amaths aaaaarrrrr~

(a+b) : (b+c) : (c+a) = 5 : 6 : 7
a=3
b=2
c=4
a+b=5
b+c=6
c+a=7
(a+b) : (b+c) : (c+a) = 5 : 6 : 7

Can you clarify whether "a", "b" and "c" are angle or length?

First thing to do this to find out a:b:c
(a+b) : (b+c) : (c+a) = 5 : 6 : 7
Let m be a number such that a+b= 5m, so b+c=6m and c+a=7m
(a+b+b+c)=11m, (b+c+c+a)=13m, (c+a+a+b)=12m
Let k be a+b+c,
2k+a+b=23m, 2k+b+c=24m, 2k+c+a=25m,
so 2k= 18m, k=9m, c=k-a-b=4m, b=k-a-c=2m, a=k-b-c=3m,
a:b:c=3:2:4

If a,b,c are length, you can use cosine law.
By cosine law, if the angle between A and B is µ
c2 = a2 + b2 - 2ab·cos(µ)
cos(µ)= (a2+b2 - c2)/2ab= (9 +4 - 16) /(2.3.2), µ=104.5 degree

If a,b,c are angles, you can use the rule that a+b+c=180 degree.
so a=60 degree, b=40 degree, c=80 degree.

I don't see why this problem is add math. This is normal maths problem simply using ratio and cosine law, no need to use compound angle formula. So pls clarify your question as I asked you in the first paragraph.


2007-03-06 21:47:45 補充：
I only give you one angle as 104.5 degree. For the other two, you can do it yourself by cosine law.

2007-03-06 22:14:10 補充：
Above solution is generally correct, except one angle is wrongly determined as 75.5 degree. The reason is that sin(75.5 degree)= sin(104.5 degree), so, if above solution is adopted, one more step should be done to deicde whether 75.5 degree or 104.5 degree is correct.

let (a+b)=5k, (b+c)=6k, (a+c)=7k
(a+b)+(b+c)+(a+c)=5k+6k+7k
2(a+b+c)=18k
a+b+c=9k
(a+b+c)-(a+b)=9k-5k
c=4k
(a+b+c)-(b+c)=9k-6k
a=3k
(a+b+c)-(a+c)=9k-7k
b=2k


(if a,b,c are the sides)
let a+b+c be s
the area of triangle
=v s(s-a)(s-b)(s-c) <--------------(Heron's formula)
=v 8.4375k^4
=k^2 v 8.4375

let x be angle A
1/2(2k)(4k)(sin x)= k^2 v 8.4375<---------1/2bc sin A
x=46.6 degree(corr to 3 sig. fig.)
.'.angle A is 46.6 degree

let x be angle B
1/2(3k)(4k)(sin x)= k^2 v 8.4375<---------1/2ac sin B
x=29.0 degree(corr to 3 sig. fig.)
.'.angle A is 29.0 degree

let x be angle C
1/2(2k)(3k)(sin x)= k^2 v 8.4375<---------1/2ab sin C
x=75.5 degree(corr to 3 sig. fig.)
.'.angle A is 75.5 degree

2007-03-06 21:53:36 補充：
應該係let x be 角A.......'.角A is 46.6度let y be 角B...........'.角B is 29.0度let z be角 C.....'.角C is 75.5度(if a,b,c are the angles)a+b+c=180度(三角形內角和)3k+2k+4k=180度k=20度a=3k度=3(20)度=60度b=2k度=2(20)度=40度c=4k度=4(20)度=80度