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

BRAINBOLT Jun 15, 2024

#1**+1 **

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! :)

NotThatSmart Jun 15, 2024