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 comers to construct a Health Centre. Itwaari 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 this proposal will be implemented.

Answer 1

Let quadrilateral $ABCD$ be the original shape of the field owned by Itwaari.

Let us join $AC$ and draw $DEparallelAC$

Join $CE$ and $EA$

Let gram panchayat construct its health center in ar$left\left(EDCright\right)$ by taking land $ODC$

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

Now he has land $EBC$

We need to prove that area of old land$=$area of new land

i.e.,area$left\left(ABCDright\right)=$area$left\left(EBCright\right)$

$triangleACD$ and $triangleACE$ have the same base $AC$ and between the same parallel lines $AC$ and $DE$

$therefore$ area$left\left(triangleACDright\right)=$area$left\left(triangleACEright\right)$ using triangles with same base and between the same parallels are equal in area.

Adding area$left\left(triangleABCright\right)$ both sides, we get

area$left\left(triangleACDright\right)+$area$left\left(triangleABCright\right)=$area$left\left(triangleACEright\right)+$area$left\left(triangleABCright\right)$

$therefore$ area$left\left(ABCDright\right)=$area$left\left(EBCright\right)$

Hence proved.

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