Question

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

Answer 1

As we observe that the greatest side of the given triangle is $\sqrt{2}$.

In given right angled triangle, say $a=x,b=\sqrt{2}x,c=x$ then

Using Pythagoras theorem,

$a^2+c^2=b^2$

$LHS=a^2+c^2=x^2+x^2=2x^{2}unit,$

$RHS=b^2=(\sqrt{2}x{)}^2=2x^{2}unit,$

$LHS=RHS$

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