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$ABCD$ is a quadrilateral.
Is $AB + BC + CD + DA > AC + BD?$
Given: $ABCD$ is a quadrilateral.
To do: To find whether $AB + BC + CD + DA > AC + BD?$
Solution:
The sum of the length of any two sides in a triangle should be greater than the length of the third side,
![](/assets/questions/media/148618-63830-1656570531.png)
Therefore
In $\Delta ABC$, $AB+BC>AC$ .......$(i)$
$\Delta ADC$, $AD+DC>AC$, .......$(ii)$
$\Delta DCB$, $DC+CB>DB$, .......$(iii)$
$\Delta ADB$, $AD+AB>DB$, .......$(iv)$
Add equation $(i)$, $(ii)$, $(iii)$ and $(iv)$
$AB+BC+AD+DC+DC+CB+AD+AB>AC+AC+DB+DB$
$(AB+AB)+(BC+BC)+(AD+AD)+(DC+DC)>2AC+2DB$
$2AB+2BC+2AD+2DC>2AC+2DB$
$2(AB+BC+AD+DC)>2(AC+DB)$
$AB+BC+AD+DC>AC+DB$
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