Mark the correct alternative in each of the following :

$I n a Δ A B C, i f (c + a + b) (a + b - c) = a b, then the measure of angle C is$

$\frac{π}{3}$

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$\frac{π}{6}$

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$\frac{2 π}{3}$

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$\frac{π}{2}$

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Solution

The correct option is A
$\frac{π}{3}$

$G i v e n : \Rightarrow (c + a + b) (a + b - c) = a b \Rightarrow c a + b c - c^{2} + a^{2} + a b - a c + a b + b^{2} - b c = a b \Rightarrow a^{2} + b^{2} - c^{2} + 2 a b = a b \Rightarrow a^{2} + b^{2} - c^{2} = - a b Now, c o s C = \frac{a^{2} + b^{2} - c^{2}}{2 a b} \Rightarrow c o s C = \frac{- a b}{2 a b} c o s C = \frac{- 1}{2} = c o s \frac{π}{3} ∴ a n g l e C = \frac{π}{3}$

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