In a $Δ A B C, (c + a + b) (a + b - c) = a b .$ The measure of $∠ C$ is

\frac{π}{3}

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\frac{π}{6}

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\frac{2 π}{3}

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none of these

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Solution

The correct option is C $\frac{2 π}{3}$
$(c + a + b) (a + b - c) = a b .$
$\Rightarrow (a + b)^{2} - c^{2} = a b \to a^{2} + b^{2} - c^{2} = - a b$ (i)
Now using cosine rule, $cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b} = - \frac{1}{2}$ using (i)
$∴ ∠ C = \frac{2 π}{3}$

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