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An equilateral triangle is inscribed in a circle of diameter D.

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An equilateral triangle is inscribed in a circle of diameter D.  [#permalink]

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New post 05 Apr 2017, 14:02
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An equilateral triangle is inscribed in a circle of diameter D. If the length of the arc bounded by the adjacent corners of the triangle is between 4π and 6π, then which of the following could be the value of D?

A)6.5
B)9
C)11.9
D)15
E)23.5

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Re: An equilateral triangle is inscribed in a circle of diameter D.  [#permalink]

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New post 05 Apr 2017, 17:22
The three arcs bounded by the three corners of the equilateral triangle together constitute the entire circle hence the combined lengths of the 3 arcs is equal to the circumference of the circle.

It is given that length of each arc is between 4π and 6π, so the total length of the 3 arcs or the circumference of the circle will be between 12π and 18π.

From the choices only D)15 works out and hence is the answer.
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Re: An equilateral triangle is inscribed in a circle of diameter D.  [#permalink]

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New post 02 Mar 2018, 09:43
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stonecold wrote:
An equilateral triangle is inscribed in a circle of diameter D. If the length of the arc bounded by the adjacent corners of the triangle is between 4π and 6π, then which of the following could be the value of D?

A)6.5
B)9
C)11.9
D)15
E)23.5


Since there are 3 arcs bounded by the adjacent corners of an equilateral triangle, and since the lengths of these 3 arcs are equal, the circumference of the circle will be between 3 x 4π = 12π and 3 x 6π = 18π. Therefore, the diameter of the circle will be between 12π/π = 12 and 18π/π = 18. So 15 could be the value of the D.

Answer: D
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Re: An equilateral triangle is inscribed in a circle of diameter D.  [#permalink]

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New post 28 Oct 2018, 12:23
stonecold wrote:
An equilateral triangle is inscribed in a circle of diameter D. If the length of the arc bounded by the adjacent corners of the triangle is between 4π and 6π, then which of the following could be the value of D?

A)6.5
B)9
C)11.9
D)15
E)23.5

\(?\,\,\,:\,\,\,D = 2r\,\,\underline {{\rm{could}}} \,\,{\rm{be}}\)

\(4\pi \,\,\, < \,\,\,{\rm{arc}}\,\,{\rm{length}}\,\,{\rm{ = }}\,\,{1 \over 3}\left( {2\pi r} \right)\,\,\, < \,\,\,6\pi \,\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,\pi \,\,\,{\rm{and}}\,\,\, \cdot \,\,3} \,\,\,\,\,4 \cdot 3\,\,\, < \,\,2r\,\, < \,\,6 \cdot 3\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left( D \right)\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: An equilateral triangle is inscribed in a circle of diameter D. &nbs [#permalink] 28 Oct 2018, 12:23
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