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Question
Mark the correct alternative in each of the following:
In a ∆ABC, if \[\left( c + a + b \right)\left( a + b - c \right) = ab\] then the measure of angle C is
Options
\[\frac{\pi}{3}\]
\[\frac{\pi}{6}\]
\[\frac{2\pi}{3}\]
\[\frac{\pi}{2}\]
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Solution
Given:
\[\left( c + a + b \right)\left( a + b - c \right) = ab\]
\[\Rightarrow \left( a + b \right)^2 - c^2 = ab\]
\[ \Rightarrow a^2 + b^2 + 2ab - c^2 = ab\]
\[ \Rightarrow a^2 + b^2 - c^2 = - ab\]
\[ \Rightarrow \frac{a^2 + b^2 - c^2}{2ab} = - \frac{1}{2}\]
\[\Rightarrow \cos C = - \frac{1}{2} = \cos\frac{2\pi}{3} \left( \text{ Using cosine rule } \right)\]
\[ \Rightarrow C = \frac{2\pi}{3}\]
Thus, the measure of angle C is \[\frac{2\pi}{3}\]
Hence, the correct answer is option (c).
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