If in any triangle, the angles are in the ratio of 1 : 2 : 1, then what will be the ratio of its sides?

$1 : \sqrt{2} : 1$

Ratio of angles of triangle,

1 : 2 : 1

Ratio = a : 2a : a

According to the question,

⇒ a + 2a + a = \({180}^\circ\)

⇒ 4a = \({180}^\circ\)

⇒ a = \({45}^\circ\)

⇒ 2a = \({90}^\circ\)

Therefore, it is a right angled triangle,

Let AB + BC = b cm

⇒ \( { (AC)}^{2 } \) = \( { (AB)}^{2 } \) + \( { (BC)}^{2 } \)

⇒ \( { (AC)}^{2 } \) = \( { 2b}^{2 } \)

⇒ AC = b\(\sqrt {2 }\)

⇒ Ratio of AB : AC : BC = b :  b\(\sqrt {2 }\) : b = 1 : \(\sqrt {2 }\) : 1