Question

The diagonals of quadrilateral PQRS intersect at point 'O' such that PO/QO=RO/SO. Show that PQRS is a trapezium.

Answer 1

Given: The diagonals of quadrilateral PQRS intersect at point 'O' such that PO/QO = RO/SO.

To find: Show that PQRS is a trapezium.

Solution:

• Now we have given that the diagonals of quadrilateral PQRS intersect at point 'O'.

• Lets construct OT parallel to PQ which meets PS at T.

• Now consider triangle PSR, we have OT || SR  .......(i)

• So ST/PT = RO/PO     .............by proportionality theorem ......(ii)

              Now we have given PO/QO = RO/SO

              We can rewrite it as:

              PO/RO = QO/SO          

              Reciprocating it, we get:

              RO/PO = SO/QO    .............(iii)

              So from (ii) and (iii), we get:

              ST/PT = SO/QO

• So this implies PQ || OT   ..........converse of proportionality theorem ..(iv)

• From (i) and (iv), we get:

              SR || PQ

• So the quadrilateral is trapezium.

Answer:

           So in solution part we proved that the quadrilateral is trapezium.