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.
Attachment:
Pentagon circle.png [ 4.69 KiB | Viewed 2439 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 2435 times ]
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Dave de Koos
GMAT aficionado