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 Post subject: GMAT Geometry
PostPosted: Mon Jan 24, 2011 6:26 pm 
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In the figure above, each square is contained within the area of the larger square. The smallest square has a side length of 2 and the ratio of the area of the shaded square to the area of the smallest square is 5/2, what is the area of the largest square if the shaded square bisects each side of the largest square at its vertices?

A. 20
B. 30
C. 40
D. 50
E. 60

(A) The area of the smallest square must be equal to 2² = 4. Since the ratio of the area of the shaded square (it consists of the shaded region and the smallest square) to the area of the smallest square is 5/2, we can solve for the area of the shaded square by solving for A in the following equation:
A/4 = 5/2
A = 4 × (5/2)
A = 10

So, the area of the shaded square is equal to 10, and each side is equal to √10. Now, all we have to do is recognize that the triangles made at each corner of the large square where, two sides are equal to half the side length and the hypotenuse is equal to the edge length of the shaded square, are right triangles and to use the Pythagorean Theorem to solve for half the edge length. Let each side of the triangle be equal to x, then 2x² = 10, so x² = 5, and x = √5. This means that the entire side length of the larger square is twice the length of a side of that triangle, or 2√5. Finally, the area of a square is equal to a side squared, so the answer must be (2√5 )² = 20, or answer choice (A).
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This question is misleading - "area of the shaded square" should not include area of the smallest square.


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 Post subject: Re: math (test 5, question 27): geometry, area
PostPosted: Mon Jan 24, 2011 6:45 pm 
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Please, pay attention to the word "square" in the term "shaded square". So "shaded square" must describe a square, not just some other type region.

On the given picture we have only 3 squares. Therefore the shaded square must be the medium one.

So "The area of the shaded square" = "the shaded area" + "the area of the smallest square".


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 Post subject: Re: math (test 5, question 27): geometry, area
PostPosted: Sun Feb 27, 2011 12:19 pm 
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The answer calculated the shaded area as 10, but the area of the medium square is 10 + 4 = 14, so the side would be square root 14, follow all the rest of of the calculation, I got 28 as the biggest square area. Please let me know if my reasoning is correct. Thanks.


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 Post subject: Re: math (test 5, question 27): geometry, area
PostPosted: Sun Feb 27, 2011 12:40 pm 
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The area of the shaded square, which equals 10, already contains the area of the smallest one (which equals 4). So there is NO need to add it the second time.

The area of the shaded square does consists of the area of the smallest square + the area of the shaded region. But shaded region is never considered in this question.


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 Post subject: Re: math (test 5, question 27): geometry, area
PostPosted: Wed Apr 20, 2011 4:18 pm 
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Answer is not here. It is 28. The area of the shaded region is 10. Agree. That doesn't make the side of the shaded SQUARE ( this square is not entirely shaded) as root of 10. The area within the shaded region is another square with area 4. Hence the middle square has an area of 10+4 = 14. Hence root of 14 is the side of the square. And so root of 28 is its length of the diagnol and the side of the outermost square. Hence 28 is the area of the largest square.


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 Post subject: Re: math (test 5, question 27): geometry, area
PostPosted: Wed Apr 20, 2011 4:33 pm 
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The question statement tells us "the ratio of the area of the shaded square to the area of the smallest square is 5/2". A square is a specific geometrical figure. The shaded region in this question is NOT a square. If the term "shaded region" was used instead, then your reasoning would be absolutely correct.

Yes, we do NOT see the shaded square on the graphics as entirely shaded one. But logically, since the shaded square must be a square in the first place, you should think of graphics as of three squares on top of each other.


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