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

Let the angle of `Delta ABC` be 4x , x and x. Then,
`4x + x + x = 180° rArr x = 30^(@)`
So, the angles are A = 120°, B = 30° and C = 30°
Now,
`a/(sinA)=b/(sinB)=c/(sinC)=k`
`rArra=(sqrt3)/2k,b=1/2k` and `c=1/2k`
`therefore` Required ratio `=a/(a+b+c)=(sqrt3)/(sqrt3+2)=sqrt3:2+sqrt3`