Question

In a quadrilateral ABCD, which of the following is true?

• A. AB + BC + CD + DA < 1.5 (BD + AC)
• B. AB + BC + CD + DA < (BD + AC)
• C. AB + BC + CD + DA < 1.75 (BD + AC)
• D. AB + BC + CD + DA < 2 (BD + AC)

Answer 1

The correct option is D AB + BC + CD + DA < 2 (BD + AC)

Construction: Join diagonals AC and BD.


In $\Delta OAB,OA+OB>AB\cdots \left(i\right)$
[sum of any two sides of a traingle is greater than the third side]
In $\Delta OBC,OB+OC>BC\cdots \left(ii\right)$
[sum of any two sides of a triangle is greater than the third side]
Similarly, in $\Delta OCD,$
$OC+OD>CD\cdots \left(iii\right)$
And, in $\Delta ODA,$
$OD+OA>DA\cdots \left(iv\right)$
On adding eqs. (i), (ii), (iii) and (iv), we get
$2\left[\right(OA+OB+OC+OD]>AB+BC+CD+DA\Rightarrow 2[(OA+OC)+(OB+OD)]>AB+BC+CD+DA\Rightarrow 2(AC+BD)>AB+BC+CD+DA[\because OA+OC=ACandOB+OD=BD]\Rightarrow AB+BC+CD+DA<2(BD+AC)$