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# A square and a circle intersect at more than one point. Does the squa

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Math Expert
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A square and a circle intersect at more than one point. Does the squa  [#permalink]

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11 Mar 2015, 03:50
2
3
00:00

Difficulty:

95% (hard)

Question Stats:

37% (01:38) correct 63% (01:53) wrong based on 170 sessions

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A square and a circle intersect at more than one point. Does the square have more area than the circle?

(1) there are exactly four intersection points
(2) at least two of the intersection points are on vertices of the square

Kudos for a correct solution.

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Re: A square and a circle intersect at more than one point. Does the squa  [#permalink]

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11 Mar 2015, 06:01
2
Hi

Statement i
circle can be outside or inside

Statement ii
Can not answer with it as circle can be outside on either side
cannot say the position can be as shown in the pic below

i + ii still not sufficient

Correct me if i am wrong..!!
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cir2.png [ 15.47 KiB | Viewed 2971 times ]

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Re: A square and a circle intersect at more than one point. Does the squa  [#permalink]

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11 Mar 2015, 06:30

from statment 1 is insuff because the circle is either smaller or larger which mean the circle could be inside or outside the square

from statment 1 is insuff because the circle is either smaller or larger which mean the circle could be inside or outside the square

or could be just half of the circle inside or out side the square

statment 1&2 the circle is intersects the square in four vertices and the circle could be larger or smaller the square so both statment are insuff
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Re: A square and a circle intersect at more than one point. Does the squa  [#permalink]

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11 Mar 2015, 06:45
'C' . The only conceivable figure in which this fits is the Circle enveloping the Square.

But of course , it can't be that easy, so .
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Posts: 50623
Re: A square and a circle intersect at more than one point. Does the squa  [#permalink]

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15 Mar 2015, 21:46
Bunuel wrote:
A square and a circle intersect at more than one point. Does the square have more area than the circle?

(1) there are exactly four intersection points
(2) at least two of the intersection points are on vertices of the square

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

Statement #1: this information, with nothing more, could mean that the circle is either smaller or larger.
Attachment:

cpotg_img20.png [ 11.21 KiB | Viewed 2820 times ]

This statement, alone and by itself, is insufficient.

Statement #2: this information, with nothing more, could mean that the circle is either smaller or larger.
Attachment:

cpotg_img21.png [ 13.37 KiB | Viewed 2808 times ]

This statement, alone and by itself, is insufficient.

Combined Statements: One possibility is the circle that intersects the square four times by passing through all four vertices:
Attachment:

cpotg_img22.png [ 6.07 KiB | Viewed 2806 times ]

That circle is clearly bigger than the square. The circle absolutely cannot pass through exactly three vertices. If it pass through two vertices, it would have to intersect the side two more times. Possibilities include the following (point C is the center of the circle).
Attachment:

cpotg_img23.png [ 15.47 KiB | Viewed 2808 times ]

Notice that, as point C approaches the top side of the square, it gets closer and closer to the circle that has this top side as a diameter, equivalent to the first circle in the statement #1 diagram. That circle is clearly has less area than the square. Well, that circle won’t work here, because it intersects at only two points, but because point C could get closer and closer to the top side without touching it, which means the area of the circle in this diagram could get closer and closer to the area of the first circle in the statement #1 diagram. This means that we could make the circle in this diagram have less area than the square has.
Thus, even with the constraints of both statements, we can construct a circle that has an area that is either greater than or less than that of the square. Even with both statements, we cannot give a definitive answer to the prompt question.

Both statements combined are insufficient.
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Re: A square and a circle intersect at more than one point. Does the squa  [#permalink]

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17 Nov 2015, 18:23
indeed, a tricky one...
one might consider statement 2 alone as sufficient to say that the square is circumscribed and thus circle has a greater area but!!! in the question it is specified that there are at least 1 point of intersection. There might be 2 points of intersection, 3, 4, 5, 6 etc.

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Re: A square and a circle intersect at more than one point. Does the squa  [#permalink]

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31 Jul 2016, 11:31
Bunuel wrote:
A square and a circle intersect at more than one point. Does the square have more area than the circle?

(1) there are exactly four intersection points
(2) at least two of the intersection points are on vertices of the square

Kudos for a correct solution.

Please refer the attached for consideration
neither option suff...
Ans E
Attachments

Untitled.png [ 8.16 KiB | Viewed 1980 times ]

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Re: A square and a circle intersect at more than one point. Does the squa  [#permalink]

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09 Jul 2018, 09:28
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Re: A square and a circle intersect at more than one point. Does the squa &nbs [#permalink] 09 Jul 2018, 09:28
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