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# A circle is inside an equalateral triangle ABC. One side of

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A circle is inside an equalateral triangle ABC. One side of [#permalink]

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07 Nov 2010, 00:18
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A circle is inside an equalateral triangle ABC. One side of the triangle is equal to 10, What is the area of the circle?
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Re: A circle is inside an equalateral triangle ABC. One side of [#permalink]

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14 Mar 2014, 05:10
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It's a simple question if one knows the formula for calculation of an inscribed circle's radius from the side of an equilateral triangle.(Thanks Bunuel!)
Radius of inscribed circle=(sqrt)3*a/6 where a is side of triangle.
After calculating radius,we can easily calculate area of inscribed circle.

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Re: A circle in a triangle [#permalink]

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07 Nov 2010, 00:31
Lets say the triangle is ABC. There for AB = BC = AC = 10
The circle has the center O and touches BC at point D.
Since ABC is equilateral,
(1) BD = DE = 5,
(2) CO bisects the angle BCA, and
(3) OD is perpendicular to DC.

From the right triangle DCO, we get sin (angle DCO) = sin 30 = 1/2. Also, sine (angle DCO) = OD/OC.
This means OD is 5, which is the radius of the circle. Therefore the area of the circle is 25π.

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Re: A circle in a triangle [#permalink]

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07 Nov 2010, 01:13
srjaidev wrote:
Lets say the triangle is ABC. There for AB = BC = AC = 10
The circle has the center O and touches BC at point D.
Since ABC is equilateral,
(1) BD = DE = 5,
(2) CO bisects the angle BCA, and
(3) OD is perpendicular to DC.

From the right triangle DCO, we get sin (angle DCO) = sin 30 = 1/2. Also, sine (angle DCO) = OD/OC.
This means OD is 5, which is the radius of the circle. Therefore the area of the circle is 25π.

Hi, it seems you were on the right track but lost track of some of the information.

You equate 1/2 to OD/OC, which is unsolvable because OD and OC are unknown.

The length we do know is DC which is 5. so using the law of tangent,

$$\frac{1}{\sqrt{3}}=\frac{OD}{DC}=\frac{OD}{5}$$

Therefore $$OD=\frac{5}{\sqrt{3}}$$

Since OD = Radius of the circle,

Area of circle = $$pi*(\frac{5}{\sqrt{3}})^2=\frac{25*pi}{3}$$

Is this right? Its late at night and I can't focus well at the moment. haha

Last edited by chaoswithin on 07 Nov 2010, 01:18, edited 1 time in total.
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Re: A circle in a triangle [#permalink]

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07 Nov 2010, 01:16
by that logic wouldnt the radius be 5/sqrt 3?
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Re: A circle in a triangle [#permalink]

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07 Nov 2010, 01:17
Yes which is what OD is equal to.
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Re: A circle in a triangle [#permalink]

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07 Nov 2010, 01:30
@Chaoswithin - ooops.. yeah.. its difficult without the figure ... yeah we don't know OC but we do know DC and therefore we could use tan 30 and get the values ... and reach the area. I believe I've given others the idea on how to move forward ..
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Re: A circle in a triangle [#permalink]

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07 Nov 2010, 04:09
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Couple of formulae that might be handy:

(1) Formula 1: The radius of Inscribed circle in a traigle, which has sides length as a, b and c, is:
Radius = $$\frac{Area of traingle}{K}$$ = $$\frac{\sqrt{K(K-a)(K-b)(K-c)}}{K}$$

(2) Formula 2: The radius of Circumcribed circle for a traingle, which has sides lengths as a, b and c, is:

Radius = $$\frac{abc}{4*Area of traingle}$$ = $$\frac{abc}{4*\sqrt{K(K-a)(K-b)(K-c)}}$$

Where, K = $$\frac{a+b+c}{2}$$

So, the radius of a inscribed circle where a=b=c=10 is:
K = $$\frac{10+10+10}{2}$$ = 15
Radius = $$\frac{\sqrt{15(15-10)(15-10)(15-10)}}{15}$$ = $$\frac{5}{\sqrt{3}}$$

