+1290 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
+953 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
+953 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
