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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

 Jun 15, 2024
 #1
avatar+1075 
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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! :)

 Jun 15, 2024

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