Now, the area of a traingle, which has radius $$\frac{5}{\sqrt{3}}$$ is: $$pi$$ * $$\frac{5}{\sqrt{3}}$$ * $$\frac{5}{\sqrt{3}}$$ = $$pi$$ * $$\frac{25}{3}$$
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Re: A circle in a triangle [#permalink]

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07 Nov 2010, 06:05
Hey nravi549: Thanks for sharing these. Though, let me add a word of caution about formulas: They cannot substitute for conceptual understanding. If you understand the concepts behind a question, knowing a formula can help you save time. That's it. It will not take you anywhere in GMAT as far as increasing your score is concerned. You may use a formula on one question, the next will be similar or higher level and will need understanding. If you answer that and subsequent questions incorrectly, you will get right back to the point from where you started.
GMAT does not test you on formulas/theorems. It tests you on your conceptual understanding of the topics and on your application skills. Whether you can twist the question to your advantage, figure out what it is testing you on and apply the basics you have learned.
When I took GMAT, I remember thinking during the exam, "After the first 4-5 questions, every question is super interesting. Every question has a trick to it. Not that it is hard to figure out, but it needs thinking and every question is new." And I don't remember using any special formulas/theorems. Just the common ones.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 20 Jul 2010 Posts: 51 Followers: 3 Kudos [?]: 62 [0], given: 51 Re: A circle in a triangle [#permalink] ### Show Tags 07 Nov 2010, 11:36 Thank you. I will take your piece of advice. Intern Joined: 05 Nov 2010 Posts: 32 Followers: 1 Kudos [?]: 22 [0], given: 0 Re: A circle in a triangle [#permalink] ### Show Tags 08 Nov 2010, 20:23 great explanations Intern Joined: 15 Mar 2014 Posts: 1 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: A circle is inside an equalateral triangle ABC. One side of [#permalink] ### Show Tags 15 Mar 2014, 05:28 Creeper300 wrote: A circle is inside an equalateral triangle ABC. One side of the triangle is equal to 10, What is the area of the circle? pi*200/9 Given side of an equilateral triangle is 10.so height is 10*[square_root 2].Also the orthocentre ,centroid coincides and centroid divides in the ratio 2:1 Hence radius of circle is10*squareroot 2/3 Area = pi*r^2=pi*200/9 Manager Joined: 22 Jan 2014 Posts: 140 WE: Project Management (Computer Hardware) Followers: 0 Kudos [?]: 63 [0], given: 143 Re: A circle is inside an equalateral triangle ABC. One side of [#permalink] ### Show Tags 16 Mar 2014, 09:54 Creeper300 wrote: A circle is inside an equalateral triangle ABC. One side of the triangle is equal to 10, What is the area of the circle? The ques is not framed properly. It says: A circle is inside an equalateral triangle ABC What is meant by 'inside' here? _________________ Illegitimi non carborundum. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7370 Location: Pune, India Followers: 2282 Kudos [?]: 15083 [0], given: 224 Re: A circle is inside an equalateral triangle ABC. One side of [#permalink] ### Show Tags 17 Mar 2014, 21:23 thefibonacci wrote: Creeper300 wrote: A circle is inside an equalateral triangle ABC. One side of the triangle is equal to 10, What is the area of the circle? The ques is not framed properly. It says: A circle is inside an equalateral triangle ABC What is meant by 'inside' here? What the question meant to say was that the circle is 'inscribed' inside the triangle. If it is not necessary for it to be inscribed, then it doesn't have a fixed area. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: A circle is inside an equalateral triangle ABC. One side of [#permalink]

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21 Jan 2016, 04:48
Creeper300 wrote:
A circle is inside an equalateral triangle ABC. One side of the triangle is equal to 10, What is the area of the circle?

The link below has the picture and explains it very well.

http://www.gmathacks.com/quant-topics/c ... angle.html
Re: A circle is inside an equalateral triangle ABC. One side of   [#permalink] 21 Jan 2016, 04:48
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