Question

If the mid-points of sides of quadrilateral are joined in order,prove that the area of parallelogram so formed will be half of area of the given quadrilateral.

Answer 1

We need to prove that $\left(\Delta POQ\right)\left(\Delta SOR\right)=\left(\Delta POS\right)\left(\Delta QOR\right)$ as area.

We draw the perpendiculars SD and QE to the diagonal PR. Then the area of triangles are
$\Delta POQ=\frac{1}{2}PO.QE$
$\Delta QOR=\frac{1}{2}OR.QE$
$\Delta POS=\frac{1}{2}OP.SD$
$\Delta ROS=\frac{1}{2}OR.SD$
Therefore
$\Delta POQ.\Delta SOR=(\frac{1}{2}OR.SD)(\frac{1}{2}PO.QE)=\frac{1}{4}\left(OR\right)\left(SD\right)\left(OP\right)\left(QE\right)$
$\Delta QOR.\Delta POS=(\frac{1}{2}OR.QE)(\frac{1}{2}OP.SD)=\frac{1}{4}\left(OR\right)\left(SD\right)\left(OP\right)\left(QE\right)$
Therefore
$\Delta POQ.\Delta SOR=\Delta QOR.\Delta POS$