C. CAU HOI VA BAITAP
c) Tfnh đ dai dudng trung tuyen m^ phat xuat tir din hA cQa tam giac.
a
GIAI
, „ . 21 + 17 + 10 _ , . a) Taco/j = = 24 (cm). Theo cdng thiic He-rdng ta cd
5 = ^24(24-21) (24-17) (24-10) =84 (cm^). _ .. , 2S 2.84 ^ ^ ^ Dodo h = — = = 8 (cm). " a 21 5 84 b) Ta cd 5 = p.r => r = — = — = 3,5 (cm). p 2A
c) Dd dai dudng trung tuyln m dugc tfnh theo cdng thiic : 1,2 , 2
2 fe +C
m =
" 2 a a ~A
^
r^ ,, 2 17^+10^ 21^ 337 ^..^ Do do m = = = 84,25
" 2 A A
=> m =yJSA,25 «9,18 (cm).
Vi du 5. Cho tam giac AfiC biet a = V6 cm, b = 2 cm, c = (1 + Vs) cm. Tfnh
cac gdc A, fi, chieu cao b^ va ban kfnh dUdng trdn ngoai tiep R ciia tam
a
giac AfiC.
GIAI
^ .,. ^,. . . b^+c^-â A + (\ + Sf-6 1 Theo dinh li cosin ta co : cos A = = f= — Theo dinh li cosin ta co : cos A = = f= —
2fec 4.(1+ V3) 2 ^ a y A = 60°.
T^ ^ D c^+â-b^ (l + V3)^+6-4 V2
Tuong tu, eosB = = -^ ?=r^^^ ? = — = — • 2ac 2.76.(1 +V3) 2 vay B = 45°.
Tacd sinB = ^ => h =c.sinB = (l + V3).sin45° = ^^^ '* (em),
c " 2
h h 0
An dung dinh If sin : = 2R=> R = = —p= = v2 (cm).
sinB 2sinB V2
VAN dl 2
Chiing minh cac he thiic ve .moi quan he giQa cac yeu to cua mot tam giac
1. Phuang phdp
Dung cac he thiic co ban dl biln đi ve nay thanh vl kia hoac chiing minh ca hai vl cimg bing mdt bilu thiic nao đ, hoac chiing minh he thiic cin chiing minh tuong duong vdi mdt he thiic da biet la dung. Khi chiing minh cin khai thac cac gia thilt va kit luan dl tim dugc cac he thiic thfch hgp lam trung gian cho qua trinh biln đị
2, Cdc vi du
Vl du 1. Cho tam giac AfiC cd G la
trgng tam. Ggi a = fiC, b = CA,
c = AB. Chimg minh rang :
GÂ + GB^ + GC^ =-(â+b^+ c^). 3
GIAI
Theo tfnh chit cua trong tam ta cd GA = - AM => GÂ = -AM^ (h.2.15).
3 9 Ap dung cdng thiic tfnh trung tuyln ciia mdt tam giac ta cd :
AM2=i 2 AB'+AC'-^ 2\ 2 ^ c +b 2 y 2 4 9 4 1 .GÂ=-AM^=-.- 9 9 2 / 2 2 Tuong tu, G8 = - GC'=^ 2 , 2 b a +c 2 2 , L 2 a c +b V 2 ^ 2\ 1,2 , 2 a b +c 2 ^ J 2 ^ 2 , 1,2 C a+fe Do đ G/^ +GB^ +GC^ = - V+^'+c^) 2 1 / 2 , , 2 , 2x =—(a +fe +c ). 3
Vl du 2. Tam giac AfiC cd fiC = a,CA = b, AB = c. Chirng minh rang
a = b cosC + c cosfị
GIAI
Theo dinh If cdsin ta cd fe = a + c - 2ac cosB. c cosB =
2 , 2 , 2
a +c -fe
2^
2 2 , , 2
Ta lai cd c =a +b - 2abcosC => fe cos C =
2 , 1,2 2
a +b -c
2^
(2)
9
Cdng tiimg vl cua (1) va (2) ta cd : fe cosC + c cosB = - ^ = ạ 2a
Vl du 3. Tam giac AfiC cd fiC = a, CA = b, Afi = c va dirdng trung tuyen
AM =c = AB. Chumg minh rang :