Question

If the sides of the triangle are $ p,q,\sqrt{{{p}^{2}}+{{q}^{2}}+pq}, $ then the greatest angle is:

• A. $ \frac{\pi }{2} $
• B. $ \frac{5\pi }{4} $
• C. $ \frac{2\pi }{3} $
• D. $ \frac{7\pi }{4} $
• E. $ \frac{5\pi }{3} $

Answer 1

Answer: (C)

The longest side of triangle is $ \sqrt{{{p}^{2}}+{{q}^{2}}+pq} $ .
Greatest angle will be opposite to longest side. Let $ \theta $ be greatest angle, then $ \cos \theta =\frac{{{p}^{2}}+{{q}^{2}}-{{p}^{2}}-{{q}^{2}}-pq}{2pq}=-\frac{1}{2} $
$ \Rightarrow $ $ \cos \theta =\cos \frac{2\pi }{3} $
$ \Rightarrow $ $ \theta =\frac{2\pi }{3} $