If the angles of a triangle are in the ratio `4:1:1,` then the ratio of the longest side to the perimeter is (a)`sqrt(3):(2+sqrt(3))` (b) `1:6` (c)`1:2+sqrt(3)` (d) `2:3`

Options

A. `sqrt3 : (2 + sqrt3)`
B. `1 : 6`
C. `1 : 2 + sqrt3`
D. `2 : 3`

Explanation

Given that `4A + A + A = 180^(@)`
or `A = 30^(@)`
Angle are `120^(@), 30^(@), 30^(@)`
`rArr (sin 120^(@))/(a) - (sin 30^(@))/(b) = (sin 30^(@))/(c) = 2R` (say)
`rArr (a)/(a + b + c) = (sin 120^(@))/(sin 120^(@) + sin 30^(@) + sin 30^(@)) = (sqrt3)/(2 + sqrt3)`