ABC is an isosceles triangle with AC = BC. If \({AB}^2\) = \(2{AC}^2\), prove that ABC is a right triangle.

Solution :

Since ABC is an isosceles triangle with AC = BC and \({AB}^2\) = \(2{AC}^2\), therefore,

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

\(\implies\)  \({AB}^2\) = \({AC}^2\) + \({BC}^2\)              (because AC = BC, given)

\(\therefore\)  \(\triangle\) ABC is right angled at C.