Question

The sides of a triangle ABC are $x,y,\sqrt{x^2+y^2+xy}$ respectively. The size of the greatest angle in radians is -

• A. $\frac{2\pi }{3}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. none of these

Answer 1

The correct option is A $\frac{2\pi }{3}$

Given, in a triangle ABC $a:b:c=x:y:\sqrt{x^2+y^2+xy}$

the angle opposite to largest side is the largest angle.

$\cos C=\frac{a^2+b^2-c^2}{2ab}$

$\cos C=\frac{x^2+y^2-(x^2+y^2+xy)}{2xy}=\frac{-1}{2}=\cos \frac{2\pi }{3}$

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

Hence, option A.