+227 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
+140 P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1 is equal to 20.
+1908 In the problem, it wanted us to find \(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\)
Notice that this is literally just the perimeter of decagon \(P_1 P_2 P_3 \dotsb P_{10}\)
The length of one side of the decagon is
\( \text{radius}/2 ( -1 + \sqrt {5}) \)
Multiplying this by 10 and we know the perimeter. We get
\( 10 (1/2) ( -1 +\sqrt 5) = 5 ( -1 + \sqrt 5) ≈ 6.18\)
Thus, 6.18 is approximately our answer.
Thanks! :)
