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In the figure above, the centers of circles W, X, Y, and Z [#permalink]
30 Nov 2007, 12:53

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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!

"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.

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 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!

"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.

"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:

WZ is the radius of W + radius of Z
ZY is the radius of Y + radius of Z
WY is the radius of Y + radius of W.

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.