+278 Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1
+1826 In order to solve this problem, we have to note something very important!
\(P_1 P_2 P_3 \dotsb P_{10}\) forms a regular decagon!
This means that \(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\) is just the perimeter of that regular decagon
In order to find one side length of the decagon, we can use the equation
\(\frac{\text{radius}}{2( -1 + \sqrt {5})}\)
Using information from the problem, we can easily find the perimeter.
\(10 (1/2) ( -1 + \sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18\)
So 6.18 is about our answer!
Thanks! :)
