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find all groups of three regular polygons with side length one that can surround one point such that each polygon shares a side with the other two. 

supermanaccz  Sep 20, 2018

Best Answer 

 #1
avatar+970 
+5

Let there be three regular polygons with \(x,y,\) and \(z\) as their number of sides. Their internal angles must sum to \(360º\) and individually be supplement to their respective external angle. Therefore, we can express the sum of the three angles as:

 

\((180º-\frac{360º}{x})+(180º-\frac{360º}{y})+(180º-\frac{360º}{z})=360º\\ 540º-\frac{360º}{x}-\frac{360º}{y}-\frac{360º}{z}=360º\\ 180º=\frac{360º}{x}+\frac{360º}{y}+\frac{360º}{z}\\ \frac2x+\frac2y+\frac2z=1\) 

 

\(3\le x \le 6\), because three sides is the least number of sides a regular polygon can have, and if \(x,y,\) or \(z\) is greater than 6, the sum will be less than one. 

 

Casework: 

 

\(x=6\Rightarrow y=6, z=6\\ x=5\Rightarrow y=5, z=10\\ x=4\Rightarrow (8,8);(6,12);(5,20)\\ x=3\Rightarrow (10,15);(9,18);(8,24);(7,42)\\\)

 

Therefore, there are 9 groups of regular polygons that can surrond a point.

 

I hope this helped,

 

Gavin. 

GYanggg  Sep 20, 2018
 #1
avatar+970 
+5
Best Answer

Let there be three regular polygons with \(x,y,\) and \(z\) as their number of sides. Their internal angles must sum to \(360º\) and individually be supplement to their respective external angle. Therefore, we can express the sum of the three angles as:

 

\((180º-\frac{360º}{x})+(180º-\frac{360º}{y})+(180º-\frac{360º}{z})=360º\\ 540º-\frac{360º}{x}-\frac{360º}{y}-\frac{360º}{z}=360º\\ 180º=\frac{360º}{x}+\frac{360º}{y}+\frac{360º}{z}\\ \frac2x+\frac2y+\frac2z=1\) 

 

\(3\le x \le 6\), because three sides is the least number of sides a regular polygon can have, and if \(x,y,\) or \(z\) is greater than 6, the sum will be less than one. 

 

Casework: 

 

\(x=6\Rightarrow y=6, z=6\\ x=5\Rightarrow y=5, z=10\\ x=4\Rightarrow (8,8);(6,12);(5,20)\\ x=3\Rightarrow (10,15);(9,18);(8,24);(7,42)\\\)

 

Therefore, there are 9 groups of regular polygons that can surrond a point.

 

I hope this helped,

 

Gavin. 

GYanggg  Sep 20, 2018
 #2
avatar+92787 
+2

Very nice, Gavin  !!!

 

cool cool cool

CPhill  Sep 20, 2018
 #3
avatar
+1

"Their internal angles must sum to 360"

 

why?

 

 

EDIT: thanks for the diagram melody :) now i understand. I thought all polygonals must surround a specific point, like a point contained within a circle. 

Guest Sep 22, 2018
edited by Guest  Sep 22, 2018
 #4
avatar+94117 
+1

I'm impressed too Gavin :)

 

Here is one of the answers Gavin found.     

 

The angles he is talking about have to be angles at a point as I will show in this diagram.

 

Melody  Sep 22, 2018
edited by Melody  Sep 22, 2018

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