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In triangle A B C ,a : b : c=4:5: 6. The...

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____.

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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.`
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