If the angles of a triangle are `30^(@) and 45^(@)`, and the included side is `(sqrt3 + 1)` cm, then

Options

A. `(sqrt3 + 1)/(2)` sq. units
B. `(sqrt3 + 1)` sq. units
C. `2(sqrt3 -1)` sq. units
D. `(2sqrt3 -1)/(2)` sq. units

Answer

A

Explanation

`angleC = 180^(@) -(30^(@) + 45^(@)) = 150^(@)`
By Sine Rule in `DeltaABC`, we get
`(a)/(sin 30^(@)) = (b)/(sin45^(@)) = (sqrt3+ 1)/(sin 105^(@))`
`rArr (a)/(1//2) = (b)/(1//sqrt2) = (sqrt3 + 1)/((sqrt3)/(2).(1)/(sqrt2) + (1)/(2).(1)/(sqrt2))`
`rArr(a)/(sqrt2) = (b)/(2) = 1`
`rArr a = sqrt2, b = 2`
`rArr Delta = (1)/(2) ab sin C`
`= (1)/(2) xx sqrt2 xx 2 sin 105^(@)`
`= sqrt2 ((sqrt3 + 1)/(2sqrt2))`
`= (sqrt3 + 1)/(2)` sq. units