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If a square is inscribed in a circle that, in turn, is inscribed in a
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08 Dec 2010, 20:04
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If a square is inscribed in a circle that, in turn, is inscribed in a larger square, what is the ratio of the perimeter of the larger square to that of the smaller square? A. \(\frac{1}{2}\) B. \(\frac{1}{\sqrt{2}}\) C. \(\sqrt{2}\) D. \(2\) E. \(3.14\) M1723 Ok, I understand this explanation from sportyrizwan: Let "a" be the diameter of the circle Therefore the side of the bigger square will be a and the perimeter 4a Also a will be the diagonal of the smaller square Therefore the side of the smaller square will be a/sqrt2 and perimeter (2a)sqrt2 Therefore the ratio of the bigger square to the smaller square is sqrt2 http://gmatclub.com/forum/squareinscribedinacircleinscribedinasquare55210.html*********** However, when I try to plug values, it doesn't work.. Let say larger square one side x=2, so P=8 Smaller square Diagonal = hypotenus = 2 So x^2+x^2=2^2 2x^2=4 x^2=2 x=sqrt2 P=4sqrt2 L/S = 8/4sqrt2 = 2 ???? It doesn't work and I can't find my mistake.. PLEASE help me to find my mistake!! Thank you very much for your help!!
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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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09 Dec 2010, 02:46
maive wrote: GMAT CLUB Tests, M17 q 23 A square is inscribed in a circle whic, in turn, is inscribed in a bigger square. What is the ratio of the perimeter of the bigger square to that of the smaller square? A. 1/2 1/rad 2 sqrt2 2 pi Look at the diagram: Attachment:
SquareCircleSquare.jpg [ 10.9 KiB  Viewed 5232 times ]
You can see that a diameter of the circle equals to a diagonal of the smaller square and a side of the bigger square: \(diameter=diagonal_{small}=side_{big}\); Now, if the side of the bigger square is \(a\) then its perimeter will be \(P=4a\); Next, a side of a smaller square will be \(\frac{a}{\sqrt{2}}\) and its perimeter \(p=\frac{4a}{\sqrt{2}}\); \(\frac{P}{p}=\sqrt{2}\) Answer: C. Or as soon as you realize that the sides of bigger and smaller squares are \(a\) and \(\frac{a}{\sqrt{2}}\) respectively, so their ration is \(\sqrt{2}\), then as P=4*side then the ratio of the sides will be the same as the ratio of the perimeters, so the same \(\sqrt{2}\).
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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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08 Dec 2010, 20:14
I think your math is off when you are trying to find a side of the smaller square.
So Large square side =2. So diameter of circle =2  therefore 2=hypotenuse of smaller square.
Since the square is two isosceles triangles  hypotenuse=x^2 + x^2 or in other words in our case 2/sqrt2 to get one of the sides.
so perimeter of larger square = 2*4= 8 perimeter of smaller square = 2/sqrt2 * 4 = 8/sqrt2.
when you then take the ratio of larger square perimeter to smaller square it looks like this:
8/(8/sqrt2) or 8 * sqrt2/8 so the 8's cancel out and your are left with sqrt 2  the answer.
hope it helps.



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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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09 Dec 2010, 05:46
maive wrote: L/S = 8/4sqrt2 = 2 ???? It doesn't work and I can't find my mistake..
PLEASE help me to find my mistake!!
Thank you very much for your help!!
You got \(\frac{L}{S} = \frac{8}{4\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}\) Sometimes, it is the silliest things that get you!
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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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09 Dec 2010, 08:14
I got it!!
Thanks a lot for your help, I really appreciate it!!



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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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25 Sep 2014, 00:16
maive wrote: If a square is inscribed in a circle that, in turn, is inscribed in a larger square, what is the ratio of the perimeter of the larger square to that of the smaller square? A. \(\frac{1}{2}\) B. \(\frac{1}{\sqrt{2}}\) C. \(\sqrt{2}\) D. \(2\) E. \(3.14\) M1723 Ok, I understand this explanation from sportyrizwan: Let "a" be the diameter of the circle Therefore the side of the bigger square will be a and the perimeter 4a Also a will be the diagonal of the smaller square Therefore the side of the smaller square will be a/sqrt2 and perimeter (2a)sqrt2 Therefore the ratio of the bigger square to the smaller square is sqrt2 http://gmatclub.com/forum/squareinscribedinacircleinscribedinasquare55210.html*********** However, when I try to plug values, it doesn't work.. Let say larger square one side x=2, so P=8 Smaller square Diagonal = hypotenus = 2 So x^2+x^2=2^2 2x^2=4 x^2=2 x=sqrt2 P=4sqrt2 L/S = 8/4sqrt2 = 2 ???? It doesn't work and I can't find my mistake.. PLEASE help me to find my mistake!! Thank you very much for your help!! Consider the diagram below: Attachment:
m1723.png [ 8.46 KiB  Viewed 4378 times ]
The side of a large square is \(a\), thus its perimeter is \(4a\); The side of a small square is \(\sqrt{(\frac{a}{2})^2+(\frac{a}{2})^2}=\frac{a}{\sqrt{2}}\), thus its perimeter is \(\frac{4a}{\sqrt{2}}\); Hence the ratio is \(\frac{(4a)}{(\frac{4a}{\sqrt{2}})}=\sqrt{2}\). Answer: C
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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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28 Oct 2016, 02:34
Bunuel : Need your help here. Why is the diagonal of the small square = a/sqrt 2?



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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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28 Oct 2016, 02:44
Vishvesh88 wrote: Bunuel : Need your help here. Why is the diagonal of the small square = a/sqrt 2? The diagonal of small square is a, not \(\frac{a}{\sqrt{2}}\). \(\frac{a}{\sqrt{2}}\) is side of a small square. Check here: ifasquareisinscribedinacirclethatinturnisinscribedina105993.html#p1419806
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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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28 Oct 2016, 03:53
Bunuel : Got it mate! thanks.



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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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19 Nov 2016, 04:29
Assume the side of the larger square to be 8, perimeter of larger square = 32.
Diameter of the circle inscribed within a square = Side of the square , therefore radius = 4. Furthermore radius of the circle is half the diagonal length of smaller square, hence 4= √2a/2 , solving for 'a', we get 4√2. Therefore, perimeter of smaller square = 16√2. Finally, 32/16√2 = √2. [C]



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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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17 Dec 2018, 09:54
maive wrote: If a square is inscribed in a circle that, in turn, is inscribed in a larger square, what is the ratio of the perimeter of the larger square to that of the smaller square?
A. \(\frac{1}{2}\) B. \(\frac{1}{\sqrt{2}}\) C. \(\sqrt{2}\) D. \(2\) E. \(3.14\) Let the side of small square be \(= 2\) Therefore the diagonal of small square \(= 2\sqrt{2}\) Diagonal of small square is \(=\) diameter of the circle Diameter of the circle is \(=\) side of the large square Therefore side of large square \(= 2\sqrt{2}\) Perimeter of small square \(= 4a = 4*2 = 8\) Perimeter of large square \(= 4a = 4*2\sqrt{2} = 8\sqrt{2}\) Ratio of perimeter of large square to perimeter of small square \(= \frac{8\sqrt{2}}{8} = \sqrt{2}\) Answer C



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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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02 Apr 2019, 10:04
1. If a square is inscribed in a circle which is inscribed in a square then outer square area will be twice that of inner square area. 2. If a circle in inscribed in a square which is inscribed in a circle then the outer circle area will be twice that of inner circle area.
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Re: If a square is inscribed in a circle that, in turn, is inscribed in a
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