If the angles of a triangle are in the ratio $1: 2: 3$, the corresponding sides are in the ratio
- $2: \sqrt{3}: 1$
- $1: \sqrt{3}: 2$
- $1: 2: 3$
- $\sqrt{3}: 2: 1$
Solution
$ \angle A: \angle B: \angle C=1: 2: 3 $ Let the ratio be ' $x$ ', then $ \angle A=x, \angle B=2 x, \angle C=3 x $ Using angle sum property, $ \begin{aligned} & \angle A+\angle B+\angle C=180^{\circ} \\ \Rightarrow & x+2 x+3 x=180 \Rightarrow x=30 \\ \therefore \quad & \angle A=30, \angle B=60, \angle C=90^{\circ} \end{aligned} $
Using sine law of triangles, $ \begin{aligned} & \quad \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \\ \Rightarrow \quad \frac{\sin 30^{\circ}}{a} & =\frac{\sin 60^{\circ}}{b}=\frac{\sin 90^{\circ}}{c} \\ \Rightarrow \quad & \frac{1}{2 a}=\frac{\sqrt{3}}{2 b}=\frac{1}{c} \Rightarrow \frac{1}{a}=\frac{\sqrt{3}}{b}=\frac{2}{c} \\ \Rightarrow \quad & a: b: c=1: \sqrt{3}: 2 \end{aligned} $
Asked in: AP EAMCET 2021 (23 Aug Shift 1)