O is any point on the diagonal PR of a parallelogram PQRS. Prove that the areas of triangles Class 9

O is any point on the diagonal PR of a parallelogram PQRS. Prove that the areas of triangles Class 9

Question 1.

O is any point on the diagonal PR of a parallelogram PQRS. Prove that the areas of triangles PSO and PQO are equal. Class 9

Solution:

Given: In a parallelogram PQRS, O is any point on the diagonal PR.

To prove: Area (∆PSO) = area (∆PQO)

Construction: Join SQ which intersects PR at B.

Proof:

We know that the diagonals of a parallelogram bisect each other.

∴ B is the midpoint of SQuestion

In ∆QPS, PB is a median.

A median divides a triangle into two triangles of equal area.

∴ Area(∆BPQ) = area(∆BPS) ...(i)

Also, in ∆QOS, OB is a median.

∴ Area (∆OBQ) = area(∆OBS) ...(ii)

Adding (i) and (ii), we get:

Area (∆BPQ) + area (∆OBQ) = area(∆BPS) + area (∆OBS)

∴ Area (∆PQO) = area(∆PSO)

Hence proved.