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ABCD is a square. The dotted lines are the lines connecting two midpoi

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ABCD is a square. The dotted lines are the lines connecting two midpoi  [#permalink]

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New post 14 Nov 2019, 01:28
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A
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  25% (medium)

Question Stats:

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[GMAT math practice question]

\(ABCD\) is a square. The dotted lines are the lines connecting two midpoints of opposite lines. Moreover, \(E, F, G, H, I, J, K\), and \(L\) are midpoints between midpoints of sides and edges again in the following figure. What is the ratio of the area of the shaded region to the area of the unshaded region?

Attachment:
11.14ps.png
11.14ps.png [ 63.93 KiB | Viewed 418 times ]


A. \(1.5:1\)

B. \(1.3:1\)

C. \(1.1:1\)

D. \(1:1\)

E. \(1:1.1\)

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Re: ABCD is a square. The dotted lines are the lines connecting two midpoi  [#permalink]

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New post 14 Nov 2019, 04:59
area of square = AD * AD
area of a triangle (shaded) = .5 * AD/2 * AD/2
total number of triangles =4
comparing the shaded and the unshaded areas of the square, we see that bot come out tp be equal.

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Re: ABCD is a square. The dotted lines are the lines connecting two midpoi  [#permalink]

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New post 17 Nov 2019, 18:42
=>


Assume \(MF = BF = BG = GN = x.\)

The area of the square of \(OGBF\) is the sum of areas of triangles \(OGB\) and \(OFB\) , \((\frac{1}{2})(x)(2x) + (\frac{1}{2})(x)(2x) = 2x^2\) and the shaded area of the triangles \(OFM\) and \(ONG\) is \((\frac{1}{2})(x)(2x) + (\frac{1}{2})(x)(2x) = 2x^2.\)

Attachment:
11.14ps(a).png
11.14ps(a).png [ 27.72 KiB | Viewed 185 times ]


So, the ratio of the area of the shaded region to the area of the unshaded region is \(1:1\).

Therefore, D is the answer.
Answer: D
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Re: ABCD is a square. The dotted lines are the lines connecting two midpoi   [#permalink] 17 Nov 2019, 18:42
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