If the sides of a triangle are in the ratio 1 : $\sqrt{2}$ : 1, show that it is a right-angled triangle.

Let ABC be the triangle.

Given,

Sides of a triangle are in the ratio 1 : $\sqrt{2}$ : 1.

Let AB = x, BC = $\sqrt{2}x$ and AC = x.

Squaring both sides we get :

AB² = x², BC² = 2x² and AC² = x².

⇒ AB² + AC² = x² + x² = 2x² = BC².

Since,

⇒ BC² = AB² + AC².

∴ Triangle ABC satisfies pythagoras theorem.

Hence, proved that ABC is a right-angled triangle.