In an equilateral triangle, the inradius, circumradius, and one of the exradii are in the ratio a)2:4:5      (b)  1:2:3          (c) 1:2:4      (d)  2:4:3

Options

A. `2: 4:5`
B. `1:2:3`
C. `1:2:4`
D. `2:4:3`

Answer

B

Explanation

We have
`Delta = (sqrt3)/(4) a^(2), s = (3a)/(2)`
`:. r = (Delta)/(s) = (a)/(2sqrt3), R = (abc)/(4Delta) = (a^(3))/(sqrt3 a^(2)) = (a)/(sqrt3)`
and `r_(1) = (Delta)/(s-a) = (sqrt3//4a^(2))/(a//2) = (sqrt3)/(2) a`
Hence, `r:R:r_(1) = (a)/(2sqrt3) :(a)/(sqrt3): (sqrt3)/(2) a = 1:2:3`