$A$ villager,Itwaari,has a plot of land in the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given an equal amount of land in lieu of his land,adjoining his plot,so as to form a triangular plot. Explain how this proposal will be implemented.

(N/A) Let the quadrilateral plot be $ABCD$. Let $E$ be the point where the diagonal $AC$ intersects the boundary of the portion to be taken. To implement the proposal,we draw a line through $D$ parallel to $AC$,which meets the extended side $BC$ at point $F$.
Now,consider $\Delta DAF$ and $\Delta DCF$. These triangles are on the same base $DF$ and between the same parallel lines $AC$ and $DF$.
Therefore,$\text{ar}(\Delta DAF) = \text{ar}(\Delta DCF)$.
Subtracting $\text{ar}(\Delta DEF)$ from both sides,we get:
$\text{ar}(\Delta DAF) - \text{ar}(\Delta DEF) = \text{ar}(\Delta DCF) - \text{ar}(\Delta DEF)$
$\Rightarrow \text{ar}(\Delta ADE) = \text{ar}(\Delta CEF)$.
This means the area of the triangular portion $\Delta ADE$ (which the Panchayat takes) is equal to the area of the triangular portion $\Delta CEF$ (which is given to Itwaari). By adding $\Delta CEF$ to the remaining part of the plot,Itwaari gets a new triangular plot $\Delta ABF$.
To verify the area: $\text{ar}(\Delta ABF) = \text{ar}(ABCE) + \text{ar}(\Delta CEF)$.
Since $\text{ar}(\Delta CEF) = \text{ar}(\Delta ADE)$,we have:
$\text{ar}(\Delta ABF) = \text{ar}(ABCE) + \text{ar}(\Delta ADE) = \text{ar}(\text{quadrilateral } ABCD)$.
Thus,the total area remains the same.