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16 Feb 2016, 18:38
Kudos
Bookmarks
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
51% (02:11) correct
49%
(01:43)
wrong
based on 181
sessions
History
Date
Time
Result
Not Attempted Yet
Attachment:
pentagons.jpg [ 5.39 KiB | Viewed 6367 times ]
A. 8
B. 9
C. 10
D. 11
E. 12
* A solution will be posted in two days.
19 Feb 2016, 14:01
Kudos
Bookmarks
I'm going to assume that the question implies that the pentagons go all the way around the circle and connect back with each other forming a closed loop of pentagons. (If the drawing were to depict this, then it would either give the answer away or be very misleading...)
Imagine if the sides of the pentagon that cross the circle were extended down to the center of the circle. Then each pentagon would be in one sector of the circle. If we could determine the central angle of the sector, we would know how many pentagons it would take to complete the circle. See the diagram below.
Pentagon circle.png [ 4.69 KiB | Viewed 6244 times ]
In a regular polygon, the measure of the interior angles is [180*(n-2)]/n, where n the number of sides of the polygon. For a pentagon, the measure of the interior angles is [180*(5-2)]/5 = 108.
If each interior angle is 108, then the exterior angles, y = 180-108 = 72
The measure of angle x is therefore 180-2*72 = 36
How many sectors of \(36^{\circ}\) will it take to complete a circle?
360/36 = 10
Answer: C
Pentagon circle 2.png [ 7.96 KiB | Viewed 6227 times ]
Imagine if the sides of the pentagon that cross the circle were extended down to the center of the circle. Then each pentagon would be in one sector of the circle. If we could determine the central angle of the sector, we would know how many pentagons it would take to complete the circle. See the diagram below.
Attachment:
Pentagon circle.png [ 4.69 KiB | Viewed 6244 times ]
In a regular polygon, the measure of the interior angles is [180*(n-2)]/n, where n the number of sides of the polygon. For a pentagon, the measure of the interior angles is [180*(5-2)]/5 = 108.
If each interior angle is 108, then the exterior angles, y = 180-108 = 72
The measure of angle x is therefore 180-2*72 = 36
How many sectors of \(36^{\circ}\) will it take to complete a circle?
360/36 = 10
Answer: C
Attachment:
Pentagon circle 2.png [ 7.96 KiB | Viewed 6227 times ]
General Discussion
20 Feb 2016, 23:22
Kudos
Bookmarks
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
pentagon.jpg [ 5.87 KiB | Viewed 6132 times ]
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.
A. 8
B. 9
C. 10
D. 11
E. 12
Attachment:
pentagon.jpg [ 5.87 KiB | Viewed 6132 times ]
