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  • 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.

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.

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Answer: 9:1

Given:- PQRS is a trapezium in which PQ || RS and PQ = 3 RS.

\Rightarrow \frac{PQ}{RS}=\frac{3}{1}

In\; \Delta POQ \; and\; ROS

\angle SOR = \angle QOP          (vertically opposite angles)

\angle SRP = \angle RPQ           (alternate angle)

As we know that if two angles of one triangle are equal to the two angles of another triangle, then the two triangles are similar by AA similarity criterion

\therefore \Delta POQ\sim \Delta ROS

By the property of area of similar triangle

\frac{ar\left ( \Delta POQ \right )}{ar\left ( \Delta SOR \right )}=\frac{\left ( PQ \right )^{2}}{\left ( RS \right )^{2}}=\left ( \frac{PQ}{RS} \right )^{2}=\left ( \frac{3}{1} \right )^{2}

\frac{ar\left ( \Delta POQ \right )}{ar\left ( \Delta SOR \right )}=\frac{9}{1}

Hence the required ratio is 9: 1.

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