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

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by Max@Math Revolution » Thu Nov 14, 2019 12:06 am
[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?

Image

A. 1.5:1
B. 1.3:1
C. 1.1:1
D. 1:1
E. 1:1.1
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by nitink » Thu Nov 14, 2019 7:50 pm
Assume that each side is 10 units. So. total area = 10 * 10 = 100


Image

from the figure, we can see that j is the mid point of DN , so DJ = 5 /2 = 2.5.

So the area of triangle JOI = 1/2 * JI * ON = 1/2 * (2.5+2.5) * 5 = 25/2.

area of shaded region (4 triangles) = 4* 25/2 = 50

area of unshaded region = total area - shaded region area = 100 -50 = 50

So ratio = 50 /50 = 1:1
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by Max@Math Revolution » Sun Nov 17, 2019 5:17 pm
=>

Assume MF = BF = BG = GN = x.
The area of the square of OGBF is the sum of areas of triangles OGB and OFB , (1/2)(x)(2x) + (1/2)(x)(2x) = 2x^2 and the shaded area of the triangles OFM and ONG is (1/2)(x)(2x) + (1/2)(x)(2x) = 2x^2.

Image

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.
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