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Show that in a quadrilateral ABCD, AB+BC+CD+DA<2(BD+AC)

Show that in a quadrilateral ABCD, AB+BC+CD+DA<2(BD+AC)

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A)

Since, the sum of lengths of any two sides in a triangle should be greater than the length of third side Therefore,

  In Δ AOB, AB < OA + OB ……….(i) 

In Δ BOC, BC < OB + OC ……….(ii) 

 In Δ COD, CD < OC + OD ……….(iii) 

 In Δ AOD, DA < OD + OA ……….(iv) 

 

⇒ AB + BC + CD + DA < 2OA + 2OB + 2OC + 2OD 

 ⇒ AB + BC + CD + DA < 2[(AO + OC) + (DO + OB)] 

 ⇒ AB + BC + CD + DA < 2(AC + BD) 

 Hence, it is proved.

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