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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 13, 2024
 #1
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First, let's note something really important for the problem. 

We have \(P_1 P_2 P_3 \dotsb P_{10}\), which actually just forms a regular decagon!

\(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\)is just the perimeter of that decagon. 

 

The length of a decagon can be found with formula \(\frac{\text{radius}}{2( -1 + \sqrt {5})}\)

 

Thus, the perimeter is \(10 (1/2) ( -1 + \sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18\)

 

So our answer is approximately 6.18

 

Thanks! :)

 Jun 13, 2024

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