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What is the area of the parallelogram inscribed in a circle?

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What is the area of the parallelogram inscribed in a circle? [#permalink] New post 31 May 2008, 04:28
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What is the area of the parallelogram inscribed in a circle?
(1) The sum of the two sides of the parallelogram is 10 cm.
(2) Area of the circle is 75 cm2.

1. statement (1) alone is sufficient but statement (2) alone is not
2. statement (2) alone is sufficient but statement (1) alone is not
3. both (1) and (2) together are sufficient but none of them alone is sufficient
4. both independently are sufficient
5. both statements (1) and (2) together are not sufficient
Answer : ( 3 )
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Re: Geometry-DS [#permalink] New post 31 May 2008, 08:23
To find the area of teh ||gram we need to know the lengths od teh sides of the ||gram

Let a and b be the sides of the ||gram

1. a+b = 10
insuff

2. we know r from stat2
insuff

using info from 1 and 2 , we can use pythogrean theorem to find a and b

a^2 + b^2 = r^2

C
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Re: Geometry-DS [#permalink] New post 31 May 2008, 08:42
goalsnr wrote:
To find the area of teh ||gram we need to know the lengths od teh sides of the ||gram

Let a and b be the sides of the ||gram

1. a+b = 10
insuff

2. we know r from stat2
insuff

using info from 1 and 2 , we can use pythogrean theorem to find a and b

a^2 + b^2 = r^2

C

There is one little flaw in your assumption that a and b are adjacent sides, what if a, and b are the parallel sides. Moreover you can never be sure that a, b, and r are part of right angle triangle.
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Re: Geometry-DS [#permalink] New post 31 May 2008, 09:34
ritula wrote:
What is the area of the parallelogram inscribed in a circle?
(1) The sum of the two sides of the parallelogram is 10 cm.
(2) Area of the circle is 75 cm2.

1. statement (1) alone is sufficient but statement (2) alone is not
2. statement (2) alone is sufficient but statement (1) alone is not
3. both (1) and (2) together are sufficient but none of them alone is sufficient
4. both independently are sufficient
5. both statements (1) and (2) together are not sufficient
Answer : ( 3 )

Can you please tell us about the source of this problem?
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Re: Geometry-DS [#permalink] New post 31 May 2008, 20:10
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abhijit_sen wrote:
ritula wrote:
What is the area of the parallelogram inscribed in a circle?
(1) The sum of the two sides of the parallelogram is 10 cm.
(2) Area of the circle is 75 cm2.

1. statement (1) alone is sufficient but statement (2) alone is not
2. statement (2) alone is sufficient but statement (1) alone is not
3. both (1) and (2) together are sufficient but none of them alone is sufficient
4. both independently are sufficient
5. both statements (1) and (2) together are not sufficient
Answer : ( 3 )

Can you please tell us about the source of this problem?
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Re: Geometry-DS [#permalink] New post 01 Jun 2008, 11:00
abhijit_sen wrote:
goalsnr wrote:
To find the area of teh ||gram we need to know the lengths od teh sides of the ||gram

Let a and b be the sides of the ||gram

1. a+b = 10
insuff

2. we know r from stat2
insuff

using info from 1 and 2 , we can use pythogrean theorem to find a and b

a^2 + b^2 = r^2

C

There is one little flaw in your assumption that a and b are adjacent sides, what if a, and b are the parallel sides. Moreover you can never be sure that a, b, and r are part of right angle triangle.



To add to this - 1. we dont know if they are Right angle triangle and 2. we can never find the height with that information to find the area of the //gram
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Re: Geometry-DS [#permalink] New post 21 Jun 2008, 23:29
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Just surfed through the forum in the search of geometry problems and found this one without solution. So, maybe somebody try? I guess it is possible to solve under 2 minutes :wink:
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Re: Geometry-DS [#permalink] New post 21 Jun 2008, 23:58
walker wrote:
Just surfed through the forum in the search of geometry problems and found this one without solution. So, maybe somebody try? I guess it is possible to solve under 2 minutes :wink:


Well, I will try to explain my theory :

my solution gave me area = 25*sqroot(3) so answer C .

