I've attached 2 drawings:

In the left sketch you can see how the triangle changes it's shape when the vertex runs along the quarter circle.

In the right sketch you find all necessary variables

Since the radius of the quarter circle is 1 the length of the arc x is measured in radians.

with:

Plug in all variables into the equation of :

(Personal remark: The domain of this function is wrong! Probably - but I haven't a proof yet - the domain is )

To get the extreme values of A(x) calculate the first derivative of A:

Now solve for x: A'(x) = 0

I've got

With you get an isosceles right triangle with

By trial and error I got

So there you have the smallest triangles at and the largest triangles if x approaches the bounds of the domain.