Mark the correct alternative in each of the following:
Question:

Mark the correct alternative in each of the following:

In a $\triangle \mathrm{ABC}$, if $(c+a+b)(a+b-c)=a b$, then the measure of angle $C$ is

(a) $\frac{\pi}{3}$

(b) $\frac{\pi}{6}$

(c) $\frac{2 \pi}{3}$

(d) $\frac{\pi}{2}$

Solution:

Given: $(c+a+b)(a+b-c)=a b$

$\Rightarrow(a+b)^{2}-c^{2}=a b$

$\Rightarrow a^{2}+b^{2}+2 a b-c^{2}=a b$

$\Rightarrow a^{2}+b^{2}-c^{2}=-a b$

$\Rightarrow \frac{a^{2}+b^{2}-c^{2}}{2 a b}=-\frac{1}{2}$

$\Rightarrow \cos C=-\frac{1}{2}=\cos \frac{2 \pi}{3} \quad$ (Using cosine rule)

$\Rightarrow C=\frac{2 \pi}{3}$

Thus, the measure of angle $C$ is $\frac{2 \pi}{3}$.

Hence, the correct answer is option (c).

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