Question

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 a triangular plot. Explain how this proposal will be implemented.

Answer 1

Let gram panchayat construct health centre in

$Ar\left(EDC\right)$ by taking land $ODC$

So, Itwaari gives up $ODC$ and takes adjacent land $DEA$ in return

Now he has land $EBC$

To Prove: Old land $=$ New land

$\therefore Ar\left(ABCD\right)=Ar\left(EBC\right)$

$△ACD$ and $△ACE$ lie on same base $AC$

and between same parallel line $AC$ and $DE$

$\therefore$ $Ar\left(ABC\right)=Ar\left(ACE\right)$

Adding $Ar\left(ABC\right))$ both sides

$Ar\left(ACD\right)+Ar\left(ABC\right)=Ar\left(ACE\right)+Ar\left(ABC\right)$

$Ar\left(ABCD\right)=Ar\left(EBC\right)$

Hence, proved