In triangle `A B C ,a : b : c=4:5: 6.` The ratio of the radius of the circumcircle to that of the incircle is____.

Explanation

Let `a = 4k, b = 5k, c = 6k`

`s = (a+b+c)/2 = (15k)/2`

`:. Delta = sqrt(s(s-a)(s-b)(s-c)) = sqrt((15k)/2(7k)/2(5k)/2(3k)/2) = (15sqrt7k^2)/4`

`:.` Inradius `(r) = Delta/s = ((15sqrt7k^2)/4)/((15k)/2) = sqrt7/2k`


Circumradius `(R) = (abc)/(4Delta) = (120k^3)/((15sqrt7k^2)/4) = 8/sqrt7k`

`:. R:r = 8/sqrt7k:sqrt7/2k = 16:7`

So, the required ratio is `16:7.`