Note: In any ΔABC we define ‘a’ as the length of the side opposite to ∠A, ‘b’ as the length of the side opposite to ∠B and ‘c’ as the length of the side opposite to ∠C.

The key point to solve the problem:

The idea of cosine formula in ΔABC

• Cos A =

• Cos B =

• Cos C =

As we have to prove under given conditions.

∵ Only cos terms are involved so we will apply cosine formula to find cos A , cos B, and cos C and we will take their ratio.

∵

∴ b + c = 12k ….eqn 1

c + a = 13k ….eqn 2

a + b = 15k ….eqn 3

But only above relation is not sufficient to find cosines as k is unknown, either we need to express k in terms of a , b or c or express a , b , c in terms of k. Later part is easier.

∴ we will find a,b,c in terms of k

Adding eqn 1,2 and 3 we have –

2 (a + b + c) = 40k

∴ a + b+ c = 20k

∴ a = 20k – (b + c) = 20k – 12k = 8k

Similarly, b = 20k – (c + a) = 20k – 13k = 7k

And c = 20k – (a + b) = 20k – 15k = 5k

Hence,

Cos A =

Cos B =

cos C =

∴

∴ ….Hence proved.