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

Explanation

Let ABCD be the plot of land of the shape of a quadrilateral. Let the portion ADE be taken over by Gram Panchayat of the village from one corner D to construct a Health Centre.
Join, AC draw a line through D parallel to AC to meet BC produced in P. Then, Itwaari must be given the land ECP adjoining his plot so as to form a triangular plot ABP as then.
`ar(DeltaADE) = ar(DeltaPEC)`
Now, `DeltaDAP and DeltaDCP` are on the same base DP and between the same parallel DP and AC.
`:. ar(DeltaDAP) = ar(DeltaDCP)`
`rArr ar(Delta DAP) - ar(Delta DEP) = ar(Delta DCP0 - ar(DeltaDEP)`
(subtracting the same area from both sides)
`rArr ar(Delta ADE) = ar(Delta PCE)`
`rArr ar(Delta DAE) + ar(ABCE) = ar(DeltaPCE) + ar(ABCE)`
(adding the same area on both sides)
`rArr ar(ABCD) = ar(DeltaABP)`