The condition on vertex $C$ is that it lies on the line $y=x$.
The area is known, and the length of the base $AB$ is known. So the height $h$ with respect to this base is known.
Find the equation of the line $AB$. I am sure that you know how to do this. Suppose that this equation turns out to be $ax+by+c=0$.
Since our point $C$ lies on the line $y=x$, we can suppose that its coordinates are $(t,t)$. We will use the formula for the perpendicular distance from a point $(u,v)$ to the line $ax+by+c=0$. It is a standard fact that this distance is equal to
We know $a$, $b$, $c$, and the distance $h$.
So we need to solve the following equation for $t$:
Or, removing absolute value signs, we need to solve the two linear equations
Remark: The idea is easily extended to solving similar problems, where $C$ lies on any line with given equation, like $2x+3y+4=0$ instead of $y=x$.