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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 21, 2024

Best Answer 

 #1
avatar+1252 
+1

Let's note something really important first. 

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

 

We can find one side of the decagon using the equation \( radius/2 ( -1 + \sqrt{5})\)

 

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

 

So our answer is about 6.18. 

 

Thanks! :)

 

~NTS

 Jun 21, 2024
edited by NotThatSmart  Jun 21, 2024
 #1
avatar+1252 
+1
Best Answer

Let's note something really important first. 

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

 

We can find one side of the decagon using the equation \( radius/2 ( -1 + \sqrt{5})\)

 

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

 

So our answer is about 6.18. 

 

Thanks! :)

 

~NTS

NotThatSmart Jun 21, 2024
edited by NotThatSmart  Jun 21, 2024

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