Answer
$(a)$
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
$(b)$
$a\approx 121.6$
$(c)$ In this kind of case there could be either two, one or no triangle with given sides and angles.
Work Step by Step
$(a)$ According to the Law of Sines, in any triangle ABC we have :
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
$(b)$ We can find $a$ using Law of Sines. In this case we have :
$\frac{\sin 85°}{a}=\frac{\sin 40°}{b}=\frac{\sin C}{100}$
If we find value of $C$ we can find $a$ too. We know that in any triangle, sum of all angles equals to $180°$, so :
$C=180°-(85°+40°)=55°$
$\frac{\sin 85°}{a}=\frac{\sin 55°}{100}$
$a=\frac{100 \sin 85°}{\sin 55°}$
$a\approx 121.6$
$(c)$ In this kind of case there could be either two, one or no triangle with given sides and angles.
