Consider six points located at $P_{0}=(0,0),P_{1}=(0,4),P_{2}=(4,0),P_{3}=(-2,-2),P_{4}=(3,3)$ and $P_{5}=(5,5)$. Let $R$ be the region consisting of $all$ points in the plane whose distance from $P_{0}$ is smaller than that from any other $P_{i}$, where $i=1,2,3,4,5.$ what is the perimeter of the region $R?$
note- I can only say that the region covered can be a circle but here the problem arises that the region will be a trapezoid . How can I solve the given problem?(the answer is $10+7\sqrt{2}$). Thank you!