In an equilateral triangle, the in-radius, circum-radius and one of the ex-radii are in the ratio

Options

A. `2 : 3 : 5`
B. `1 : 2 :3`
C. `1 : 3 : 7`
D. `3 : 7 : 9`

Explanation

Let a be the iength of each side of the triangle.
We have`,Delta=(sqrt3)/4a^(2)` and `s=(3a)/2`
`thereforer=(Delta)/s=a/(2sqrt3),R=(abc)/(4Delta)=(a^(3))/(sqrt3a^(2))=a/(sqrt3)`
and `r_(1)=(Delta)/(s-a)=(sqrt3//4a^(2))/(a//2)=(sqrt3)/2a`
`thereforer : R : r_(1)=a/(2sqrt3):a/(sqrt3):(sqrt3)/2a=1:2:3`