# How to calcultate perimeter of region R such that it contains the given points?

Consider $$6$$ points located at $$P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)$$. Let $$R$$ be the region consisting of all points in the plane whose distance from $$P_0$$ is smaller than that from any other $$P_i$$, $$i=1,2,3,4,5$$. Find the perimeter of the region $$R$$.

I thought of calculating the the circumference of circle with lowest radius from the given points, but apparently it doesn't gives the answer. Where am I going wrong?

• If you want to contain all the points, you need to use the circle with the largest radius (rather than smallest radius). Mar 23 '19 at 15:42
• @JacobJones that doesn't give the answer. The region formed will be a trapezium, but I don't know how they got that. Mar 23 '19 at 15:50
• I misread the question. Start by finding the midpoint between $P_0$ and the other points. I believe these midpoints will form the shape containing the points you are interested in. Mar 23 '19 at 15:53
• @JacobJones That doesn't do too. Mar 23 '19 at 16:00
• You're right, I didn't think about that enough. I believe this solves it though. Find the midpoints between $P_0$ and the other points. At that midpoint, draw a line perpendicular to the line from $P_0$ to the corresponding point. The shape traced by these perpendicular lines contains all points closer to $P_0$ than to another $P_i$. Mar 23 '19 at 16:09

Find the midpoints between $$P_0$$ and the other points (dark blue in the image). At that midpoint, draw a line perpendicular to the line from $$P_0$$ to the corresponding point. The shape traced by these perpendicular lines contains all points closer to $$P_0$$ than to another $$P_i$$ (the light green shaded region). 