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PS: Equilateral Triangle inscribed in circle

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kevch25 Junior | Next Rank: 30 Posts Default Avatar
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PS: Equilateral Triangle inscribed in circle

Post Wed Feb 28, 2007 10:04 pm
An 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

I thought I had this question easily figured out, but it's not any of the answer choices, so I'm pretty certain I'm doing something wrong Embarassed. ...so your explanations are appreciated and I'm sure I'm overlooking something pretty elementary.

My logic is this:

Equilateral triangle = 60 degrees for all 3 angles. So the arc ABC must be sitting at 60 degrees. But it is on the outer part of the circle, so 60 * 2 will bring us to the center = 120 degrees. 120 / 360 means the arc is 1/3 of the circle. We know the arc is 24, so 24 * 3 is 72.
The circumference of the circle is pi * d so the diameter is 72 / pi which is approximately equal to 23.

Please tell me where i'm leading myself astray.

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jayhawk2001 Community Manager
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Post Wed Feb 28, 2007 10:20 pm
kevch25, you have most of it correct except for the angle.

Length of arc ABC is 240 degrees not 120 degrees.
Arc AB is 120 deg and Arc BC is 120 deg. ABC hence is 240 deg.

The rest follows as per your logic i.e.

240 / 360 * x = 24
x = 36

So, D = 36 / pi = 11

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kevch25 Junior | Next Rank: 30 Posts Default Avatar
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Post Wed Feb 28, 2007 10:50 pm
jayhawk2001 wrote:
kevch25, you have most of it correct except for the angle.

Length of arc ABC is 240 degrees not 120 degrees.
Arc AB is 120 deg and Arc BC is 120 deg. ABC hence is 240 deg.

The rest follows as per your logic i.e.

240 / 360 * x = 24
x = 36

So, D = 36 / pi = 11
oh man, thank you for catching that. I was about to go to sleep with the nightmare of not knowing what I did wrong on this!!

Thanks again for the quick reply.

-Kevin

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Post Fri Sep 29, 2017 1:20 pm

To find the appropriate diameter (d) of the circle, the circumference (c) (complete distance around the circle) has to be first evaluated. To do this, Mathematically,
c = πd
the entire circle is 360 degree (ABCA), where in the equilateral triangle AB=BC=CA
Given the length of the arc ABC = 24.
It can be seen that each side of the triangle is 120 degree because the triangle has 3 sides and it is 360 degree. i.e 360/3 = 120
Now the length of arc AB and BC = 24
then, length of arc AB, BC and CA = 36
Equating degrees with the length of the arc, we can see that
24 corresponds to 240 degree
34 corresponds to 360 degree.
the total circumference C= 36
but, c = πd
we can diameter d= c/π
d= 36/3.142
d= 11.45
approximately, d=11

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