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 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.

Answer 1

Hint: In this question, first of all draw the diagram of the original shape of the field and then cut the land which would be given to build the Health Centre. And add the area of plot given by Gram Panchayat with the remaining Itwaari`s plot. Now show that the area of the new plot is equal to the area of the old plot.

Complete step-by-step answer:
Let quadrilateral \[ABCD\] be the original shape of the field owned by Itwaari.
Let us join \[AC\] and draw \[DE\parallel AC\]
Joining \[CE\] and \[EA\] we have figure as shown in below:

Let Gram Panchayat construct its Health centre in area of \[\Delta EDC\] 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., \[ar\left( {ABCD} \right) = ar\left( {ACE} \right)\]
Triangles \[ACD\] and \[ACE\] have the same base \[AC\] and lies between the same parallel lines \[AC\& DE\]
Therefore, \[ar\left( {ACD} \right) = ar\left( {ACE} \right)\] by the property that triangles with the same base and between the same parallels are equal in area.
Adding \[ar\left( {ABC} \right)\] both sides we have
\[
   \Rightarrow ar\left( {ACD} \right) + ar\left( {ABC} \right) = ar\left( {ACE} \right) + ar\left( {ABC} \right) \\
   \Rightarrow ar\left( {ABCD} \right) = ar\left( {EBC} \right) \\
  \therefore ar\left( {ABCD} \right) = ar\left( {EBC} \right) \\
\]
Hence proved.

Note: Triangles with the same base and between the same parallels are equal in area. Here the area of the plot given to the Health centre by Itwaari and the area of plot given by Gram Panchayat are equal in areas but the adjacent corners have changed.