Question

In a parallelogram PQRS, M and N are the midpoints of the sides PQ and PS respectively. If area of ΔPMN is 20 square units, find the area of the parallelogram PQRS.

Answer 1

Construction: Join SM and SQ.

In a parallelogram PQRS, SQ is the diagonal.
So, it bisects the parallelogram.

∴ Area(DPSQ) = `(1)/(2) xx "Area(parallelogram PQRS)"`

SM is the median of ΔPSQ.

∴ Area(ΔPSM) = `(1)/(2) xx "Area(ΔPSQ)"`

= `(1)/(2) xx (1)/(2) xx "Area(parallelogram PQRS)"`

= `(1)/(4) xx "Area(parallelogram PQRS)"`

Again, MN is the median of ΔPSM.

∴ Area(ΔPMN) = `(1)/(2) xx "Area(ΔPSM)"`

= `(1)/(2) xx (1)/(4) xx "Area(parallelogram PQRS)"`

= `(1)/(8) xx "Area(parallelogram PQRS)"`

⇒ 20 = `(1)/(8) xx "Area(parallelogram PQRS)"`

⇒ Area(parallelogram PQRS) = 160 square units.