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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

 Aug 15, 2024
 #1
avatar+1926 
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mmm...take a look at the points given. 

\(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\)

 

This is essentially just the perimeter of the decagon \(P_1P_2P_3P_4P_5...P_{10}\)

 

The lenngth of one side is essentially

\(radius/2 ( -1 +\sqrt 5)\)

 

Now that we know one side, we can find the perimeter through the equation

\(10 (1/2) ( -1 + \sqrt 5) = 5 ( -1 + \sqrt 5) ≈ 6.18\)

 

Thus, the answer is 6.18, 

 

Thanks! :)

 Aug 15, 2024
edited by NotThatSmart  Aug 15, 2024

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