Question

Three friends walk away from a point in three different directions such that the path of each is equally inclined to those of the other two. find the angle their parts make with other.

Answer 1

Let O is the point from away three friends A , B and C in different directions OA , OB and OC respectively .
Let a is the angle between OA and OC , then according to condition of question angle between OA and OB will also a . And angle between OB and OC will also a .
Now, you can see that,
Angle between OA and OC + angle between OA and OB + angle between OB and OC = a complete circle = 360°
⇒ a + a + a = 360°
⇒ 3a = 360° ⇒ a = 120°
Hence, angle between their path is 120°

Answer 2

Let the three friends walk away from a point along OA, OB, and OC.
∴ ∠AOB = ∠AOC ...(i)

Also, OB makes equal angles with OA and OC.
∴ ∠AOB = ∠BOC ... (ii)
From Eqs. (i) and (ii), we get
∠AOB = ∠BOC = ∠AOC ...(iii)

Now, ∠AOB + ∠BOC + ∠AOC = 360°
[angles around a point is 360°]
=> ∠AOB + ∠AOB + ∠AOB = 360° [from Eq. (iii)]
=> 3∠AOB = 360°
=> ∠AOB = 120°

Hence,∠AOB = ∠BOC = ∠AOC = 120°
[from Eq. (iii)]