Question

A farmer was a having a field in the form of a parallelogram $PQRS$. He took a point $A$ on $RS$ and joined it to point $P$ and $Q$. In how many parts the field is divided? What are the shapes of these parts? The farmer want to sow wheat and pulses in equal portions of the field separately. How should he do it?

Answer 1

The field is divided in three triangles.

Since $△APQ$ and parallelogram $PQRS$ are on the same base $PQ$ and between the same parallels $PQ$ and $RS$.

$\therefore ar\left(APQ\right)=\frac{1}{2}ar\left(PQRS\right)$

$2ar\left(APQ\right)=ar\left(PQRS\right)$

But,

$ar\left(PQRS\right)=ar\left(APQ\right)+ar\left(PSA\right)+ar\left(ARQ\right)\left(\text{From fig.}\right)$

$2ar\left(APQ\right)=ar\left(APQ\right)+ar\left(PSA\right)+ar\left(ARQ\right)$

$ar\left(APQ\right)=ar\left(PSA\right)+ar\left(ARQ\right)$

Thus, $\text{area of}△APQ=\text{area of}△PSA+\text{area of}△ARQ$

To sow wheat and pulses in equal portions of the field separately, farmer should sow wheat in $△APQ$ and pulses in other two triangles or pulses in $△APQ$ and wheat in other two triangles.