# Q: 6       A farmer was having a field in the form of a parallelogram PQRS. She took any point A on RS and joined it to points P and 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?

M manish

We have a field in the form of parallelogram PQRS and a point A is on the side RS. Join AP and AQ. The field is divided into three parts i.e, $\Delta$APS, $\Delta$QAR and $\Delta$PAQ.

Since $\Delta$APQ and parallelogram, PQRS is on the same base PQ and between same parallels RS and PQ.
Therefore, $ar(\Delta APQ) = \frac{1}{2}ar(PQRS)$............(i)
We can write above equation as,

ar (||gm PQRS) - [ar ($\Delta$APS) + ar($\Delta$QAR)] = 1/2 .ar(PQRS)
$\Rightarrow ar(\Delta APS)+ar(\Delta QAR) = \frac{1}{2}ar(PQRS)$
from equation (i),
$\Rightarrow ar(\Delta APS)+ar(\Delta QAR) =ar(\Delta APQ)$

Hence, she can sow wheat in $\Delta$APQ and pulses in [$\Delta$APS + $\Delta$QAR] or wheat in [$\Delta$APS + $\Delta$QAR] and pulses in  $\Delta$APQ.

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