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# A farmer was having a field in the form of a parallelogram PQRS. She took any point $ A $ on $ \mathrm{RS} $ and joined it to points $ \mathrm{P} $ and $ \mathrm{Q} $. In how many parts the fields is divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?

**Given:**

A farmer was having a field in the form of a parallelogram PQRS. She took any point \( A \) on \( \mathrm{RS} \) and joined it to points \( \mathrm{P} \) and \( \mathrm{Q} \).

The farmer wants to sow wheat and pulses in equal portions of the field separately.

To do:

We have to find the number of parts the field has to be divided and the shapes of these parts.

**Solution:**

From the figure, we can observe that,

The field is divided into three parts each in a triangular shape.

$\triangle APQ$ and parallelogram $PQRS$ lie on the same base $PQ$ and between the same parallels $PQ$ and $SR$.

This implies,

$ar (\triangle APQ) = \frac{1}{2} ar (PQRS)$...….(i)

Now,

$ar (\triangle APQ)+ar (\triangle PAS)+ar (\triangle QAR)=ar (PQRS)$

$\Rightarrow \frac{1}{2} ar (PQRS)+ar (\triangle PAS)+ar (\triangle QAR)=ar (PQRS)$ [From(i)]

$ar (\triangle PAS)+ar (\triangle QAR)=ar (PQRS)-\frac{1}{2} ar (PQRS)$

$ar (\triangle PAS)+ar (\triangle QAR)=\frac{1}{2} ar (PQRS)$............(ii)

From (i) and (ii), we can conclude that,

The farmer must sow wheat or pulses in $\triangle PAQ$ or either in both $\triangle PSA$ and $\triangle QAR$.

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