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Circle O is inscribed in equilateral triangle ABC, which is

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Circle O is inscribed in equilateral triangle ABC, which is itself inscribed in circle P. What is the area of circle P?

(1) The area of circle O is \(4\)pie.

(2) The area of triangle ABC is \(12\sqrt{3}\).
[Reveal] Spoiler: OA

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Last edited by Bunuel on 20 May 2014, 03:27, edited 1 time in total.
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Re: Interesting Geometry Problem: Veritas [#permalink]

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Hussain15 wrote:
Circle O is inscribed in equilateral triangle ABC, which is itself inscribed in circle P. What is the area of circle P?

(1) The area of circle O is \(4\)pie.

(2) The area of triangle ABC is \(12\sqrt{2}\).


For equilateral triangle:
• The radius of the circumscribed circle is \(R=a*\frac{\sqrt{3}}{3}\), (where \(a\) is the side of equilateral triangle);
• The radius of the inscribed circle is \(r=a*\frac{\sqrt{3}}{6}\);
• The area of equilateral triangle is \(A=a^2*\frac{\sqrt{3}}{4}\).

We are asked to calculate area of bigger circle P - \(area_P=\pi{R^2}\). Note that knowing any of the following: the side of equilateral triangle \(a\), radius of the smaller circle O (as it gives \(a\)) or the radius of P itself is sufficient to calculate area of P.

(1) The area of circle O is \(4\pi\) --> we can find \(r\) --> we can find \(a\) --> we can find \(R\). Sufficient.

(2) The area of triangle ABC is \(12\sqrt{3}\) --> we can find \(a\) --> we can find \(R\). Sufficient.

Answer: D.
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if you don't have enough time to calculate or don't remember formulas, here is fast "intuitive" approach:

Let's imagine this highly fixed structure. If you change any linear size or area, the structure just scales. We can't change any part of the system without proportionally changing all others parts. Once you get this "intuitive" idea, any linear size or area of any part of the structure defines all other linear sizes and areas of the system. For instance, if we know the height of the triangle, it's enough to find all other parameters in the system. Both statements give us information about one of the parts of the system. So, it's D.

P.S. It's a lot of text but it took 10-20sec.
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I wish I could understand the system approach. Perhaps its the thinking of a MBA student, which I am unable to get. ;)
I try to go through it again. Let's see!!!

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Re: Interesting Geometry Problem: Veritas [#permalink]

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Sorry Hussain15, it's just what I was thinking when took a look at the problem. If it doesn't work for you, just leave it.
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Re: Interesting Geometry Problem: Veritas [#permalink]

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New post 26 Jun 2010, 15:56
D it is.

If a circle is inscribed in an equilateral triangle , you can find radius if the side if a triangle /height of the triangle is given or you can find side of a triangle if radius of the inscrbed circle is given

Even if you dont remember formulas as spelled out by Bunnel..you just need to remember the above fact.
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Re: Interesting Geometry Problem: Veritas [#permalink]

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PriyaRai wrote:
D it is.

If a circle is inscribed in an equilateral triangle , you can find radius if the side if a triangle /height of the triangle is given or you can find side of a triangle if radius of the inscrbed circle is given

Even if you dont remember formulas as spelled out by Bunnel..you just need to remember the above fact.



Even I don't remember all the formulas used above, i was able to get the answer as D with little logic and knowledge.

Who wants to know all the formulas, some time you can do without that.

there's a saying:

Who wants to know the price of everything and value of nothing.

:)
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Re: Interesting Geometry Problem: Veritas [#permalink]

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walker wrote:
if you don't have enough time to calculate or don't remember formulas, here is fast "intuitive" approach:

Let's imagine this highly fixed structure. If you change any linear size or area, the structure just scales. We can't change any part of the system without proportionally changing all others parts. Once you get this "intuitive" idea, any linear size or area of any part of the structure defines all other linear sizes and areas of the system. For instance, if we know the height of the triangle, it's enough to find all other parameters in the system. Both statements give us information about one of the parts of the system. So, it's D.

