Question

The sides of a triangle are in the ratio $1:\sqrt{3}:2$.The angles are in the ratio

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

Answer 1

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

$a=1,b=\sqrt{3},c=2$

$\Rightarrow a^2+b^2=c^2$from above values of $a,b,c$

$\therefore \angle C={90}^{\circ }$

$\Rightarrow \frac{1}{2}=\frac{a}{c}=\frac{\sin A}{\sin C}$

$\Rightarrow \sin A=\frac{1}{2}$ since $\angle C={90}^{\circ }$ and $\sin \angle C=\sin {90}^{\circ }=1$

$\Rightarrow \angle A={30}^{\circ }$

By Angle sum property, $\angle A+\angle B+\angle C={180}^{\circ }$

$\Rightarrow \angle B={180}^{\circ }-\angle A-\angle C$

$\Rightarrow \angle B={180}^{\circ }-{30}^{\circ }-{90}^{\circ }$

$\Rightarrow \angle B={180}^{\circ }-{120}^{\circ }={60}^{\circ }$

$\therefore \angle A:\angle B:\angle C={30}^{\circ }:{60}^{\circ }:{90}^{\circ }$

$\therefore \angle A:\angle B:\angle C=1:2:3$