The points $A(0,3),$ $B(4,1)$ and $C(p,q)$ lie in the plane. The isosceles triangle $\mathrm{△}ABC$ has $AB$ as its base. You are given that

1. Find the possible locations of the point $C.$

2. Find the exact area of the triangle $\mathrm{△}ABC.$

There are two possible locations for $C$ and they both lie on the perpendicular bisector of $AB.$

You are looking for the point on the perpendicular bisector which is $\sqrt{85}$ away from $A.$

For (b), there's no easy way out - once you have found $C$, you need to compute the base and height using the distance formula and use that to find the area of the triangle.