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555-605 Level|   Geometry|      
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banksy
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Equilateral triangle is inscribed in a circle. If length of arc ABC is Updated on: 04 Dec 2019, 00:57
Originally posted by banksy on 05 Mar 2011, 08:50.
Last edited by Bunuel on 04 Dec 2019, 00:57, edited 2 times in total.
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B
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35% (medium)

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68% (01:19) correct 32% (02:00) wrong based on 102 sessions

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Equilateral triangle is inscribed in a circle. If length of arc ABC is 2pi, what is the radius of the circle?

(A) 1
(B) 3/2
(C) 2
(D) 5/3
(E) 3
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B
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Equilateral triangle is inscribed in a circle. If length of arc ABC is 05 Mar 2011, 09:16
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banksy
164 Equilateral triangle is inscribed in a circle. If length of arc ABC is 2pi, what is the radius of the circle?
(A) 1
(B) 3/2
(C) 2
(D) 5/3
(E) 3

Arc ABC is \(\frac{2}{3}\) of the circumference (as ABC is equilateral triangle then (arc AB)=(arc BC)=(arc AC), so (arc AB)+(arc BC)=(arc ABC)=2/3 of circumference, so \(arc \ ABC=2\pi\) basically means that the circumference is \(3\pi\) --> \(circumference=2\pi{r}=3\pi\) --> \(r=\frac{3}{2}\).

Answer: B.

Similar question: geometry-circle-triangle-from-mba-com-97393.html
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Equilateral triangle is inscribed in a circle. If length of arc ABC is Updated on: 05 Mar 2011, 09:51
Originally posted by banksy on 05 Mar 2011, 09:28.
Last edited by banksy on 05 Mar 2011, 09:51, edited 1 time in total.
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Yep, thanks Bunuel.
But I don't understand why it is not possible to use another method to solve it....
it drives me crazy cos I don't understand what is wrong with it...please look..

I used formula:
Circumference of sector= (angle in front of the sector/ 360) * 2*pi*R

so if we know that ABC equilateral, so all angles=60 and all circumferences in front of A ,B and C are same. so if circumference in front of 2 angles = 2pi, so circumference in front of one angle is pi.
so
pi= 60/360*2pi*R
so pi=1/6*2pi*R
so pi=1/3pi*R
R=3....

i hope it is clear what I wrote here.
so what is wrong with it?
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Equilateral triangle is inscribed in a circle. If length of arc ABC is 05 Mar 2011, 09:37
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banksy
Yep, thanks Bunuel.
But I don't understand why it is not possible to use another method to solve it....
it drives me crazy cos I don't understand what is wrong with it...please look..

I used formula:
Circumference of sector= (angle in front of the sector/ 360) * 2*pi*R

so if we know that ABC equilateral, so all angles=60 and all circumferences in front of A ,B and C are same. so if circumference in front of 2 angles = 2pi, so circumference in front of one angle is pi.
so
pi= 60/360*2pi*R
so pi=1/6*2pi*R
so pi=1/3pi*R
R=3....

i hope it is clean what I wrote here.
so what is wrong with it?

In your formula 60 degrees is the measure of the inscribed angle and you should use the measure of the central angle which is twice the inscribed angle so 120 degrees.

Check this: math-circles-87957.html

• Arc Length The formula the arc measure is: \(L=2\pi{r}\frac{C}{360}\), where C is the central angle of the arc in degrees. Recall that \(2\pi{r}\) is the circumference of the whole circle, so the formula simply reduces this by the ratio of the arc angle to a full angle (360). By transposing the above formula, you solve for the radius, central angle, or arc length if you know any two of them.
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Equilateral triangle is inscribed in a circle. If length of arc ABC is 05 Mar 2011, 09:41
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oh, now it is clear)thank you very much!!!=)))
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Equilateral triangle is inscribed in a circle. If length of arc ABC is 12 Oct 2018, 06:42
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Bunuel
banksy
164 Equilateral triangle is inscribed in a circle. If length of arc ABC is 2pi, what is the radius of the circle?
(A) 1
(B) 3/2
(C) 2
(D) 5/3
(E) 3

Arc ABC is \(\frac{2}{3}\) of the circumference (as ABC is equilateral triangle then (arc AB)=(arc BC)=(arc AC), so (arc AB)+(arc BC)=(arc ABC)=2/3 of circumference, so \(arc \ ABC=2\pi\) basically means that the circumference is \(3\pi\) --> \(circumference=2\pi{r}=3\pi\) --> \(r=\frac{3}{2}\).

Answer: B.

Similar question: https://gmatclub.com/forum/geometry-circ ... 97393.html

Bunuel
i think the official answer is wrong. It should be B but it says C.
regards
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Equilateral triangle is inscribed in a circle. If length of arc ABC is 30 Nov 2021, 06:01
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Attachment:
temp.JPG
temp.JPG [ 32.55 KiB | Viewed 3650 times ]

Angle made by Arc ABC with the center of the circle is 240°. (Watch this video to understand how)

Length of Arc ABC = \(\frac{240°}{360°}\) * 2 \(\pi\) r = \(\frac{2}{3}\) * 2 \(\pi\) r = \(\frac{4}{3}\) * \(\pi\) r = 2 \(\pi\) (given)

=> \(\frac{4}{3}\) r = 2
=> r = \(\frac{6}{4}\) = \(\frac{3}{2}\)

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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Equilateral triangle is inscribed in a circle. If length of arc ABC is 28 Jun 2022, 15:49
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Area of a circle = 2 x pi x radius squared
Perimeter of a circumference = 2 x pi x radius
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982DF557-B75B-43D5-90C7-872BAB0798AE.jpeg [ 240.42 KiB | Viewed 2787 times ]

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