Question

Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ = 3 RS, Then the ratio of areas of triangles POQ and ROS is:​

Answer 1

Please find the attachment

Answer 2

Given :-

• PQRS is a Trapezium.
• Diagonals of a trapezium PQRS intersect each other at the point O.
• PQ || RS.
• PQ = 3 RS.

Construction :-

• Refer to image.

Solution :-

in ∆POQ and ∆ROS we have ,

→ ∠OQP = ∠OSR (PQ || RS, therefore, Alternate interior angle.)

Similarly,

→ ∠OPQ = ∠ORS (PQ || RS, therefore, Alternate interior angle.)

Therefore,

→ ∆POQ ~ ∆ROS (By AA similarity).

So,

By Theorem :-

• If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

Hence,

→ (Area ∆POQ) : (Area ∆ROS) = PQ² : RS²

→ (Area ∆POQ) : (Area ∆ROS) = 3² : 1²

→ (Area ∆POQ) : (Area ∆ROS) = 9 : 1 (Ans.)