Question

The perimeter of a triangle ABC is 6 times the arithmetic mean of the sines of its angles. If the side b is 2, then the angle B is

Answer 1

Answer:

P(ABC) = 6× Arithmetic mean if angles

Arithmetic mean = Angle2 +A2+A3 / 3

Sum of 3 angles of a triangle = 180°

So , Arithmetic mean = 180/3=60

Perimeter (ABC)= 6× 60= 360units.

Answer 2

Let a, b and c be the sides, and A , B , C are the angles of triangle ABC. Accordingly:-

a+b+c=6×(sin A+sin B+sinC)/3

We know that:-

Sin A/a=sin B/b=sin C/c=k.

Therefore put sin A=ak , sin B=bk ,sinC=ck

(a+b+c)=2(ak+bk+ck)

(a+b+c)=2k(a+b+c)

=> 1=2k or k=1/2

Now sin B/b=k , put a=2 , k=1/2

sin B/1=1/2

sin B=1/2

sin B=sin 30

B = 30°