Anlges of a triangle are in 4:1:1 ratio .The reatio between its greatest side and perimeter is

Options

A. `(3)/(2+sqrt(3)`
B. `(sqrt(3))/(2+sqrt(3))`
C. `(sqrt(3))/(2-sqrt(3))`
D. `(1)/(2+sqrt(3))`

Explanation

Angles are in ratio 4:1:1
`rArr " angles are " 120^@,30^@,30^@`
If sides opposite to these angles are a,b,c respectively , then a will be the gratest side .Now from
sine formula `(a)/(sin 120^@) =(b)/(sin30^@)= (c ) /(sin 30 ^@)`
`rArr a/(sqrt(3)//2)=b/(1//2)=c/(1//2)`
`rArr a/sqrt(3)=b/1=c/1= k (say ) `
than `a= sqrt(3K), ` perimeter `= (2+sqrt(3))k`
`therefore " required ratio" = (sqrt(3)k)/((2+sqrt(3))k)=(sqrt(3))/(2+sqrt(3))`