Diagonals of a trapezium PQRS intersect each other at at the point 0, PQ‖RS and PQ = 3RS. Find the ratio of the areas of ∆POQ and ∆ROS.

Question

Diagonals of a trapezium PQRS intersect each other at at the point 0, PQ ‖ RS and PQ = 3RS. Find the ratio of the areas of ∆POQ and ∆ROS.

Philoid · Accepted Answer

Given,

PQRS is a trapezium in which PQ‖RS and PQ = 3RS

In ∆POQ and ∆ROS,

∠SOR = ∠QOP [vertically opposite angles]

∠SRP = ∠RPQ [alternate angles]

∴ ∆POQ ∼ ∆ROS [by AAA similarity criterion]

By property of area of similar triangle,

[from Equation (i)]

Hence, the required ratio is 9:1.