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geometry question

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geometry question

by EddieU » Mon May 30, 2011 1:27 pm
Hi, hopefully someone can help me out with this question:

In the figure attached, equilateral triangle ABC is inscribed in the circle. If the length of arc ABC is 24, what is the approximate diameter of the circle?

Thanks in advance.
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equilateral triangle in circle
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by SoCan » Mon May 30, 2011 1:44 pm
The vertices of an equilateral triangle divide the circumference into 3 equal arcs. Therefore, the arc ABC is 2/3 of the circumference. 24/(2/3) = 36.

This is kind of tricky, because we're used to seeing circumference expressed in terms of pi. If I were going too quickly, and 36 were an option, I wouldn't be surprised if I picked that and moved on. Luckily it's not an answer, so it's easy to realize that you need to divide 36 by pi. pi~3, 36/3 = 12. Since pi is slightly larger than 3, we know the answer must be slightly smaller than 12, so the answer is 11.
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