If the lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is:

Step-by-step solution

Let a < b < c are three sides and θ is the smallest angle. Hence three angles are θ , π - 3 θ , 2 θ Given, 2 b = a + c ⇒ 2 sin ⁡ B = sin ⁡ A + sin ⁡ C (using sine rule) ⇒ 2 s i n 3 θ = s i n θ + s i n 2 θ ⇒ 2 3 s i n θ - 4 s i n 3 θ = s i n θ 1 + 2 c o s θ ⇒ 6 - 8 1 - c o s 2 θ = 1 + 2 c o s θ ⇒ c o s θ = 3 4 , - 1 2 ( - 1 2 is rejected since θ is acute) ∴ a : b : c = s i n A : s i n B : s i n C = s i n θ : s i n 3 θ : s i n 2 θ = 1 : 3 - 4 s i n 2 θ : 2 c o s θ = 1 : 4 c o s 2 θ - 1 : 2 c o s θ = 4 : 5 : 6