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Question

Find the area of WXYZ from the figure given below provided that E, F, G, H are the respective mid points.
 


A
Ar(WXYZ)=15Ar(ABCD)
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B
Ar(WXYZ)=12Ar[AEW]+12Ar[FXD]+12Ar[GYC]+12Ar[BHZ]
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C
Ar[WXYZ]=12Ar[ABCD]
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D
Ar(WXYZ)=14Ar(ABCD)
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Solution

The correct option is A Ar(WXYZ)=15Ar(ABCD)
Ar[WXYZ]=15Ar[ABCD]
Trapezium across the sides of the square WXYZ
Note the triangle at all the 4 corners ΔAEW,ΔFXD,ΔCYG,ΔBZH
Hypotenuse of each of these triangles AE, FD, GC, BH is same as the slatted sides of the trapezium.
This is because  E, F, G and H are midpoints AF = FD likewise for the others as well.
Now by placing the triangles in such a way it coincides with the trapezium we get

Now we notice we have 5 smaller squares.
Hence, areas of all the squares will be equal since they have the same side length .
Ar[WXYZ]=15Ar[ABCD]

Mathematics

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