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# In the figure above, the centers of circles W, X, Y, and Z

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In the figure above, the centers of circles W, X, Y, and Z [#permalink]

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30 Nov 2007, 13:53
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In the figure above, the centers of circles W, X, Y, and Z are joined to form a quadrilateral. Circle W is identical to circle Y and circle X is identical to circle Z, and each of the circles is tangent to either two or three other circles, as shown. Is quadrilateral WXYZ a square?

(1) The circumference of circle W is equal to the circumference of circle X.
(2) The sum of the radii of circles W and Y is one-half the sum of the diameters of circles X and Z.

Would someone show me how to solve this problem? I have the OA, but I got this problem wrong and don't have the explanations, so I will need your help! Appreciate it!
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30 Nov 2007, 14:17
the OA is D, but can you show me how?
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30 Nov 2007, 15:41
"Circle W is identical to circle Y and circle X is identical to circle Z" so we have two equal Y circles and two two equal X circles.
And It means that the quadrilateral is a rhombus (due to symmetry).

1. means that radii X and Y are equal. so, XY=2Ry, YC=Ry
a=YC/XY=Ry/2Ry=1/2 (for square - 1/√2) - It's not a square - suff.

2. means the same as 1. that radii X and Y are equal. - suff.
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01 Dec 2007, 02:25
1. means that radii X and Y are equal. so, XY=2Ry, YC=Ry
a=YC/XY=Ry/2Ry=1/2 (for square - 1/√2) - It's not a square - suff.

There is something i didn't understand. I've highlighted some of your points in bold and have underlined some of them. First of all, what did you mean by XY? does that mean multiplying their areas? radii? and what did you mean by 2Ry? what does the small y stand for? same thing applies to YC and Ry. Can you please explain what those variables, including whether they're capital letters, stand for?

thanks
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01 Dec 2007, 03:33
tarek99 wrote:
1. means that radii X and Y are equal. so, XY=2Ry, YC=Ry
a=YC/XY=Ry/2Ry=1/2 (for square - 1/√2) - It's not a square - suff.

...

Can you please explain what those variables, including whether they're capital letters, stand for?

thanks

Sorry for that.

XY - a segment between the center of X and the center of Y circles. The segment is a side of the rhombus.

YC - a segment between the center of Y and the center of the rhombus (C point).

Ry - a radius of Y circle.
Rx - a radius of X circle.
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01 Dec 2007, 16:26
I think the best way to imagine this answer is to remember:

Square has all four sides and diagonals equal in length.

1) All four sides equal are equal(radius of all four sides is same) but its clear from diagram that diagonals will not be of same length.

2) Exactly same explaination.

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02 Dec 2007, 13:21
thanks walker! first of all, in any square, the ratio of its diagonal to its side should be x*sqrt(2) to x. So the ratio of half of the diagonal to the whole side of a square is sqrt(2) / 2.

According to statement 1, half of this diagonal is y's radius, we'll call it x. the side of this figure is 2x (because the equal radiis of X and Y are added together). So the ratio of half of this diagonal to the whole side is x/2x or 1/2. Obviously this ratio is not equal to that of a square, therefore this statement is suff. to say "no."

Statement 2 gives us the same information, and therefore would approach it the same way. Answer is D. Great! thanks again walker!
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07 Dec 2007, 12:32
walker wrote:
"Circle W is identical to circle Y and circle X is identical to circle Z" so we have two equal Y circles and two two equal X circles.
And It means that the quadrilateral is a rhombus (due to symmetry).

1. means that radii X and Y are equal. so, XY=2Ry, YC=Ry
a=YC/XY=Ry/2Ry=1/2 (for square - 1/√2) - It's not a square - suff.

2. means the same as 1. that radii X and Y are equal. - suff.

I'm getting D but I think you guys are off on your reasoning unless I'm missing something.

The original diagram has 4 circles w,x,y,z and w=y and x=z.

Stmt 1 says that the circumference of w=x. The only way for the circumference to be equal is if the circles are the same size. So we have w=x and we already know w=y and x=y so they must all be equal. If all the circles are equal then it's a square.

Stmt 2 says that the sum of the radii of circles W and Y is one-half the sum of the diameters of circles X and Z. We know w=y and x=z from the stem. 2r=d. So if the sum of the radii of w and y is half the diameter of x and z then the radii are equal. If the radii are equal then again all the circles are equal and it a square.

Each stmt alon is suff
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07 Dec 2007, 12:56
gixxer1000 wrote:
walker wrote:
"Circle W is identical to circle Y and circle X is identical to circle Z" so we have two equal Y circles and two two equal X circles.
And It means that the quadrilateral is a rhombus (due to symmetry).

1. means that radii X and Y are equal. so, XY=2Ry, YC=Ry
a=YC/XY=Ry/2Ry=1/2 (for square - 1/√2) - It's not a square - suff.

2. means the same as 1. that radii X and Y are equal. - suff.

I'm getting D but I think you guys are off on your reasoning unless I'm missing something.

The original diagram has 4 circles w,x,y,z and w=y and x=z.

Stmt 1 says that the circumference of w=x. The only way for the circumference to be equal is if the circles are the same size. So we have w=x and we already know w=y and x=y so they must all be equal. If all the circles are equal then it's a square.

Stmt 2 says that the sum of the radii of circles W and Y is one-half the sum of the diameters of circles X and Z. We know w=y and x=z from the stem. 2r=d. So if the sum of the radii of w and y is half the diameter of x and z then the radii are equal. If the radii are equal then again all the circles are equal and it a square.

Each stmt alon is suff

But given the diagram, equal circles doesn't imply a square. The arguments above show this--another way to see it is to notice that:

If all radii are equal, then WZ = ZY = WY. So ZWY is an equilateral triangle. So WZY is a 60-degree angle. So WXYZ can't be a square.

Both statements imply that the radii are equal. So both statements imply that the figure can't be a square.
07 Dec 2007, 12:56
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