If the lengths of the two equal sides of an iscoseles triangle

I think this happens when the angle between the two equal sides is 90 degrees, or a right-angle isosceles triangle.. Then by Pythagoras' Theorem, you have:

x^2 + x^2 =Hypotenuse^2

2x^2 = Hypotenuse^2 Take the sqrt of both sides

Hypotenuse =x*sqrt(2)

We can prove the guest's answer as follows :

The area, A, of the triangle can be expressed as

A  = (1/2) x^2 sin (θ )        where θ  is the included angle between the two equal sides

But....the sine is at a maximum at 90°.....so the area is maximized when θ  =  90°