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To construct this shape, draw a circle. Place the compass on a point on the circle and draw an arc of the same radius as the circle. Now place the compass at the intersection of the arc and the circle and draw an arc. Repeat this process until you have gone around the entire circle. Which of the following shapes is created by connecting all arc and circle intersections with a ruler?

A) Regular pentagon

B) Regular hexagon

C) Regular Octagon

D) Regular Dodecagon

Does anyone else get an octagon?

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What are your thoughts on this? Have you tried the construction? – robjohn Dec 18 '13 at 9:32
I'm trying to do it without a compass (on the actual test, we will not be allowed to use one). I tried it and got an octagon (but then again, I'm not accurate). – User69127 Dec 18 '13 at 9:43
up vote 3 down vote accepted

The answer is $B$, regular hexagon. You'll find six equilateral triangles if you connect all points by lines. Your way to construct the figures gives these equilateral triangles. Do you understand why?

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Yes I understand why...i'm trying to do it without a compass (since I won't be able to use one on the test). Yes, ridiculous if you ask me. But that's California subject matter testing for you! – User69127 Dec 18 '13 at 9:45
Well, I think first you should use compass to draw a correct figure. Then, note that you always use the same radious. Then? – mathlove Dec 18 '13 at 9:47
yes...I'm going to have to buy me one now... thanks for reassuring me the answer. – User69127 Dec 18 '13 at 9:48

Since the edges of the polygon have the same length as the radius of the circle, the angle subtended by an edge from the center of the circle is one angle of an equilateral triangle; that is, $60^\circ$.

$\hspace{3.2cm}$enter image description here

Since there are $360^\circ$ in a circle, so there are $6$ sides to the polygon.

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