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

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Math Expert
Joined: 02 Sep 2009
Posts: 56244
The shaded region in the figure above represents a rectangular frame  [#permalink]

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17 Sep 2018, 22:02
00:00

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

Attachment:

image024.gif [ 2.1 KiB | Viewed 375 times ]

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Intern
Joined: 31 Aug 2018
Posts: 3
GMAT 1: 610 Q43 V31
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

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27 Sep 2018, 01:10
Bunuel Can you please provide an explanation?
Intern
Joined: 17 May 2018
Posts: 49
The shaded region in the figure above represents a rectangular frame  [#permalink]

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27 Sep 2018, 03:58
1
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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Joined: 30 May 2017
Posts: 19
Re: The shaded region in the figure above represents a rectangular frame  [#permalink]

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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
Re: The shaded region in the figure above represents a rectangular frame   [#permalink] 27 Sep 2018, 08:48
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