In the figure above, rectangle ABCD has perimeter 22; the length of DC

Question

https://gmatclub.com/forum/download/file.php?id=29251 In the figure above, rectangle ABCD has perimeter 22; the length of DC is 7. What is the area of DEF? A. 12 B. 8 \sqrt{3} C. 24 D. 32 E. 64 2016-02-07_2128.png

OptimusPrepJanielle · Answer

Hi Bunuel, I think we need the area of ADEF here and not ODEF.
Can you please check it once.

KarishmaB · Answer

I am assuming we are looking for the area of the triangle DEF (if we needed the area of quad ADEF, we would have written it as ADFE so guessing that O is a typo).

Since ABCD is a rectangle and CD is 7, AB is also 7.
The perimeter of ABCD is 22 so BC + AD = 22 - 7*2 = 8
Since BC = AD, AD = 4

Triangle ADE is a 30-60-90 triangle so sides are in the ratio  1:\sqrt{3}:2 .
Since AD = 4, DE =  4*\sqrt{3}

Tringle DEF is 45-45-90 triangle so the two legs are equal in length. Since DE is  4*\sqrt{3} , EF is also  4*\sqrt{3} .

Area of DEF =  (1/2)*4*\sqrt{3}*4*\sqrt{3} = 24

Answer (C)

Bunuel · Answer

Edited. It's DEF not ODEF. Thank you.

Icecream87 · Answer

DA = 4 
 DE = EF = 4\sqrt{3} 
so answer C

cnbcravi · Answer

30 ,60 ,90 triangle and 45 , 45 , 90 triangle are quit common from gmat point. first we will find the othr side and then using this combination we can find requird value. Since ABCD is a rectangle and CD is 7, AB is also 7.
The perimeter of ABCD is 22 so BC + AD = 22 - 7*2 = 8
Since BC = AD, AD = 4

Triangle ADE is a 30-60-90 triangle so sides are in the ratio 1:3√:21:3:2.
Since AD = 4, DE = 4∗3√4∗3

Tringle DEF is 45-45-90 triangle so the two legs are equal in length. Since DE is 4∗3√4∗3, EF is also 4∗3√4∗3.

Area of DEF = (1/2)∗4∗3√∗4∗3√=24(1/2)∗4∗3∗4∗3=24 
, so correct answer is c. please give kudos ..it motivates you...

arjunperiyasamys · Answer

use of pythogorous triplets:
rectangle ABCD:
14 + 2x = 22
2x = 8
x = 4

using pythogorous triplets 30:60:90 = x: xsqrt3 : 2x

we know side DE is 4sqrt3

using pythogorous triplets 45:45:90 = x: x : xsqrt2 = : 4sqrt3 : 4sqrt6

Area = 1/2 x base x height = 24

bumpbot · Answer

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In the figure above, rectangle ABCD has perimeter 22; the length of DC

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In the figure above, rectangle ABCD has perimeter 22; the length of DC

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