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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, 15:02
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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, 18:22
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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, 10:43
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1
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, 13:23
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.

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

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New post 06 Jan 2019, 13:14
Hi All,

We're told that in the figure above, an EQUILATERAL triangle is inscribed in a circle and the arc bounded by adjacent corners of the triangle is between 4π and 6π long. We're asked which of the following COULD be the diameter of the circle. As scary as this question might look, it's based on a couple of standard Geometry rules, so you can answer it with just a little work.

To start, an equilateral triangle has 3 equal angles - and since the triangle is inscribed in the circle, each of the three 'arc pieces' is equal in length. Thus, the total of those three arcs (re: the circumference of the circle) is between (3)(4π) and (3)(6π). If the total circumference is between 12π and 18π, then we can 'work backwards' to find the possible diameter....

12π = 2π(R) = πD..... Diameter = 12
18π = 2π(R) = πD..... Diameter = 18

There's only one answer that's between 12 and 18...

Final Answer:

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

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New post 23 Mar 2019, 00:47
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


step 1:
Sum of the 2 angles formed by a chord at the circumference in the two opposite segments of a circle is equal to 180 degrees.
60+x=180
ie., x=120
step 2:
length of an arc=(x/360)*2πr
Take length of arc=5π( as it is between 4π and 5π)
5π=(120/360)*2π*r
15π=2πr
15=2r=D

That's it...
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Re: An equilateral triangle is inscribed in a circle of diameter D.   [#permalink] 23 Mar 2019, 00:47
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