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by GMATGuruNY » Sun Jan 05, 2014 3:44 am
Equilateral triangle ABC is inscribed in a circle. If the length of arc ABC is 24, what is the approximate diameter of the circle?

A) 5
B) 8
C) 11
D) 15
E) 19
Figures in PS problems are generally DRAWN TO SCALE.
It should be clear from the figure that arc AB = arc BC = arc AC.
Thus, arc ABC -- which has a length of 24 -- constitutes 2/3 of the circumference:
24 = (2/3)c
c = 36.

Since c = πd ≈ 3d, we get:
3d ≈ 36
d ≈ 12.

The correct answer is C.
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by Uva@90 » Sun Jan 05, 2014 4:05 am
GMATGuruNY wrote: Figures in PS problems are generally DRAWN TO SCALE.
It should be clear from the figure that arc AB = arc BC = arc AC.
Thus, arc ABC -- which has a length of 24 -- constitutes 2/3 of the circumference:
24 = (2/3)c
c = 36.

Since c = πd ≈ 3d, we get:
3d ≈ 36
d ≈ 12.

The correct answer is C.
Hi Mitch,
Is the below one correct ?
The radius of the circumscribed circle is R= (SQRT(3)/3) *a
The radius of the inscribed circle is r= (SQRT(3)/6)*a, where a is the side of the equilateral triangle.

If it is, then from the question we know a = 8
then R = SQRT(3)/3 *8
so diameter is 9.23
so closest ans is 8.
so we should pick B only right instead of C

Could you please explain this.

Thanks in advance.

Regards,
Uva.
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by GMATGuruNY » Sun Jan 05, 2014 4:24 am
The radius of the inscribed circle is r= (SQRT(3)/6)*a, where a is the side of the equilateral triangle.
from the question we know a = 8
The question stem indicates that arc ABC = 24.
This does not imply that each side of the triangle has a length of 8.
Last edited by GMATGuruNY on Sun Jan 05, 2014 4:39 am, edited 1 time in total.
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by Uva@90 » Sun Jan 05, 2014 4:28 am
GMATGuruNY wrote:
Uva@90 wrote:a is the side of the equilateral triangle.[/color]
from the question we know a = 8
The question stem does not indicate that a=8.
The shortest distance between two points is a straight line.
Thus, an arc that connects points A and B must be longer than a straight line that connects points A and B.
Since arc AB = 8, line segment AB must be LESS than 8.
Thus, a<8.
OMG!!
I Completely misunderstood the question.
Question states LENGTH OF ARC =24.
I understood as length of line segment =24 i.e sum of each sides of triangle =24.

Thanks a ton Mitch.

Regards,
Uva.
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