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September 29th, 2014, 01:16 AM   #1
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Polygon of n sides PROBLEM!!

This is an interesting sum that I got from one of my friends. I don't know how to go about in this sum. please help. thanks!
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Last edited by skipjack; September 30th, 2014 at 05:42 AM.
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September 29th, 2014, 04:47 AM   #2
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That sum is difficult to read. Can you type it out please?

Also, what is your approach? How are you going to find the length of the individual line segments in the sum?

Last edited by skipjack; September 29th, 2014 at 07:38 AM.
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September 29th, 2014, 05:14 AM   #3
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Suppose A1,A2,A3-------An is an n-sided regular polygon such that:
1/A1A2 = 1/A1A3+1/A1A4. Determine the number of sides of the polygon.


no approach yet trying hard...
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Last edited by secanth; September 29th, 2014 at 05:22 AM.
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September 29th, 2014, 07:06 AM   #4
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Just to simplify, assume A1A2 has length 1. Let's write the lengths of the others as b and c so you now have 1 = 1/b + 1/c.

Now suppose you knew the value of n. How would you find the length of b (A1A3)?
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September 29th, 2014, 09:43 AM   #5
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Note that A2A3 is parallel to A1A4. If you prefer a reasonably accurate diagram, note that n turns out to be 7.
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September 29th, 2014, 10:05 AM   #6
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I would try drawing the circumcircle (assuming radius 1). I can then make the required lengths to be the hypotenuse of isoceles triangles each with shorter sides of 1 and angles between those sides of $\frac{2k\pi}{n}$ (k = 1, 2, 3).

It is then easy to work out the three required lengths in terms of $n$.
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September 29th, 2014, 07:00 PM   #7
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Polygon.GIF
The above rough diagram might make it easier to find correct expressions. If still stuck, there is a complete answer findable by searching in meritnation.com. If there's a way to do the problem without using trigonometry, I can't find it.

Last edited by skipjack; September 30th, 2014 at 07:14 AM.
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