Question
In a $$\vartriangle ABC,\left( {c + a + b} \right)\left( {a + b - c} \right) = ab.$$ The measure of $$\angle C$$ is
A.
$$\frac{\pi }{3}$$
B.
$$\frac{\pi }{6}$$
C.
$$\frac{2\pi }{3}$$
D.
None of these
Answer :
$$\frac{2\pi }{3}$$
Solution :
$$\eqalign{
& 2s\left( {2s - 2c} \right) = ab\,\,\,{\text{or, }}\frac{{s\left( {s - c} \right)}}{{ab}} = \frac{1}{4} \cr
& {\text{or, }}{\cos ^2}\frac{C}{2} = \frac{1}{4}\,\,\,{\text{or,}}\,\,\cos \frac{C}{2} = \frac{1}{2}\,\,\,\left( {\because \,\,\,\frac{C}{2}\,{\text{must be acute}}} \right). \cr} $$