C. CAU HOI VA BAITAP
b) AB= IAB I= 79^4 = Vi3, 4
Tacd BC= 6 ; va w 25 13 BC = |BC| = J36 + — = —. V 4 2
Nhan xet. Cd thi chiing minh tam giac ABC vudng tai A bing each chiing minh ring BC^ = AB'^ + AC^.
Vl du 2. Tfnh gdc giCra hai vecto a v^ b trong cac trudng hgp sau : a) a = (1 ; -2), b = (-1 ; - 3 ) ; b) a = (3 ; -4), b = (4 ; 3); c) a = (2 ; 5), b = (3 ; -7). GlAl , r 7, ạb l.(-l) + (-2).(-3) 5 V2 a) cos(a, fe) = 1^1 |_| = ; ;—=— = —7^ = — • .lal.lfel V1 + 4.V1 + 9 V50 2 vay ( a , fe) = 45°. , ^ .- -;, ạfe 3.4 + (-4).3 0 . . b) COS(a, fe) = rrn-pj - i r = — = 0. lal.lfel V9 + I6.VI6 + 9 25 vay ( a , fe) =90°. , - - ạb 2.3 + 5.(-7) c) cos(a, fe) = ,_, ,_, = -29 >/2 fllJfel V4 + 25.V9 + 49 29V2 2 vay ( a , fe) =135°.
Vi du 3. Trong mat phing Ox/cho hai diem Ă2 ; 4) va 6(1.; 1). Tim toa do diem C sao cho tam giac AfiC la tam giac vudng can tai fị
GIAI
Gia sit diim C cin tim cd toa đ la (x ; y). Di A ABC vudng can tai B ta
phai cd: • B1.'B6=O IBA|=|BC| vdi BA =(1 ;3)va BC ( x - 1 ; y - l ) . Dilu đ cd nghia la : | l . ( ^ - l ) + 3.(>'-l) = 0 \l^+3''=(x-lf+(y-l)^
l(3-3j)2+(j-l)2=io r;c = 4-3>' . r;c = 4-3>' .
[ l 0 / - 2 0 > ' = 0.
Giai he phuong tiinh tren ta tim dugc toa đ hai diim C va C thoa man dilu kien cua bai toan :
C = (4 ; 0) va C = (-2 ; 2) (h.2.13).
C. CAU HOI VA BAI TAP
2.13. Cho hai vecto a vk b diu khac vecto 0. Tfch vd hudng ạ fe khi nao
duong, khi nao am va khi nao bing 0 ?
2.14. Ap dung tfnh chit giao hoan va tfnh chat phan phd'i cua tfch vd hudng hay chiing minh eac két qua sau day :
(a + b) =\a\ +|fe| +2ạb ; (a-b) =\a\ +\b\ -2ạb ;
(a + fe)(a-fe) = |a| -jfej .
2.15. Tam giac ABC vudng can tai A vacdAB = AC^= ạ Tfnh:
a) AB.A6 ; b) 'B1.'B6 ; c) 'AB.'B6. 2.16. Cho tam giac ABC cd AB = 5 cm, BC = 7 cm, CA = 8 cm. 2.16. Cho tam giac ABC cd AB = 5 cm, BC = 7 cm, CA = 8 cm.
a) Tfnh AB.AC rdi suy ra gia tri ciia gdc A ; b) Tfnh €l.CB.
2.17. Tam giac ABC cd AB = 6 cm, AC = 8 cm, BC = 11 cm. a) Tfnh AB.AC vk chiing td ring tam giac ABC cd gdc A tụ
b) Tren canh AB \ky diim M sao cho AM = 2 cm va ggi  la trung diim
2.18. Cho tam giac ABC can (AB = AC). Ggi H la trung diim ciia canh BC, D la hinh chilu vudng gdc ciia H txtn canh AC, M la trung diim cua doan HD. Chung minh ring AM vudng gdc vdi BD.
2.19. Cho hai vecto a va fe cd |a| = 5, |fe| = 12 va |a + fe| = 13. Tfnh tfch vd
hudng ạ(a + fe) va suy ra gdc giiia hai vecto a va a + fẹ
2.20. Cho tam giac ABC. Ggi H la true tam cua tam giac va M la trung diim cua
. — . 1 9
canh BC. Chiing minh ring MH.MA = - BC .
2.21. Cho tam giac diu ABC canh ạ Tfnh ~ABAC vk JB.^.
2.22. Cho tii giac ABCD cd hai dudng cheo AC vk BD vudng gdc vdi nhau va cit nhau tai M. Ggi P la trung diim cua canh AD. Chiing minh ring MP vudng gdc vdi BC khi va chi khi 1AAM6 = JiBMD.
2.23. Trong mat phing Oxy cho tam giac ABC vdi A = (2 ; 4), B = (-3 ; 1) va
C = (3;-l).Tfnh:
a) Toa do diim D di ttr giac ABCD la hinh binh hanh ;
b) Toa đ chan Á ciia dudng cao ve tit dinh Ạ
2.24. Trong mat phing Oxy, cho tam giac ABC vdi A = (-1 ; 1), B = (1 ; 3) va C = (1 ; -1). Chiing minh tam giac ABC la tam giac vudng can tai Ạ
2.25. Trong mat phing Oxy cho bdn diim Ă-l ; 1), B(0 ; 2), C(3 ; 1) va D(0 ; -2). Chiing minh ring tir giac ABCD la hinh thang can.
2.26. Trong mat phing Oxy cho ba diim Ă-l ; -1), B(3 ; 1) va C(6 ; 0).
a) Chiing minh ba diim A, B, C khdng thing hang.
b) Tfnh gdc B cua tam giac ABC.
2.27. Trong mat phing Oxy cho hai diim Ă5 ; 4) va B(3 ; -2). Mdt diim M di đng tren true hoanh Ox. Tim gia tri nhd nhit cua I MA + MB|.
2.28. Trong mat phing Oxy cho bdn diim Ă3 ; 4), B(4 ; 1), C(2 ; -3), D(-\ ; 6); Chiing minh ring tii giac ABCD ndi tilp dugc trong mdt dudng trdn.