P.S. It's a lot of text but it took 10-20sec.


I am myself a proponent of exactly this thinking. It makes perfect sense and takes a few seconds. And, you get very good at it with practice.
Something akin to this for the intuitively inclined:
"If there is only one way in which you can draw a geometry diagram with certain specifications, you will be able to find all other sides and angles."
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Re: Interesting Geometry Problem: Veritas [#permalink]

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New post 17 Jul 2011, 12:55
Thanks for the question and the intuitive approach to solve it! I'll try to practice this approach on similar questions.
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Re: Interesting Geometry Problem: Veritas [#permalink]

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New post 17 Jul 2011, 13:06
Answer D.

A) if the area of circle is given. you can (r1)of the inscribed circle and from that the sides of the triangle. Sides of triangle can give you the radius (r2) of the outer circle, enough to answer the question

B) area of triangle will give you the side and also the radius (r2) or circumscribed circle.

so answer D
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Re: Interesting Geometry Problem: Veritas [#permalink]

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Bunuel wrote:
Hussain15 wrote:
Circle O is inscribed in equilateral triangle ABC, which is itself inscribed in circle P. What is the area of circle P?

(1) The area of circle O is \(4\)pie.

(2) The area of triangle ABC is \(12\sqrt{2}\).


For equilateral triangle:
• The radius of the circumscribed circle is \(R=a*\frac{\sqrt{3}}{3}\), (where \(a\) is the side of equilateral triangle);
• The radius of the inscribed circle is \(r=a*\frac{\sqrt{3}}{6}\);
• The area of equilateral triangle is \(A=a^2*\frac{\sqrt{3}}{4}\).

We are asked to calculate area of bigger circle P - \(area_P=\pi{R^2}\). Note that knowing any of the following: the side of equilateral triangle \(a\), radius of the smaller circle O (as it gives \(a\)) or the radius of P itself is sufficient to calculate area of P.

(1) The area of circle O is \(4\pi\) --> we can find \(r\) --> we can find \(a\) --> we can find \(R\). Sufficient.

(2) The area of triangle ABC is \(12\sqrt{2}\) --> we can find \(a\) --> we can find \(R\). Sufficient.


Answer: D.


Dear Members,

Has anyone noticed that both the statements contradict each other

From statement 1 , we get \(a = 4\sqrt3\) or \(a^2 = 48\)

from statement 2 we get \(12\sqrt2 = a^2 *\frac{\sqrt3}{4}\)

or \(a^2 = 48*\frac{\sqrt2}{sqrt3}\)

both the statements should give the same value for a and \(a^2\) ,( side of the triangle).

Let me know if I am misinterpreting anything.

Although the answer is still D, both the statements shouldn't give different values for a.
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Re: Interesting Geometry Problem: Veritas [#permalink]

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New post 20 May 2014, 02:37
qlx wrote:
Bunuel wrote:
Hussain15 wrote:
Circle O is inscribed in equilateral triangle ABC, which is itself inscribed in circle P. What is the area of circle P?

(1) The area of circle O is \(4\)pie.

(2) The area of triangle ABC is \(12\sqrt{2}\).


For equilateral triangle:
• The radius of the circumscribed circle is \(R=a*\frac{\sqrt{3}}{3}\), (where \(a\) is the side of equilateral triangle);
• The radius of the inscribed circle is \(r=a*\frac{\sqrt{3}}{6}\);
• The area of equilateral triangle is \(A=a^2*\frac{\sqrt{3}}{4}\).

We are asked to calculate area of bigger circle P - \(area_P=\pi{R^2}\). Note that knowing any of the following: the side of equilateral triangle \(a\), radius of the smaller circle O (as it gives \(a\)) or the radius of P itself is sufficient to calculate area of P.