First of all, this parallelogram is inscribed in a circle it will always be rectangle or square ,since it will share the same center as circle .

property of parellelogram - two opposite sides are parallel - for two opposite sides to be parallel in circle , they have to be equidistant from center and so both the diagonal will be same .
If it were given that a trapezoid is inscribed inscribed in a circle our answer would be E .


Now looking at the conditions given

1) a+b=10 , infinite possibilities , insuff

2) pi*r^2 = 75 ; r~5

(2r)^2 = a^2 +b^2

100 = a^2 + b^2 , again infinite possibilities , insuff

combine 1 and 2

1. (a+b)^2 =100 = a^2+b^2+2a^b =100 , but according to condition 2: a^2 + b^2 =100 , this leads us to 2ab=0 which is not possible , hence the first condition is talking about two opposite sides with the same length , so 2a=10 a=5

b^2=75 ; b=5*sqroot(3) and so area is 25*root(3)
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Re: Geometry-DS [#permalink] New post 22 Jun 2008, 00:23
Hi, i think you are assuming that the points of the parallelogram lie on the circle ..... which might not be the case.
It is possible to have a parallelogram inscribed in a circle with sides 6 & 4 or 5 & 5 (remember the diameter of the circle is aprox. 10cm) .
Hence i think the ans is E
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Re: Geometry-DS [#permalink] New post 22 Jun 2008, 05:33
i too get E..

Walker what do you think?
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Re: Geometry-DS [#permalink] New post 22 Jun 2008, 07:47
saurabhkowley18 wrote:
Hi, i think you are assuming that the points of the parallelogram lie on the circle ..... which might not be the case.
It is possible to have a parallelogram inscribed in a circle with sides 6 & 4 or 5 & 5 (remember the diameter of the circle is aprox. 10cm) .
Hence i think the ans is E



Inscribed means all the point lie on a circle .
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Re: Geometry-DS [#permalink] New post 23 Jun 2008, 04:00
ritula wrote:
What is the area of the parallelogram inscribed in a circle?
(1) The sum of the two sides of the parallelogram is 10 cm.
(2) Area of the circle is 75 cm2.

1. statement (1) alone is sufficient but statement (2) alone is not
2. statement (2) alone is sufficient but statement (1) alone is not
3. both (1) and (2) together are sufficient but none of them alone is sufficient
4. both independently are sufficient
5. both statements (1) and (2) together are not sufficient
Answer : ( 3 )


individually in suff
combining
parallel sides are equidistant from the center
so let the distance of one side from center be x and other be y

2x = one side and 2y =other

using pythogoreon theorem x^2+y^2=r^2

form 1 2x+2y=10-----(1)

from 2 : pi r^2=75

r^2=75/pi

squaring 1
(2x+2y)2=100
4(x^2+y^2)+8xy=100
=4(r^2)+8xy=100

=4(75/pi)+8xy=100
we can get the value of xy
as for both square or rectangle area is product of 2 adj sides this is the area
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Re: Geometry-DS [#permalink] New post 23 Jun 2008, 05:45
What is the area of the parallelogram inscribed in a circle?
(1) The sum of the two sides of the parallelogram is 10 cm.
(2) Area of the circle is 75 cm2.

1. statement (1) alone is sufficient but statement (2) alone is not
2. statement (2) alone is sufficient but statement (1) alone is not
3. both (1) and (2) together are sufficient but none of them alone is sufficient
4. both independently are sufficient
5. both statements (1) and (2) together are not sufficient


IMO --- E.....
first lets sort out the info ... parallelogram inscribed in a circle means that figure has to be rectangle...

1) The sum of the two sides of the parallelogram is 10 cm.

let the rectangle be ABCD... with AB> BC....

from the above statement its not correct to assume that two sides means AB and BC...
2 sides can also mean AD and BC.... so 1 is insufficient...

2) Area of the circle is 75 cm2.
so Pi*r*r = 75
so r *r = 75 / 3.14 ====> r= some value...

AC or BD, which are the diagonals will be the diameter of the circle... so BD = AC = 2r
now , we know the diagonal's length .... but we still dontknow what the lengths of the sides are...
even if we combine both we wont be able to get to one solution...
so E........................

whats the OA???
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Re: Geometry-DS [#permalink] New post 23 Jun 2008, 06:30
Either way you can answer the question with statement (1) and (2) : answer is definitely (C).

