Solve this :

1 .   PQRS   is   a   trapezium   in   which   PQ   is   parallel   to   SR   and   SR > PQ .   A   line   through   P   parallel   to   QR   cuts   the   diagonal   QS   at   X .   Prove   that   ar ( XQR ) = ar ( PQS ) . [ hint : Join   PR ]

Dear Student,

Please find below the solution to the asked query:

We have our diagram , As : 

Here we have join PR .

We know area of triangle = 12×Base × Height

Given : A line through P parallel to QR cuts the diagonal QS at X. So  QR | | PX

Here ,

Area of XQR =  Area of PQR                     --- ( 1 ) (  As both lies in same parallel lines " QR | | PX " , So height is same and have same base = QR )

And

Area of PQR =  Area of PQS                     --- ( 2 ) (  As both lies in same parallel lines " PQ | | SR "(Given ),So height is same and have same base = PQ )

We know " Euclid's first axiom : Things which are equal to the same thing are also equal to one another.  " , So from equation 1 and 2 we get :

Area of XQR =  Area of PQS                                         ( Hence proved )



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