In the given figure,ABCD is a quadrilateral and E,F G and H are respectively the mid-points of its sides. Prove that the area of the parallelogram EFGH formed by joining the mid-points of the sides of the quadrilateral is half the area of the quadrilateral.
The question can be easily proved if the ABCD is a parallelogram but as the ABCD is a quadrilateral it is being difficult. How to go about it?
Best Answer
$EF$ is the basis media of triangle $ABC$, and then $A(BEF)=1/4A(ABC)$. Idem $A(GHD)=1/4A(ACD)$.
Now, note that $A(ACD)+A(ABC)=A(ABCD)\to A(BEF)+A(GHD)=1/4A(ABCD)[*_1]$.
Idem, $ A(HEA)+A(GFC)=1/4A(ABCD)[*_2]$.
Adding $ [*_1] +[*_2]$ the problem is solved