Let a , b and c be the length of sides of a triangle A B C such that a + b 7 = b + c 8 = c + a 9 . If r and R are the radius of incircle and radius of circumcircle of the triangle A B C , respectively, then the value of R r is equal to

Step-by-step solution

Let a + b 7 = b + c 8 = c + a 9 = k so a + b = 7 k ,   b + c = 8 k ,   c + a = 9 k Solving this we get a = 4 k ,   b = 3 k ,   c = 5 k and calculating semi-perimeter S = a + b + c 2 ⇒ S = 6 k We know that r R = 4 sin A 2 sin B 2 sin C 2 ⇒    r R = 4 s - b s - c b c × s - a s - c a c × s - a s - b a b ⇒ r R = 4 · s - a s - b s - c a b c ⇒ r R = 4 · 2 k 3 k k 4 k 3 k 5 k ⇒ R r = 5 2