In square ABCD above, if DE = EB and DF = FC, then the area of the sha

Question

https://gmatclub.com/forum/download/file.php?id=36145 In square ABCD above, if DE = EB and DF = FC, then the area of the shaded region is what fraction of the area of square region ABCD? (A) 1/16 (B) 1/8 (C) 1/6 (D) 1/4 (E) 1/3 2017-07-07_1134.png

shreyashree · Accepted Answer

Consider any square. Draw the diagonals. Join the midpoints of the opposite sides.

You will see that there are eight smaller triangles each similar to the other. The area of all these smaller triangles is same. 
Lets say the area of one small triangle is x. Thus the sum of the areas of all 8 triangles would be 8x which would also be the area of the square itself .

Triangle EFC's area shown in the figure above is thus 1/8th of the area of the square.

Thus option B would be correct.

Regards, 
Shreya

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sashiim20 · Answer

Let the side of square be  = 2 . Therefore,  AB = BC = CD = AD = 2

Area of square  ABCD = 2^2 = 4

Given;  DF = FC . Therefore;  DF = FC = 1

Diagonal of square  = (side) \sqrt{2} . Therefore,  BD = 2\sqrt{2}

DE = EB = \frac{(2\sqrt{2})}{ 2}   => DE = EB = \sqrt{2}

Joining  EC  to  A , we get  AC  as another diagonal. Therefore,  EC = DE = EB . Hence  EC = \sqrt{2}

Using Pythagorean theorem in  	riangle EFC , we get;

EF^2 + FC^2 = EC^2

EF^2 = EC^2 - FC^2

EF^2 =  \sqrt{2^2}   - 1^2

EF^2 = 2 - 1 = 1

EF = \sqrt{1} = 1

Shaded region is  	riangle EFC , Which is right angled triangle.

Area of shaded region  = \frac{1}{2} EF * FC = \frac{1}{2} * 1*1 = \frac{1}{2}

Required fraction = Area of the shaded region / Area of square

Required fraction  = \frac{(1/2)}{4} = \frac{1}{8}

Answer (B)...

prerakgoel03 · Answer

B is the correct answer as Area of square is Xsquare and area of triangle is xsquare/8 so fraction will be 1/8.

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amanvermagmat · Answer

We are given that E is the mid point of diagonal DB.

Now lets say area of square ABCD is 1.

Then, area of triangle DCB = 1/2 of full square = 1/2
Now, since E is mid point of DB, area of triangle CED = 1/2 of triangle DCB = 1/4 
Also, since F is mid point of CD, area of triangle EFC (Shaded region) = 1/2 of triangle CED = 1/8

Thus shaded area is 1/8 of total square's area. Hence B  answer.

Shiv2016 · Answer

I solved this question using:
- 45-45-90 triangle

EF is perpendicular to DC and thus angle DFC is 90.

In triangle BDC, BC= DC (sides of a square). Therefore, angle DBC is 45.

Now triangle DFE is an isosceles triangle in which DF=FE. So sides are in ratio 1:1:root 2

Area of triangle (shaded)= 1/2 *x^2
Area of square= 4x^2

The answer is 1/8

bumpbot · Answer

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In square ABCD above, if DE = EB and DF = FC, then the area of the sha

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In square ABCD above, if DE = EB and DF = FC, then the area of the sha

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