Question

The sides of a triangle have lengths a²+b², 2ab, a²–b², where a > b and a, b ∈ R⁺. Prove that the angle opposite to the side having length a²+b² is a right angle.

Answer 1

If we assume that the angle opposite to the side having length a²+b² is a right angle.

Then a²+b² is the hypotenuse.

therefore, (a²+b²)² = (2ab)² + (a²–b²)²

or, LHS = $a^{4}$ + 2a²b² + $b^{4}$

or, RHS = 4a²b² + $a^{4}$ - 2a²b² + $b^{4}$

or, RHS = $a^{4}$ + 2a²b² + $b^{4}$ = LHS. [Proved]