In any triangle, if the angles are in the ratio 1 : 2 : 3, then what will be the ratio of the sides opposite to them?

$1 : \sqrt{3} : 2$

Formula to be used here,

\(\frac{a}{sin A}\) = \(\frac{b}{sin B}\) = \(\frac{c}{sin C}\)

⇒ Let A = p, B = 2p, C = 3p

⇒ \(\angle\)A + \(\angle\)B + \(\angle\)C = \({180}^\circ\)

⇒ p + 2p + 3p = \({180}^\circ\)

⇒ 6p = \({180}^\circ\)

⇒ p = \({30}^\circ\)

\(\angle\)A = \({30}^\circ\),  \(\angle\)B = \({60}^\circ\),  \(\angle\)C = \({90}^\circ\)

Now,

\(\frac{a}{sin 30}\) = \(\frac{b}{sin 60}\) = \(\frac{c}{sin 90}\)

⇒ a = 1, b = \(\sqrt {3 }\), C = 2,

⇒ The ratio a, b and c = 1 : \(\sqrt {3 }\) : 2.