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The Pythagorean equation relates the sides of a right triangle in a simple way, so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum.

Given:

AC = BC

ABC is an isosceles triangle

\[A{B^{2\;}} = {\text{ }}2{\text{ }}A{C^2}\]

Consider a right angled triangle ABC as shown in the figure

For ABC to be a right triangle, it should satisfy Pythagoras theorem

i.e \[

A{B^{2\;}} = {\text{ }}A{C^{2\;}} + {\text{ A}}{C^2} \\

\\ \] \[A{B^{2\;}} = {\text{ }}A{C^{2\;}} + {\text{ }}B{C^2}\]

Now, AC = BC (given)………………(1)

On substitution, we find

\[ \Rightarrow \]\[A{B^{2\;}} = {\text{ }}2{\text{ }}A{C^2}\]

\[ \Rightarrow \]\[A{B^{2\;}} = {\text{ }}A{C^{2\;}} + {\text{ A}}{C^2}\]

\[ \Rightarrow \]\[A{B^{2\;}} = {\text{ }}A{C^{2\;}} + {\text{ }}B{C^2}\]

Here AB is the largest side ,i.e Hypotenuse of triangle ABC.

So, the given sides form Pythagorean triplets.

Hence ABC to be a right triangle right angled at C.

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