中四數學+附加數問題

自由人

2008-03-02 8:56 pm

1)The three sides of a triangle are (x/y)+(y/z),(y/z)+(z/x) and (z/x)+(x/y), where x,y,z are positive numbers.Prove that the area of this triangle is sqrt[(x/y)+(y/z)+(z/x)]
2)If sin(A+B)=1/2 and sin(A-B)=1/3, find the ratio of tanA:tanB.

回答 (1)

myisland8132

2008-03-02 10:18 pm

✔ 最佳答案

1
s
=1/2[(x/y)+(y/z)+(y/z)+(z/x)+(z/x)+(x/y)]
=(x/y)+(y/z)+(z/x)

s-[(x/y)+(y/z)]=(z/x)
s-[(y/z)+(z/x)]=(x/y)
s-[(z/x)+(x/y)]=(y/z)

the area of this triangle is
sqrt[s(z/x)(x/y)(y/z)]
=sqrt[[(x/y)+(y/z)+(z/x)](z/x)(x/y)(y/z)]
=sqrt[(x/y)+(y/z)+(z/x)]

2
sin(A+B)=1/2 and sin(A-B)=1/3,

sinAcosB+cosAsinB=1/2
sinAcosB-cosAsinB=1/3

2cosAsinB=1/6
cosAsinB=1/12..(1)

Similarly
2sinAcosB=5/6
sinAcosB=5/12..(2)

(2)/(1)
tanA/tanB=5

the ratio of tanA:tanB is 5:1

2008-03-02 14:19:19 補充：
Heron's Formula:

http://mathworld.wolfram.com/HeronsFormula.html

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