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If n regular pentagons are tangent each other in points of a

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If n regular pentagons are tangent each other in points of a

by Max@Math Revolution » Tue Feb 16, 2016 5:40 pm
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If n regular pentagons are tangent each other in points of a circle as above figure, n=?

A. 8
B. 9
C. 10
D. 11
E. 12


* A solution will be posted in two days.
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by Max@Math Revolution » Sat Feb 20, 2016 10:22 pm
If n regular pentagons are tangent each other in points of a circle as above figure, n=?

A. 8
B. 9
C. 10
D. 11
E. 12

Image


Just like the picture above, there is a regular pentagon ABCDE. If you extend two lines, it becomes like the picture above. However, since an interior angle of one regular pentagon is 180(5-2)/5=108, angle A=E=B=108 and angle A+E+B=324. Since sum of interior angles of a rectangle ABFE is 360, angle F is 360-324=36. Then, angle F is a central angle, which is 360/36=10. Therefore, the answer is C.
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