(1) The area of circle O is \(4\pi\) --> we can find \(r\) --> we can find \(a\) --> we can find \(R\). Sufficient.

(2) The area of triangle ABC is \(12\sqrt{2}\) --> we can find \(a\) --> we can find \(R\). Sufficient.


Answer: D.


Dear Members,

Has anyone noticed that both the statements contradict each other

From statement 1 , we get \(a = 4\sqrt3\) or \(a^2 = 48\)

from statement 2 we get \(12\sqrt2 = a^2 *\frac{\sqrt3}{4}\)

or \(a^2 = 48*\frac{\sqrt2}{sqrt3}\)

both the statements should give the same value for a and \(a^2\) ,( side of the triangle).

Let me know if I am misinterpreting anything.

Although the answer is still D, both the statements shouldn't give different values for a.


The DS question asks for data sufficiency and not the final answer. Two statements may give two different answers or same answers is not of any merit in these questions.

Don't fall for such traps.

Cheers!!!
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Re: Interesting Geometry Problem: Veritas [#permalink]

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New post 20 May 2014, 03:27
mittalg wrote:
qlx wrote:
Bunuel wrote:

For equilateral triangle:
• The radius of the circumscribed circle is \(R=a*\frac{\sqrt{3}}{3}\), (where \(a\) is the side of equilateral triangle);
• The radius of the inscribed circle is \(r=a*\frac{\sqrt{3}}{6}\);
• The area of equilateral triangle is \(A=a^2*\frac{\sqrt{3}}{4}\).

We are asked to calculate area of bigger circle P - \(area_P=\pi{R^2}\). Note that knowing any of the following: the side of equilateral triangle \(a\), radius of the smaller circle O (as it gives \(a\)) or the radius of P itself is sufficient to calculate area of P.

(1) The area of circle O is \(4\pi\) --> we can find \(r\) --> we can find \(a\) --> we can find \(R\). Sufficient.

(2) The area of triangle ABC is \(12\sqrt{2}\) --> we can find \(a\) --> we can find \(R\). Sufficient.


Answer: D.


Dear Members,

Has anyone noticed that both the statements contradict each other

From statement 1 , we get \(a = 4\sqrt3\) or \(a^2 = 48\)

from statement 2 we get \(12\sqrt2 = a^2 *\frac{\sqrt3}{4}\)

or \(a^2 = 48*\frac{\sqrt2}{sqrt3}\)

both the statements should give the same value for a and \(a^2\) ,( side of the triangle).

Let me know if I am misinterpreting anything.

Although the answer is still D, both the statements shouldn't give different values for a.


The DS question asks for data sufficiency and not the final answer. Two statements may give two different answers or same answers is not of any merit in these questions.

Don't fall for such traps.

Cheers!!!


That's not true. On the GMAT, two data sufficiency statements always provide TRUE information and these statements never contradict each other or the stem.
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Re: Interesting Geometry Problem: Veritas [#permalink]

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New post 20 May 2014, 03:28
qlx wrote:
Bunuel wrote:
Hussain15 wrote:
Circle O is inscribed in equilateral triangle ABC, which is itself inscribed in circle P. What is the area of circle P?

(1) The area of circle O is \(4\)pie.

(2) The area of triangle ABC is \(12\sqrt{2}\).


For equilateral triangle:
• The radius of the circumscribed circle is \(R=a*\frac{\sqrt{3}}{3}\), (where \(a\) is the side of equilateral triangle);
• The radius of the inscribed circle is \(r=a*\frac{\sqrt{3}}{6}\);
• The area of equilateral triangle is \(A=a^2*\frac{\sqrt{3}}{4}\).

We are asked to calculate area of bigger circle P - \(area_P=\pi{R^2}\). Note that knowing any of the following: the side of equilateral triangle \(a\), radius of the smaller circle O (as it gives \(a\)) or the radius of P itself is sufficient to calculate area of P.

