已知三角形ABC的角A、B、C所对的边分别是a、b、c,设向量m=(a,b),向...
问题描述:
已知三角形ABC的角A、B、C所对的边分别是a、b、c,设向量m=(a,b),向...
已知三角形ABC的角A、B、C所对的边分别是a、b、c,设向量m=(a,b),向量n=(sinB,sinA),向量p=(b-2,a-2).(1)若向量m平行于向量n,判断三角形ABC的形状.(2)若向量m垂直于向量p,边长c=2,角C=3分之派,求三角形ABC的面积.
答
m=(a,b),n=(sinB,sinA),p=(b-2,a-2)
(1)
if m // n
=> a/b = sinB/sinA
a/sinB = b/sinA
=> sinA = sinB
=> A =B
三角形 =等腰三角形
(2)
m垂直于p,c=2,C=π/3
m垂直于p
=> m.p=0
(a,b).(b-2,a-2)=0
a(b-2)+b(a-2) =0
ab-(a+b)=0
(a+b)^2 = (ab)^2
a^2+b^2 +2ab-(ab)^2 + 4 = 4 = c^2
by cosine-rule
2ab - (ab)^2 + 4 = -2abcosC
= -ab
(ab)^2 -3ab-4 =0
(ab-4)(ab+1) =0
ab = 4 or -1(rejected)
三角形ABC的面积
=(1/2)ab sinC
= (1/2).4 .(√3/2)
=√3