With usual notations if the angles of a triangle are in the ratio $1: 2: 3$, then their corresponding sides are in the ratio.

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Aicharya · Accepted Answer

Let the angles be $\mathrm{x}, 2 \mathrm{x} 3 \mathrm{x}$ $	herefore \mathrm{x}+2 \mathrm{x}+3 \mathrm{x}=180^{\circ} \Rightarrow 6 \mathrm{x}=180^{\circ} \Rightarrow \mathrm{x}=30^{\circ}$ Thus angles of the triangle are $30^{\circ}, 60^{\circ}, 90^{\circ}$. $\begin{aligned} & 	ext { Now, } \frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}}=\frac{\mathrm{c}}{\sin \mathrm{C}} \ & 	herefore \frac{\mathrm{a}}{\sin 30^{\circ}}=\frac{\mathrm{b}}{\sin 60^{\circ}}=\frac{\mathrm{c}}{\sin 90^{\circ}} \Rightarrow \frac{\mathrm{a}}{\left(\frac{1}{2}ight)}=\frac{\mathrm{b}}{\left(\frac{\sqrt{3}}{2}ight)}=\frac{\mathrm{c}}{(1)} \ & 	herefore 2 \mathrm{a}=\frac{2 \mathrm{~b}}{\sqrt{3}}=\mathrm{c} \Rightarrow=\frac{\mathrm{c}}{2} 	ext { and } \mathrm{b}=\frac{\sqrt{3} \mathrm{c}}{2} \ & 	herefore \mathrm{a}: \mathrm{b}: \mathrm{c}=\frac{\mathrm{c}}{2}: \frac{\sqrt{3} \mathrm{c}}{2}: \mathrm{c}=1: \sqrt{3}: 2 \end{aligned}$

With usual notations if the angles of a triangle are in the ratio $1: 2: 3$, then their corresponding sides…

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