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Question
Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ = 3 RS. Find the ratio of the areas of triangles POQ and ROS.
Solution
Given PQRS is a trapezium in which PQ || RS and PQ = 3 RS
⇒ `("PQ")/("RS") = 3/1` ...(i)
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,
`("ar(∆POQ)")/("ar(∆SOR)") = ("PQ")^2/("RS")^2`
= `("PQ"/"RS")^2`
= `(3/1)^2` ...[From equation (i)]
⇒ `("ar(∆POQ)")/("ar(∆SOR)") = 9/1`
Hence, the required ratio is 9 : 1.
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