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# Help

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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.

Sep 20, 2018

#1
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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.

Sep 20, 2018

#1
+988
+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
#2
+105238
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Very nice, Gavin  !!!

CPhill  Sep 20, 2018
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"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
+105894
+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