$ABCD$ is a quadrilateral.
Is $AB + BC + CD + DA > AC + BD?$

Academic Mathematics NCERT Class 7

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,

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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Updated on: 10-Oct-2022

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