Sides of a Triangle are in the ratio $4:5:6$ . If R denotes the circumradius and r denotes inradius , then find $\dfrac{R}{r}$.
To find the inradius
Calculate the semi-perimeter
semi-perimeter, S, (4 + 5 +6) / 2 = 15/2 = 7.5
Use Heron's Formula to find the Area, A
A = sqrt [ S ( S -A)(S- B) (S - C) ] = sqrt [ 7.5 * 1.5 * 2.5 * 3.5 ]
A^2 = (7.5 * 1.5 * 2.5 * 3.5) = 98.4375
Area = rS
A/S = r = A /7.5
Circumradius = product of the sides / (4 * area) = 4*5*6 / (4 * area) = 120 / (4A) = 30 / A = R
So
R /r = (30 / A)
_________ = (30 / A) ( 7.5 / A) = 225 / A^2 = 225/ 98.4375 ≈ 2.29
A / 7.5