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# Geometry

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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

#1
+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

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

Thanks! :)

Jun 8, 2024

#1
+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

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