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

jaekg Oct 10, 2024

#1**0 **

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.

AUnVerifedTaxPayer Oct 10, 2024

#2**+1 **

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! :)

NotThatSmart Oct 12, 2024