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the diagonals of a quadrilateral PQRS intersect each other at point O such that PO/QR=OQ/OS.Prove that PQRS is atrapezium
 
Asked by Pawan | 24 Aug, 2014, 02:53: PM
answered-by-expert Expert Answer
A quadrilateral PQRS whose diagonals PR and QS intersect each other at O such that fraction numerator O S over denominator O Q end fraction equals fraction numerator O R over denominator O P end fraction
To prove : Quadrilateral PQRS is a trapezium
Construction : Draw OT || PQ meeting PS at T.

Proof : In trianglePSR, OT || SR....(i)
fraction numerator S T over denominator P T end fraction equals fraction numerator R O over denominator P O end fraction                                         (By Basic proportionaity theorem)..........(ii)
Also, fraction numerator S O over denominator O Q end fraction equals fraction numerator R O over denominator P O end fraction                                (Given)...........(iii)
From (ii) and (iii), we get

fraction numerator S T over denominator P T end fraction equals fraction numerator S O over denominator Q O end fraction
 SR || OT                                                   (Converse of Basic proportionality theorem)......(iv) 
From (i) and (iv), we get
SR || PQ
Hence quadrilateral PQRS is trapezium.

 

Answered by Anuja Salunke | 25 Aug, 2014, 10:07: AM

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