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

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

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

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...

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

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