Question

Question 12
A villager Itwaari has a plot of land of 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 the 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.

Answer 1

Let quadrilateral ABCD be the original shape of the field. The proposal may be implemented as follows.
Join diagonal BD and draw a line parallel to BD through point A. Let it meet the extended side CD of ABCD at point E.
Join BE and AD. Let them intersect each other at O.
Then, portion $\Delta AOB$ can be cut from the original field so that the new Shape of the field will be $\Delta BCE$. (See figure) We have to prove that the area of $\Delta AOB$ (portion that was cut so as to construct Health Centre) is equal to the area of $\Delta DEO$ (portion added to the field so as to make the area of the new field so formed equal to the area of the original field).

It can be observed that $\Delta DEB$ and $\Delta DAB$ lie on the same base BD and are between the same parallels BD and AE.
$\therefore Area\left(\Delta DEB\right)=Area\left(\Delta DAB\right)$
Then, $Area\left(\Delta DEB\right)-Area\left(\Delta DOB\right)=Area\left(\Delta DAB\right)-Area\left(\Delta DOB\right)$
[Subtracting Area $\left(\Delta DOB\right)$ from both sides]
$\Rightarrow Area\left(\Delta DEO\right)=Area\left(\Delta AOB\right)$