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# What is the perimeter of an equilateral triangle inscribed

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Joined: 16 Apr 2012
Posts: 7
What is the perimeter of an equilateral triangle inscribed  [#permalink]

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Updated on: 09 Aug 2012, 15:05
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Difficulty:

45% (medium)

Question Stats:

68% (01:52) correct 32% (02:13) wrong based on 188 sessions

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What is the perimeter of an equilateral triangle inscribed in a circle of radius 4 ?

A. $$6\sqrt{2}$$
B. $$6\sqrt{3}$$
C. $$12\sqrt{2}$$
D. $$12\sqrt{3}$$
E. $$24$$

Originally posted by arthuro69 on 09 Aug 2012, 13:06.
Last edited by Bunuel on 09 Aug 2012, 15:05, edited 1 time in total.
Renamed the topic and edited the question.
Magoosh GMAT Instructor
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Posts: 4472
Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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09 Aug 2012, 16:29
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arthuro69 wrote:
What is the perimeter of an equilateral triangle inscribed in a circle of radius 4 ?

A. $$6\sqrt{2}$$
B. $$6\sqrt{3}$$
C. $$12\sqrt{2}$$
D. $$12\sqrt{3}$$
E. $$24$$

Hi, there. I'm happy to help.

The full solution to the problem is in the attached pdf.

If the details of the 30-60-90 triangle are not familiar to you, I recommend brushing up with this post:
http://magoosh.com/gmat/2012/the-gmats- ... triangles/

Let me know if there are any further questions.

Mike
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Mike McGarry
Magoosh Test Prep

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Joined: 03 Oct 2010
Posts: 4
Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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04 Sep 2012, 03:29
The radius of circum circle of an equilateral triangle = a/sqrt(3). a is the side of triangle.
Here: a/sqrt(3) = 4
a = 4*sqrt(3).
perimeter 3a = 3*4*sqrt(3) = 12sqrt(3).
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Joined: 02 Sep 2009
Posts: 58453
Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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04 Sep 2012, 03:32
arthuro69 wrote:
What is the perimeter of an equilateral triangle inscribed in a circle of radius 4 ?

A. $$6\sqrt{2}$$
B. $$6\sqrt{3}$$
C. $$12\sqrt{2}$$
D. $$12\sqrt{3}$$
E. $$24$$

All you need to know about triangles for the GMAT: math-triangles-87197.html

Hope it helps.
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GMAT 1: 680 Q50 V31
Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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06 Jul 2014, 11:30
Let x be the side of triangle.

Using cosine rule:
(x/2)/R = cos 30
=> x = 2Rcos 30 = 4√3
=> Perimeter = 3X = 12√3
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Posts: 4009
Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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19 Apr 2018, 14:23
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arthuro69 wrote:
What is the perimeter of an equilateral triangle inscribed in a circle of radius 4 ?

A. $$6\sqrt{2}$$
B. $$6\sqrt{3}$$
C. $$12\sqrt{2}$$
D. $$12\sqrt{3}$$
E. $$24$$

So, here's what the diagram looks like.

If we draw lines from the center to each vertex, we get the following:

Now we'll draw a line from the center that is PERPENDICULAR to one side of the tirangle.

We now have a SPECIAL 30-60-90 right triangle.

Here's the base version of this SPECIAL TRIANGLE

We can see that the each 30-60-90 triangle in the diagram is TWICE as big as the base version. So, each side opposite the 60º angle must have length 2√3

This means ONE side of the equilateral triangle has length 4√3, so the PERIMETER = 4√3 + 4√3 + 4√3 = 12√3

Cheers,
Brent
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Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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16 Jul 2019, 10:45
arthuro69 wrote:
What is the perimeter of an equilateral triangle inscribed in a circle of radius 4 ?

A. $$6\sqrt{2}$$
B. $$6\sqrt{3}$$
C. $$12\sqrt{2}$$
D. $$12\sqrt{3}$$
E. $$24$$

way simpler to simply know a formula :

Whenever a Circle circumscribes an equilateral triangle, the The radius of the circle can be defined by : Radius = $$\frac{Side Of The Triangle}{\sqrt{3}}$$.

that way, if we solve the problem : 4 = $$\frac{s}{\sqrt{3}}$$ . Hence, side = $$4\sqrt{3}$$.

Now, the perimeter = 3 x $$4\sqrt{3}$$ = 12$$\sqrt{3}$$ .
The answer is the option D.
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Re: What is the perimeter of an equilateral triangle inscribed   [#permalink] 16 Jul 2019, 10:45
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