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 plt. Explain how his proposal will be implemented.

Explanation

Construction
Join the diagonal `BD`.
Draw `AE` parallel to `BD`.
Join `BE`, that intersected `AD` at `O`.
We get,
`△BCE` is the shape of the original field
`△AOB` is the area for constructing health centre.
`△DEO` is the land joined to the plot.
To prove:
`ar(△DEO) = ar(△AOB)`
Proof:
`ΔDEB` and `ΔDAB` lie on the same base `BD`, in-between the same parallels `BD` and `AE.`
Therefore,
`Ar(ΔDEB) = ar(ΔDAB)`
`=> ar(ΔDEB) – ar(ΔDOB) = ar(ΔDAB) – ar(ΔDOB)`
`=> ar(ΔDEO) = ar(ΔAOB)`