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The shaded region in the figure above represents a rectangular frame

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The shaded region in the figure above represents a rectangular frame  [#permalink]

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New post 17 Sep 2018, 22:02
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C
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E

Difficulty:

  75% (hard)

Question Stats:

50% (04:22) correct 50% (03:20) wrong based on 24 sessions

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The shaded region in the figure above represents a rectangular frame with length 12 inches and width 8 inches. The frame encloses a rectangular picture that has same area as the frame itself. If the diagonal of the picture represents 1/2 of the diagonal distance from one outer corner of the frame to the opposite corner, what is the perimeter of the picture, in inches?


A. 30

B. 20√2

C. 2√52

D. 109

E. √52

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Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

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New post 27 Sep 2018, 01:10
Bunuel Can you please provide an explanation?
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New post 27 Sep 2018, 03:58
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Indeed, I don't find a result that fits with your possible solutions.

Let l = length of picture and w = width of picture

area of the frame = area of the big rectangle - area of the picture = 12*8 - l*w

We're told that the picture has the same area as the frame:
l*w = 12*8 - l*w
2*l*w = 12*8
l*w = 6*8 = 48

Then we're told that the diagonal of the picture is half the length of the diagonal of the frame:
diagonal of frame = \(\sqrt{12^2 + 8^2} = \sqrt{208} = 2*\sqrt{52}\)
diagonal of picture = \(\sqrt{52}\)
or
l^2 + w^2 = 52

We have l*w and we have L^2 + w^2. Let's work with (l+w)^2.

(l+w)^2 = l^2 + w^2 + 2*w*l = 52 + 96 = 148

By applying a square root on both sides:

\(l+w = \sqrt{148} = \sqrt{4*37} = 2*\sqrt{37}\)

We need 2*(l+w) to find the perimeter of the picture. That would be \(4*\sqrt{37}\).

This seems too lengthy for a GMAT problem (maybe there is a shorter way?) but also doesn't match any of the possible solutions.
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Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

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New post 27 Sep 2018, 08:48
Without seing the solution proposed, I followed the same logic and got the exact same result.

The only thing that I can add, is that 37 is close to 36 which is 6^2.

So the solution has to be close to 4*6=24

Considering that 2^(1/2)s more or less equal to 1.2; 20*2^(1/2) = 24 Answer B
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Re: The shaded region in the figure above represents a rectangular frame   [#permalink] 27 Sep 2018, 08:48
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