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Q.

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

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a

1:3

b

3:1

c

1:9

d

9:1

answer is D.

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Detailed Solution

Given a trapezium PQRS in which PQ is parallel to RS and PQ = 3RS.
The diagonals of trapezium intersect at O.
In △POQ and △ROS
∠SOR = ∠QOP (Vertically opposite angles)
∠SRP = ∠RPQ (Alternate angles)
△POQ ∼ △ROS (By AAA similarity criterion)

\frac{ar (ΔPOQ)}{ar (ΔSOR)} = \frac{{PQ}^{2}}{{RS}^{2}} = {(\frac{PQ}{RS})}^{2} {= (\frac{3 RS}{RS})}^{2} = \frac{9}{1}

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