Question

State whether the statement is true/false

In a quadrilateral ABCD, AB + BC + CD + DA < 2 (BD + AC).

• A. True
• B. False

Answer 1

The correct option is A True

Sum of lengths of any two sides of a triangle should be greater than the third side.

$In△AOB,$

$AO+BO>AB-\left(1\right)$

$In△BOC,$

$BO+CO>BC-\left(2\right)$

$In△COD,$

$CO+DO>CD-\left(3\right)$

$In△AOD,$

$AO+DO>AD-\left(4\right)$

$Adding\left(1\right)\left(2\right)\left(3\right)\&\left(4\right)$

$⟹AB+BC+CD+DA<2(OA+OB+OC+OD)$

$⟹$ $AB+BC+CD+DA<2(OA+OA)+2(OB+OD)$

$⟹AB+BC+CD+A<2(AC+BD)$

Hence proved.