Question

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

• A. $Ar\left(WXYZ\right)=\frac{1}{5}Ar\left(ABCD\right)$
• B. $Ar\left(WXYZ\right)=\frac{1}{2}Ar\left[AEW\right]+\frac{1}{2}Ar\left[FXD\right]+\frac{1}{2}Ar\left[GYC\right]+\frac{1}{2}Ar\left[BHZ\right]$
• C. $Ar\left[WXYZ\right]=\frac{1}{2}Ar\left[ABCD\right]$
• D. $Ar\left(WXYZ\right)=\frac{1}{4}Ar\left(ABCD\right)$

Answer 1

The correct option is A $Ar\left(WXYZ\right)=\frac{1}{5}Ar\left(ABCD\right)$

$Ar\left[WXYZ\right]=\frac{1}{5}Ar\left[ABCD\right]$
Trapezium across the sides of the square WXYZ
Note the triangle at all the 4 corners $\Delta AEW,\Delta FXD,\Delta CYG,\Delta BZH$
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 $\therefore$ 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 .
$\therefore$ $Ar\left[WXYZ\right]=\frac{1}{5}Ar\left[ABCD\right]$