+1222 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
+1910 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! :)
