A farmer has a field in the form of a parallelogram PQRS as shown in the figure. He took the mid- point A on RS and joined it to points P and Q. In how many parts of field is divided? What are the shapes of these parts?

The farmer wants to sow groundnuts which are equal to the sum of pulses and paddy. How should he sow? State reasons?

It can be seen from the figure that the field is divided in three triangular parts ΔSPA, ΔAPQ and ΔARQ

Extend the segment PQ to B and drop a perpendicular from point R on the extended line

Thus the segment RC becomes the height of ΔAPQ and also the height of parallelogram PQRS

consider parallelogram PQRS

base = PQ

height = RC

area of parallelogram = base × height

area(PQRS) = PQ × RC …(i)

For ΔAPQ

Base = PQ

Height = RC ...(because even if we drop a perpendicular from point

A on base PQ it would be of the same length as RC

since SR||PB)

Area of ΔAPQ = × AP × RC

Using (i)

⇒ Area of ΔAPQ = × Area of parallelogram PQRS

⇒ 2 × area(ΔAPQ) = Area of parallelogram PQRS …(ii)

Since Area(PQRS) = area(ΔPSA) + area(ΔAPQ) + area(ΔAQR)

Using equation (ii) we get

2 × area(ΔAPQ) = area(ΔPSA) + area(ΔAPQ) + area(ΔAQR)

⇒ area(ΔAPQ) = area(ΔPSA) + area(ΔAQR) …(iii)

let the number of groundnuts be g, pulses be p_u and paddy be p_a

given g = p_u + p_a

compare this with equation (iii) we get

area(ΔAPQ) = g

area(ΔPSA) = p_u

area(ΔAQR) = p_a

therefore, the farmer must sow ground nuts in the region under the area(ΔAPQ), the pulses in the region under the area(ΔPSA) and the paddy in the region under the area(ΔAQR)