Proving right isosceles triangle

Question

The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z satisfies that z=√(2xy) , and the triangle XYZ is an isosceles. (This is my answer and I want to know your comments if my proof appropriate for the problem.)

Answer 1

Given:

• Right $\triangle XYZ$
  
• Side lengths $x,y$ and hypotenuse $z$.
• $z = \sqrt{2xy}$

To Prove:

• $\triangle XYZ$ is an isosceles triangle.

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Your analysis correctly showed that $z = \sqrt{x^2 + y^2}.$

But then, your analysis showed that if the triangle is isosceles then $z = \sqrt{2xy}$.

This is the reverse of what you need to prove, which is that if $z = \sqrt{2xy}$, then the triangle is isosceles.

The actual required completion of the problem is similar to the work that you showed, trying to complete the problem.

$\sqrt{x^2 + y^2} = z = \sqrt{2xy} \implies x^2 + y^2 = 2xy \implies $

$(x - y)^2 = (x^2 + y^2 - 2xy) = 0 \implies (x - y) = 0 \implies $

$(x = y) \implies $ the triangle is isosceles.