BTGmoderatorDC wrote:
In the figure above, an equilateral triangle is inscribed in a circle. If the arc bounded by adjacent corners of the triangle is between
4Ï€ and
6Ï€ long, which of the following could be the diameter of the circle?
(A) 6.5
(B) 9
(C) 11.9
(D) 15
(E) 23.5
Source: Manhattan Prep
$$?\,\,\,:\,\,\,D = 2r\,\,\,\,\underline {{\rm{could}}} \,\,\,{\rm{be}}$$
$$4\pi \,\,\, < \,\,\,{\rm{arc}}\,\,{\rm{length}}\,\,{\rm{ = }}\,\,{1 \over 3}\left( {2\pi r} \right)\,\,\, < \,\,\,6\pi \,\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,\pi \,\,\,{\rm{and}}\,\,\, \cdot \,\,3} \,\,\,\,\,4 \cdot 3\,\,\, < \,\,2r\,\, < \,\,6 \cdot 3\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left( D \right)$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
