Question

ABCD is quadrilateral Is AB+BC+CD+DA< 2(AC+BD)?

Answer 1

AC is Hypotenuse in triangle ABC.
so,
AB<AC---1
BC<AC---2
because hypotenuse is the biggest one in Right triangle.And ABCD is quardrilateral so ABC is Right triangle.
similarly,
DA<BD---3
CD<BD---4
Now from 1,2,3& 4
AB+BC+CD+DA<(AC+AC)+(BD+BD)
=>AB+BC+CD+DA<2(AC+BD)

Answer 2

Answer:

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.