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# 164 Equilateral triangle is inscribed in a circle. If length

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164 Equilateral triangle is inscribed in a circle. If length [#permalink]  05 Mar 2011, 07:50
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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
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Re: 164 Equilateral triangle is inscribed in a circle. If length [#permalink]  05 Mar 2011, 08:16
Expert's post
banksy wrote:
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 $$3\pi$$ --> $$circumference=2\pi{r}=3\pi$$ --> $$r=\frac{3}{2}$$.

Similar question: geometry-circle-triangle-from-mba-com-97393.html
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Re: 164 Equilateral triangle is inscribed in a circle. If length [#permalink]  05 Mar 2011, 08:28
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?

Last edited by banksy on 05 Mar 2011, 08:51, edited 1 time in total.
Math Expert
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Posts: 28201
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Kudos [?]: 44855 [1] , given: 6629

Re: 164 Equilateral triangle is inscribed in a circle. If length [#permalink]  05 Mar 2011, 08:37
1
KUDOS
Expert's post
banksy wrote:
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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Re: 164 Equilateral triangle is inscribed in a circle. If length [#permalink]  05 Mar 2011, 08:41
oh, now it is clear)thank you very much!!!=)))
Re: 164 Equilateral triangle is inscribed in a circle. If length   [#permalink] 05 Mar 2011, 08:41
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