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# If a circle is inscribed in an equilateral triangle, what is the area

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If a circle is inscribed in an equilateral triangle, what is the area [#permalink]

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29 Feb 2016, 10:42
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If a circle is inscribed in an equilateral triangle, what is the area of the triangle NOT taken up by the circle?

(1) The area of the circle is 12π
(2) The length of a side of the triangle is 12
[Reveal] Spoiler: OA

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If a circle is inscribed in an equilateral triangle, what is the area [#permalink]

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03 Mar 2016, 00:25
If a circle is inscribed in an equilateral triangle then this property exist-

R = A * [ (\sqrt{3}) / 6 ]
A= Side of equilateral triangle

S1= R is given
S2= A is given

Now we can easily calculate corresponding areas of circle and triangle

IMO- D
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Prakhar

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If a circle is inscribed in an equilateral triangle, what is the area [#permalink]

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03 Mar 2016, 02:47
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Bunuel wrote:
If a circle is inscribed in an equilateral triangle, what is the area of the triangle NOT taken up by the circle?

(1) The area of the circle is 12π
(2) The length of a side of the triangle is 12

Following is the formula that tells the relation between the radius of the circle and the side of the triangle:

r = a * ($$\sqrt{3}$$/ 6)
So if we have either of the side or the radius, we can find the other thing.
The relation can be found by using the figure below:

Attachment:

circle in triangle.JPG [ 16.87 KiB | Viewed 2286 times ]

Statement 1: The area of the circle is 12π
We can find the radius of the circle and hence the side of the triangle and the corresponding area
Therefore we can find the difference between the areas
SUFFICIENT

Statement 2: The length of a side of the triangle is 12
We can find the radius of the circle and hence the area of the circle
Therefore we can find the difference between the areas
SUFFICIENT

Option D
NOTE: We do not need to find the area. There is not need to do the calculations.
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If a circle is inscribed in an equilateral triangle, what is the area [#permalink]

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03 Mar 2016, 03:47
TeamGMATIFY wrote:
Bunuel wrote:
If a circle is inscribed in an equilateral triangle, what is the area of the triangle NOT taken up by the circle?

(1) The area of the circle is 12π
(2) The length of a side of the triangle is 12

Following is the formula that tells the relation between the radius of the circle and the side of the triangle:

r = a * ($$\sqrt{3}$$/ 6)
So if we have either of the side or the radius, we can find the other thing.
The relation can be found by using the figure below:

Attachment:
circle in triangle.JPG

Statement 1: The area of the circle is 12π
We can find the radius of the circle and hence the side of the triangle and the corresponding area
Therefore we can find the difference between the areas
SUFFICIENT

Statement 2: The length of a side of the triangle is 12
We can find the radius of the circle and hence the area of the circle
Therefore we can find the difference between the areas
SUFFICIENT

Option D
NOTE: We do not need to find the area. There is not need to do the calculations.

HI TeamGMATIFY,

Actually I already learnt this formula but haven't tried to figure it out the logic behind this.
The figure provided by you makes a triangle of 30-60-90 = 1x : Root 3* x : 2x

So the opposite length of side 30 degree angle --> R
=> 1x= R or x=R

And opposite length of side 60 degree angle--> A/2
=> root 3 * x = A/2

By this I am getting relation as --
R= A / (root 3 * 2) which is not the same

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Prakhar

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Re: If a circle is inscribed in an equilateral triangle, what is the area [#permalink]

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03 Mar 2016, 04:38
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Expert's post
PrakharGMAT wrote:

HI TeamGMATIFY,

Actually I already learnt this formula but haven't tried to figure it out the logic behind this.
The figure provided by you makes a triangle of 30-60-90 = 1x : Root 3* x : 2x

So the opposite length of side 30 degree angle --> R
=> 1x= R or x=R

And opposite length of side 60 degree angle--> A/2
=> root 3 * x = A/2

By this I am getting relation as --
R= A / (root 3 * 2) which is not the same

Once you have got R= A / (root 3 * 2)
You need to rationalize the denominator, or simply remove the under root from the denominator
Multiply $$\sqrt{3}$$ on both numerator and denominator and you will get

R= A $$\sqrt{3}$$/ 6

Does this help?
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Re: If a circle is inscribed in an equilateral triangle, what is the area [#permalink]

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17 Aug 2017, 10:57
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Re: If a circle is inscribed in an equilateral triangle, what is the area [#permalink]

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23 Aug 2017, 08:41
Hello,

Can someone elaborate on this question without the use of a formula?

Thanks

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Re: If a circle is inscribed in an equilateral triangle, what is the area   [#permalink] 23 Aug 2017, 08:41
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