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 w

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)`