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A square tile, a regular hexagon tile, and a regular n-sided polygon tile share a common vertex. There are no gaps or overlaps between the tiles. What is n?

 

The exterior angles of a k-sided polygon form an arithmetic sequence. The smallest and largest interior angles of the polygon are 136 and 176. What is k?

 Feb 3, 2019
 #1
avatar+104830 
+1

A square tile, a regular hexagon tile, and a regular n-sided polygon tile share a common vertex. There are no gaps or overlaps between the tiles. What is n?

 

The interior angle of the hexagon =120°

The interior angle of the square = 90°

So....the interior angle of the remaining regular polygon = 360 - 120 - 90  =   150“

 

So  we can solve this

 

(n - 2)180

________  =      150

     n

 

(n - 2) 180  = 150n

 

180n - 360 =  150n

 

180n - 150n = 360

 

30n =  360

 

n = 12

 

 

cool cool cool

 Feb 3, 2019
 #2
avatar+104830 
+1

The exterior angles of a k-sided polygon form an arithmetic sequence. The smallest and largest interior angles of the polygon are 136 and 176. What is k?

 

The sum of the exterior angles of any polygon = 360°

 

So....the exterior angle of the 176°   interior angle = 180 - 176 =  4°

And the exterior angle of the 136° interior angle = 180 - 136  = 44°

 

So.........using the sum for an arithmetic series we have that

 

360 = (k/2) (4 + 44)

 

360 = k  (48/2)

 

360 =  24k

 

15 = k

 

 

cool cool cool

 Feb 3, 2019

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