Question

A farmer was having a field in the form of a parallelogram $ \mathrm{PQRS}$. She took any point $ \mathrm{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?

Answer 1

As we observe from the figure, the field is divided into three parts each in triangular shape.

$\mathrm{\Delta PSA},\mathrm{\Delta PAQ}\mathrm{and}\mathrm{\Delta QAR}$ are the triangles.

We know that,

$\mathrm{Area}\mathrm{of}(\mathrm{\Delta PSA}+\mathrm{\Delta PAQ}+\mathrm{\Delta QAR})=\mathrm{Area}\mathrm{of}\mathrm{PQRS}—\left(\mathrm{i}\right)$

$\mathrm{Area}\mathrm{of}\mathrm{\Delta PAQ}=\frac{1}{2}\mathrm{area}\mathrm{of}\mathrm{PQRS}—\left(\mathrm{ii}\right)[\mathrm{When}\mathrm{the}\mathrm{triangle}\mathrm{and}\mathrm{parallelogram}\mathrm{are}\mathrm{on}\mathrm{the}\mathrm{same}\mathrm{base}\mathrm{and}\mathrm{between}\mathrm{the}\mathrm{same}\mathrm{parallel}\mathrm{lines}\mathrm{then},\mathrm{Area}\mathrm{of}\mathrm{triangle}=\frac{1}{2}\mathrm{Area}\mathrm{of}\mathrm{parallelogram}.$

From (i) and (ii) we get,

$\mathrm{Area}\mathrm{of}\mathrm{\Delta PSA}+\mathrm{Area}\mathrm{of}\mathrm{\Delta QAR}=\frac{1}{2}\mathrm{area}\mathrm{of}\mathrm{PQRS}—\left(\mathrm{iii}\right)$

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

The farmer must sow wheat or pulses in $\mathrm{\Delta PAQ}$ or either in both $\mathrm{\Delta PSA}\mathrm{and}\mathrm{\Delta QAR}.$