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

Options

A. area of the triangle is `(1)/() (sqrt3 + 1)` sq. units
B. area of the triangle is `(1)/(2) (sqrt3 -1)` sq. units
C. ratio of greater side to smaller side is `(sqrt3 + 1)/(sqrt2)`
D. ratio of greater side to smaller side is `(1)/(4sqrt3)`

Answer

A, C

Explanation

We have `(sin A)/(a) = (sin B)/(b) = (sinC)/(c)`
or `(sin 105^(@))/(sqrt3 +1) = (sin 30^(@))/(b) = (sin 45^(@))/(c)`
or `b = ((sqrt3+1))/(2 sin 105^(@)), c = (sqrt3+1)/(sqrt2 sin 105^(@))`

Area of `DeltaABC = (1)/(2) bc sin A = (1)/(2) bc sin 105^(@)`
`= (1)/(2) ((sqrt3+1)^(2))/(2 sqrt2 . sin^(2) 105^(@)) sin 105^(@) = (sqrt3 +1)/(2)`
Also, `(a)/(b) = (sin 105^(@))/(sin 30^(@)) = (sqrt3 +1)/(sqrt2)`