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

`a = 4k, b = 5k, c = 6k`
`s = (15)/(2) k`,
`s - a = (7)/(2) k`,
`s - b = (5)/(2) k`,
`s - c = (3)/(2) k`
`rArr Delta^(2) = 15 xx 7 xx 5 xx 3 ((k)/(2))^(4)`
or `Delta = 15 sqrt7 ((k)/(2))^(2)`
Now, `r = (Delta)/(s) = (15 sqrt7 (k//2)^(2))/((15//2) k) = sqrt7 (k)/(2)`
and `R = (abc)/(4 Delta) = (4 xx 5 xx 6 xx k^(3))/(4 xx 15 sqrt7 xx (k^(2)//4)) = (8)/(sqrt7) k`
`:. (R)/(r) = (16)/(7)`