(1) The area of circle O is \(4\pi\) --> we can find \(r\) --> we can find \(a\) --> we can find \(R\). Sufficient.

(2) The area of triangle ABC is \(12\sqrt{2}\) --> we can find \(a\) --> we can find \(R\). Sufficient.


Answer: D.


Dear Members,

Has anyone noticed that both the statements contradict each other

From statement 1 , we get \(a = 4\sqrt3\) or \(a^2 = 48\)

from statement 2 we get \(12\sqrt2 = a^2 *\frac{\sqrt3}{4}\)

or \(a^2 = 48*\frac{\sqrt2}{sqrt3}\)

both the statements should give the same value for a and \(a^2\) ,( side of the triangle).

Let me know if I am misinterpreting anything.

Although the answer is still D, both the statements shouldn't give different values for a.


You are right. I guess the second statement should read: he area of triangle ABC is \(12\sqrt{3}\).

Edited the question. Thank you.
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Circle O is inscribed in equilateral triangle ABC, which is [#permalink]

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New post 20 Oct 2015, 08:28
how do you solve, in theory, to come to the conclusion that radius of the circumscribed circle is a*sqrt(3)/3?
is it because the diameter will be 1/2 time of the side?
in this case, the diameter will be a/2, and R is a/4, where a is the side of the equilateral triangle.

knowing that the triangle is inscribed in a circle, we can draw 2 radii which will connect with one side of the triangle, creating a 30-30-120 triangle. Then, we can draw a perpendicular, and get 2 triangles of 30-60-90, in which the longest leg will be a/2, where a is the side of the equilateral triangle.

is my way of thinking right?
in case we know area of the small circle, we can find the side of the equilateral triangle, and thus, can find the radius of the big circle.
in case we know the area of the equilateral triangle, we can deduct that A=[S^2 sqrt(3)]/4. Now, we can find the side of the equilateral triangle, and hence, find the radius of the big circle.

I believe this is more a 700 level question :)
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mvictor wrote:
how do you solve, in theory, to come to the conclusion that radius of the circumscribed circle is a*sqrt(3)/3?
is it because the diameter will be 1/2 time of the side?
in this case, the diameter will be a/2, and R is a/4, where a is the side of the equilateral triangle.

knowing that the triangle is inscribed in a circle, we can draw 2 radii which will connect with one side of the triangle, creating a 30-30-120 triangle. Then, we can draw a perpendicular, and get 2 triangles of 30-60-90, in which the longest leg will be a/2, where a is the side of the equilateral triangle.

is my way of thinking right?
in case we know area of the small circle, we can find the side of the equilateral triangle, and thus, can find the radius of the big circle.
in case we know the area of the equilateral triangle, we can deduct that A=[S^2 sqrt(3)]/4. Now, we can find the side of the equilateral triangle, and hence, find the radius of the big circle.

I believe this is more a 700 level question :)


Check out these posts. They discuss relations between circles and inscribed polygons (including equilateral triangle)

http://www.veritasprep.com/blog/2013/06 ... d-circles/
http://www.veritasprep.com/blog/2013/07 ... relations/

Once you understand these relations, you will jump to (D) immediately.
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Re: Circle O is inscribed in equilateral triangle ABC, which is [#permalink]

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New post 22 Oct 2015, 12:32
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Circle O is inscribed in equilateral triangle ABC, which is itself inscribed in circle P. What is the area of circle P?

(1) The area of circle O is 4 pie.

(2) The area of triangle ABC is 123 √ .

In the original condition, the there is only one variable (radius), and we need one equation to solve for the question.
2 equations are given from the 2 conditions, so there is high chance (D) will be our answer; in fact, (D) is our answer.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: Circle O is inscribed in equilateral triangle ABC, which is   [#permalink] 22 Oct 2015, 12:32
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Circle O is inscribed in equilateral triangle ABC, which is

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