Two sides of a tariangle are given by the roots of the equation `x^(2) -2sqrt3 x+2 =0.` The angle between the sides is `(pi)/(3).` Find the perimeter of `Delta.`

Options

A. `6+sqrt3`
B. `2sqrt3+sqrt6`
C. `2sqrt3+sqrt(10)`
D. none of these

Explanation

Let ABC be the triangle such that its sides a= BC and b =CA are the roots of the equation `x^(2) - 2sqrt3x + 2 = 0.`
`thereforea+b=2sqrt3` and ab=2
It is also given that
`angleC=pi/3`
`rArrcosC=1/2`
`rArr(a^(2)+b^(2)-c^(2))/(2ab)=1/2`
`rArra^(2)+b^(2)-c^(2)=ab`
`rArr(a+b)^(2)-c^(2)=3ab`
`rArr(2sqrt3)^(2)-c^(2)=3xx2rArrc^(2)=6rArrc=sqrt6`
`therefore` Perimeter of `Delta ABC=a+b+c=2sqrt3+sqrt6`