Question

Sides of a $△$ are in ratio $1:\sqrt{3}:2$, the angles of $△$ are ratio

• A. $1:3:5$
• B. $2:3:4$
• C. $3:2:1$
• D. $1:2:3$

Answer 1

The correct option is D $1:2:3$

Ans. $\left(d\right)$ Let the sides $a,b,c$ be $1\lambda ,\sqrt{3}$ and $2\lambda$ respectively
Clearly $c^2=a^2+b^2$ $\therefore C={90}^o=\frac{\pi }{2}$
$\cos { A } =\dfrac { { b }^{ 2 }+{ c }^{ 2 }-{ a }^{ 2 } }{ 2bc } =\dfrac { \lambda ^{ 2 }\left( 3+4-1 \right) }{ \lambda ^{ 2 }
4\sqrt { 3 } } =\dfrac { \sqrt { 3 } }{ 2 } $
$\therefore A=\pi /6$
and hence $B=\frac{\pi }{2}-A=\frac{\pi }{3}$ $\therefore A:B:C=1:2:3$
Alt: By sine rule we can say that
$\sin A:\sin B:\sin C=\frac{1}{2}:\frac{\sqrt{3}}{2}:1$
$\therefore A=\frac{\pi }{6},B\frac{\pi }{3}$ and $C=\frac{\pi }{2}$
or $A:B:C=1:2:3$