September 29th, 2014, 12:16 AM  #1 
Newbie Joined: Sep 2014 From: india Posts: 8 Thanks: 0  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! Last edited by skipjack; September 30th, 2014 at 04:42 AM. 
September 29th, 2014, 03:47 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra 
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 06:38 AM. 
September 29th, 2014, 04:14 AM  #3 
Newbie Joined: Sep 2014 From: india Posts: 8 Thanks: 0 
Suppose A1,A2,A3An is an nsided regular polygon such that: 1/A1A2 = 1/A1A3+1/A1A4. Determine the number of sides of the polygon. no approach yet trying hard... Last edited by secanth; September 29th, 2014 at 04:22 AM. 
September 29th, 2014, 06:06 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
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)? 
September 29th, 2014, 08:43 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,820 Thanks: 2159 
Note that A2A3 is parallel to A1A4. If you prefer a reasonably accurate diagram, note that n turns out to be 7.

September 29th, 2014, 09:05 AM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra 
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 $n$. 
September 29th, 2014, 06:00 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,820 Thanks: 2159  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 06:14 AM. 

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geometry, geometryrelated, n sides, polygon, problem, sides 
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