Question

State True or False

If $PQRS$ is a quadrilateral in which diagonal $PR$ and $QS$ intersect at $O$. Then The sum of all the sides is greater than the sum of its diagonal i.e. $PQ+QR+RS+SP>PR+QS$

• A. True
• B. False

Answer 1

The correct option is A True

$\Rightarrow$ In triangle, sum of two sides is more than the third side.

So, in $△PQR,$

$\Rightarrow$ $PQ+QR>PR$ ----- ( 1 )

In $△PSR,$

$\Rightarrow$ $PS+SR>PR$ -----( 2 )

Adding equation ( 1 ) and ( 2 ) we get,

$\Rightarrow$ $PQ+QR+RS+SP>2PR$ ------ ( 3 )

Now, in $△QRS,$

$\Rightarrow$ $QR+RS>QS$ ------ ( 4 )

In $△QSP,$

$\Rightarrow$ $PQ+SP>QS$ ----- ( 5 )

Adding ( 4 ) and ( 5 ) we get,

$\Rightarrow$ $PQ+QR+RS+SP>2QS$ ----- ( 6 )

Adding ( 3 ) and ( 6 ) we get,

$\Rightarrow$ $2(PQ+QR+RS+SP)>2(PR+QS)$

$\therefore$ $PQ+QR+RS+SP>PR+QS$