfiC cd Ă1 ; -1), fi(5 ; -3) dinh C tren Oy va trgng
GIAI
Vi C nim tren Oy ntn ta cd C(0 ; y). Vi trgng tam G nim tren Ox ntn ta cd G{x; 0). Theo cdng thiic toa đ cua trgng tam tam giac ta cd
- l - 3 + >'
= 0 ^ J = 4. vay C cd toa đ la (0 ; 4).
Vi du 3. Cho Ă-2 ; 1), fi(4 ; 5). Tim toa do trung diim / ciia doan thing Afi va tim toa do diim C sao cho tCif giac OACfi la hinh binh hanh, O la gdc toa dọ
GlAl
Theo cdng thiic toa đ trung diim ta cd - 2 + 4
Xj = = 1; 1 + 5 -
y,= — = 3 .
2 - ' 2 Vaytoađ/lăl ;3)(h.l.33). Vaytoađ/lăl ;3)(h.l.33).
Tii giac OACB la hinh binh hanh khi va chi khi / la trung diim ciia OC. Do đ
c. 1.36. 1.36. -^ = 1 => jc^ = 2 . 2 ^ 2 =3=>>'c = 6. vay toa đ C la (2 ; 6).