A villager Ramayya has a plot of land in the shape of a quadrilateral. The gram panchayat of the village decided to take over some portion of his plat from one of the corners of construct a school. Ramayya agrees to the above proposal with the condition that he should be given equal amount of land in exchange of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented. (Draw a rough sketch of plot).

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Solution

Let $A B C D$ be the quadrilateral plot produced $B A$ to meet $D E$ draw parallel to $E A$ at $E$ to join $E C$
The $△ B A C$ and $△ A D C$ lie on the same parallel $D E$ and $C A$
$a r (E A C) - a r (A D C)$
Now $A r (A B C D) = a r (A B C) + a r (A C D)$
$a r (A B C) + a r (E A C) = a r (E B C)$
$A r (A B C D) = a r (A B C) + A r (A D C) = a r (E B C)$
$∴$ $A R (A B C D) = A r (E B C)$

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Let ABCD be the quadrilateral plot produced BA to meet DE draw parallel to EA at E to join ECThe △BAC and △ADC lie on the same parallel DE and CA ar(EAC)−ar(ADC)Now Ar(ABCD)=ar(ABC)+ar(ACD)ar(ABC)+ar(EAC)=ar(EBC)Ar(ABCD)=ar(ABC)+Ar(ADC)=ar(EBC)∴ AR(ABCD)=Ar(EBC)