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Equilateral triangle ABC is inscribed within a circle as shown above.

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Bunuel
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Equilateral triangle ABC is inscribed within a circle as shown above. [#permalink] New post   06 Jun 2017, 09:51
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Equilateral triangle ABC is inscribed within a circle as shown above. If the circle has an area of 36π, what is the length of minor arc AC?

A. 3π

B. 4π

C. 5π

D. 6π

E. 9π

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naorba
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Equilateral triangle ABC is inscribed within a circle as shown above. [#permalink] New post   06 Jun 2017, 10:11
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Bunuel wrote:

Equilateral triangle ABC is inscribed within a circle as shown above. If the circle has an area of 36π, what is the length of minor arc AC?

A. 3π

B. 4π

C. 5π

D. 6π

E. 9π

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Attachment:
EquilateralCircle.png





Triangle ABC is equilateral, thus minor arcs AC, AB, and BC are equal and each is 1/3rd of the circumference (since these 3 arcs make the whole circumference).

were given that the area of the Circle is 36π, using the area formula we can derive the radius and then the circumference.

\(π*r^2\) = 36π, then the radius r is 6

using the circumference formula of a Circle
\(2*π*r = 2*π*6 = 12π\)

then minor arc AC: \(\frac{12π}{3} = 4π\)

Answer (B)

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Re: Equilateral triangle ABC is inscribed within a circle as shown above. [#permalink] New post   30 Nov 2021, 07:23
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Angle made by Minor Arc AC with the center of the circle is 120°.
(Watch this video to understand how)

Length of Minor Arc AC = \(\frac{120°}{360°}\) * 2 \(\pi\) r = \(\frac{1}{3}\) * 2 \(\pi\) r = \(\frac{2}{3}\) * \(\pi\) r

Area of Circle = \(\pi r^2\) = 36 \(\pi\) (given)
=> \(r^2\) = \(6^2\)
=> r = 6

=> Length of Minor Arc AC = \(\frac{2}{3}\) * \(\pi\) r = \(\frac{2}{3}\) * \(\pi\) 6
= 4 \(\pi\)

So, Answer will be B
Hope it helps!

Watch the following video to Learn Properties of Equilateral Triangle inscribed in a Circle

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Re: Equilateral triangle ABC is inscribed within a circle as shown above. [#permalink] New post   08 Dec 2019, 20:29
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Bunuel wrote:

Equilateral triangle ABC is inscribed within a circle as shown above. If the circle has an area of 36π, what is the length of minor arc AC?

A. 3π

B. 4π

C. 5π

D. 6π

E. 9π

Show Spoiler
Attachment:
EquilateralCircle.png


Minor arc AC is 1/3 of the circumference.

Since the area of the circle is 36π, the radius is 6, and thus, the circumference is 12π. Minor arc AC is therefore 1/3 x 12π = 4π.

Answer: B
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Re: Equilateral triangle ABC is inscribed within a circle as shown above. [#permalink] New post   29 Dec 2019, 02:11
Minor arc AC is 1/3 of the circumference
Can somebody explain this ?
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Re: Equilateral triangle ABC is inscribed within a circle as shown above. [#permalink] New post   20 Feb 2020, 03:25
Another method:
Center be o, then Oa=Ob=Oc= radius= 6 ( using pir^2=36pi).
Now we can create triangle OAB which is issocles since Oa=OB ( radius).
Now angles oab and oba =30 degree, therefore angle Aob =120 degress , this is right because angle created in the center is double of the angles created in either of the arcs, thus acb = 60 degree, using the formula
theta/360*2pir will give you 4pi
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Re: Equilateral triangle ABC is inscribed within a circle as shown above. [#permalink]
20 Feb 2020, 03:25
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