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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 8, 2024

Best Answer 

 #1
avatar+759 
+1

Let's first notice something really important. 

 

The list \(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\) are just the sides of a regular decagon!

 

The sidelength of one side the decagon can be represented as \(\frac{\text{radius}}{2 ( -1 + \sqrt{5})}\)

 

Plugging in all the values we know, we have

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

 

So about 6.18 is our answer!

 

Thanks! :)

 Jun 8, 2024
 #1
avatar+759 
+1
Best Answer

Let's first notice something really important. 

 

The list \(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\) are just the sides of a regular decagon!

 

The sidelength of one side the decagon can be represented as \(\frac{\text{radius}}{2 ( -1 + \sqrt{5})}\)

 

Plugging in all the values we know, we have

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

 

So about 6.18 is our answer!

 

Thanks! :)

NotThatSmart Jun 8, 2024

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