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Q: 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 an 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.

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We have a quadrilateral shaped plot ABCD. Draw DF || AC and AF || CF.
Now, \DeltaDAF and \DeltaDCF are on the same base DF and between same parallels AC and DF.
\therefore  ar (\DeltaDAF) = ar( \DeltaDCF)
On subtracting  \DeltaDEF from both sides, we get

ar( \DeltaADE) = ar( \DeltaCEF)...............(i)
The portion of  \DeltaADE can be taken by the gram panchayat and on adding the land \DeltaCEF to his (Itwaari) land so as to form a triangular plot.(\DeltaABF)

We need to prove that ar( \DeltaABF) = ar (quad. ABCD) 

Now, adding ar(quad. ABCE) on both sides in eq (i), we get

ar ( \DeltaADE) + ar(quad. ABCE) = ar( \DeltaCEF) + ar(quad. ABCE)
ar (ABCD) = ar( \DeltaABF)


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