Answer :


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.


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