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the area of the middle region?

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the area of the middle region?

by sanju09 » Tue Oct 18, 2011 2:42 am
A triangle is cut into three regions of equal height by two lines of lengths 3 and 6 that are parallel to the base of length 9. If the height of the triangle is h, what is the area of the middle region?
(A) 0.5 h
(B) 3 h
(C) 18 h
(D) 1.5 h
(E) 9 h
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by Anurag@Gurome » Tue Oct 18, 2011 2:51 am
sanju09 wrote:A triangle is cut into three regions of equal height by two lines of lengths 3 and 6 that are parallel to the base of length 9. If the height of the triangle is h, what is the area of the middle region?
Refer to the following figure,
Image

MN = 6, and PQ = 3


Area of the triangle AMN = (2h/3)*(6)/2 = 2h
Area of the triangle APQ = (h/3)*(3)/2 = 0.5h

Hence, area of the middle region = area of PQMN = (2h - 0.5h) = 1.5h

The correct answer is D.
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by mrandom » Tue Oct 18, 2011 2:52 am
[img]https://img542.imageshack.us/img542/2161/gmatanswer1.jpg[/img]

As the regions are of equal heights, the height of the middle region is h/3.

The area would be h/3 * (3+6)/2 = 9h / 6 = 1.5h, so the answer is (D)
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by neelgandham » Tue Oct 18, 2011 3:01 am
From the details in the question, we can conclude that the three triangles(APQ, AMN, ABC) are similar triangles with the ratio of sides 1:2:3, so the areas are in the ratio 1:4:9.

Area of tri APQ = 1x
Area of tri AMN = 4x
Area of the mid region = [spoiler]4x-x = 3x = (3 * 0.5h) = 1.5h, The correct answer is D.[/spoiler]

Just another approach ! @Anurag sir, correct me if I am wrong
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by sanju09 » Tue Oct 18, 2011 3:24 am
Great work Anurag and neelgandham, this question was prepared with the inquisitiveness to know whether the wordings are appropriate to describe the figure uniquely or is there still any ambiguity apart from the topsy-turvy answer choices. Anurag drew the figure exactly what the question intended to. D is the right answer and neelgandham's approach is too good.
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