If you assume that the question talk about two different sides of the parallelogram (which would just be logical since the question states "the two sides of the parallelogram"), then (1) and (2) are sufficient (see answers above).

If you assume that the question can talk about two opposite sides of the parallelogram (which is in fact a rectangle as shown above as well), it is even more simple. It means that theses sides are 5 cm long (since they are equal and their sum is 10 cm). Try to inscribe a rectangle with one side of 5cm in a circle of area 75cm^2 and you'll see there is just one possible answer ;)
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Re: Geometry-DS [#permalink] New post 24 Jun 2008, 09:24
ritula wrote:
OA is E



I still believe it is C !! chances OA is wrong ?I looked at my solution and did not find anything wrong. Can someone look at my solution and point out a flaw, if answer has to be E ?
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Re: Geometry-DS [#permalink] New post 24 Jun 2008, 09:36
rpmodi wrote:
ritula wrote:
OA is E



I still believe it is C !! chances OA is wrong ?I looked at my solution and did not find anything wrong. Can someone look at my solution and point out a flaw, if answer has to be E ?



The problem here is Stat1 is ambigious.
if a and b are adjacent sides of the parallelogram then we can use deduce C as the answer

But this info is not explicitly stated..so we cannot sum the sides are adjacent. Hence E.
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Re: Geometry-DS [#permalink] New post 24 Jun 2008, 11:08
goalsnr wrote:
rpmodi wrote:
ritula wrote:
OA is E



I still believe it is C !! chances OA is wrong ?I looked at my solution and did not find anything wrong. Can someone look at my solution and point out a flaw, if answer has to be E ?



The problem here is Stat1 is ambigious.
if a and b are adjacent sides of the parallelogram then we can use deduce C as the answer

But this info is not explicitly stated..so we cannot sum the sides are adjacent. Hence E.


Yeah But I have a reasoning in my explanation , why this summation has to be of two opposite sides and not two adjacent sides
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Re: Geometry-DS [#permalink] New post 24 Jun 2008, 12:14
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rpmodi wrote:
Yeah But I have a reasoning in my explanation , why this summation has to be of two opposite sides and not two adjacent sides


I thought you had a clever approach, but there is one small mistake. When you calculated the radius, you deduced that it was approximately 5, and that the diameter was approximately 10. You then used the equation a^2 + b^2 = 100 and the equation a^2 + 2ab + b^2 = 100 to deduce that 2ab = 0. The problem, of course, is that the diameter is not exactly equal to 10, so a^2 + b^2 is not exactly equal to 100. The diameter is equal to 10*sqroot(3/Pi) < 10, and a^2 + b^2 is equal to something a bit smaller than 100, so 2ab is not equal to zero.
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Re: Geometry-DS [#permalink] New post 24 Jun 2008, 12:22
IanStewart wrote:
rpmodi wrote:
Yeah But I have a reasoning in my explanation , why this summation has to be of two opposite sides and not two adjacent sides


I thought you had a clever approach, but there is one small mistake. When you calculated the radius, you deduced that it was approximately 5, and that the diameter was approximately 10. You then used the equation a^2 + b^2 = 100 and the equation a^2 + 2ab + b^2 = 100 to deduce that 2ab = 0. The problem, of course, is that the diameter is not exactly equal to 10, so a^2 + b^2 is not exactly equal to 100. The diameter is equal to 10*sqroot(3/Pi) < 10, and a^2 + b^2 is equal to something a bit smaller than 100, so 2ab is not equal to zero.



Ahh ! great catch .....looks like ur first post on this forum :-) well here you go, your first Kudos +1 .....
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Re: Geometry-DS [#permalink] New post 24 Jun 2008, 23:36
rpmodi wrote:
Yeah But I have a reasoning in my explanation , why this summation has to be of two opposite sides and not two adjacent sides

I see no reason why it has to be two opposite sides (the wording in the question tends to point the contrary even if it is ambiguous).

Why do you say it can't be two adjacent sides ?
Re: Geometry-DS   [#permalink] 24 Jun 2008, 23:36
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