Question

PQRS IS A QUAD. IN WHICH DIAGONALS PR AND QS INTERSECT AT O.PROVE THAT PQ+QR+RS+SP<2(PR+QS)

Answer 1

given that ,
 PQRS is a quadrilateral  in which diagonal PR and QS intersect at a O . 
to prove - PQ +QR +RS+SP < 2 ( PR + QS ) 
 proof -
       we know that sum of any two side of a triangle is greater than the third side .
.'. in Δ PQO , 
       PO+QO>PQ ,  .......................(i)
    in Δ SOP  
        SO + PO >PS , .........................(ii)
   in Δ SOR 
       SO + OR > RS  ...........................(iii)
   in Δ QOR , 
     QO + OR > QR ...........................(iv)
on adding eqn. i , ii , iii & iv 
    we get ,
PO+QO+SO+PO+SO+OR+QO+OR > PQ+PS+SR+QR 
also ⇒ 2 ( PO + QO + SO + OR ) > PQ+PS+SR + QR 
       = 2( PR + QS ) > PQ+PS+RS + QR  ( proved) ....
 
 
 

Answer 2

given that , PQRS is a quadrilateral  in which diagonal PR and QS intersect at a O . to prove - PQ +QR +RS+SP < 2 ( PR + QS )  proof -       we know that sum of any two side of a triangle is greater than the third side ..'. in Δ PQO ,        PO+QO>PQ ,  .......................(i)    in Δ SOP          SO + PO >PS , .........................(ii)   in Δ SOR        SO + OR > RS  ...........................(iii)   in Δ QOR ,      QO + OR > QR ...........................(iv)on adding eqn. i , ii , iii & iv     we get ,PO+QO+SO+PO+SO+OR+QO+OR > PQ+PS+SR+QR also ⇒ 2 ( PO + QO + SO + OR ) > PQ+PS+SR + QR