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# 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 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.

**Given:**

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 equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot.

**To do:**

We have to explain how the proposal will be implemented.

**Solution:**

Let $ABCD$ be the plot of land of the shape of a quadrilateral.

Let $ADE$ be the portion taken over by the Gram Panchayat of the village from one corner $D$ to construct a Health Centre.

Join $AC$ and draw a line through $D$ parallel to $AC$ to meet $BC$ produced at $P$. This implies,

Itwaari must be given the land $ECP$ adjoining his plot so as to from a triangular plot $ABP$.

$ar (\triangle ADE) = ar (\triangle PEC)$

$\triangle DAP$ and $\triangle DCP$ are on the same base $DP$ and between the parallels $DP$ and $AC$.

Therefore,

$ar (\triangle DAP) = ar(\triangle DCP)$

Subtracting $ar (\triangle DEP)$ from both sides, we get,

$ar (\triangle DAP) - ar (\triangle DEP) = ar (\triangle DCP) - ar (\triangle DEP)$

$ar (\triangle ADE) = ar (\triangle PCE)$

Adding $ar (ABCE)$ on both sides, we get,

$ar (\triangle ADE) + ar (ABCE) = ar (\triangle PCE) + ar (ABCE)$

$ar (ABCD) = ar (\triangle ABP)$

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