Question

Find the number of $\Delta$'s that could be constructed by the measures $A={30}^{\circ },a=7,b=8$

Answer 1

Using cosine rule,
$\cos A=\cos {60}^0=\frac{b^2+c^2-a^2}{2bc}=\frac{15+c^2}{16c}=\frac{\sqrt{3}}{2}$
$\Rightarrow c^2-8\sqrt{3}c+15=0$
Discriminant of above quadratic is, $\Rightarrow D=64\times 3-60>0$
Hence there will be two real values of $c$.
So there can be two triangle possible with the given set of information.