Question

A villager Tiwari 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. Tiwari 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: Here, in this problem we will use the property of triangle that is when two triangles lie on the common base and between parallel lines then the area of both the triangles will be equal.

Complete Step-by-step Solution

Let ABCD be the original shape of the field which is a quadrilateral. Join AC and draw ED parallel to AC. Join CE and EA.


Let the area taken for health centre from the plot be DCO and area given to Tiwari in lieu be EAC.
Now, we have to show that $ar(DCO) = ar(EAC)$
The triangle DCA and triangle EAC lie between parallel lines ED and AC with same base as AC.
$ar(DCO) = ar(EAC)$
Area of old plot is given by,
$\begin{array}{l}
A = ar(ABCD)\\
A = ar(DCA) + ar(ACB)\\
A = ar(EAC) + ar(ACB)\\
A = ar(ECB)
\end{array}$
Therefore, $ar(ECB)$ is the area of the new plot.


Note: Here, the area taken for the health centre from the plot has to be a triangle (although we can extend it to make a parallelogram) otherwise, the final plot of Tiwari would not be a triangle. Make a rough diagram of the plot, health centre and final triangular plot and then by using properties of triangles show that the area of Tiwari’s plot taken for making the health centre is same as the area of the new portion added into the plot.