Question

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

Answer 1

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

Therefore, In ΔABC, AB + BC > AC ……….(i)

In ΔADC, AD + DC > AC ……….(ii)

In ΔDCB, DC + CB > DB ……….(iii)

In ΔADB, AD + AB > DB ……….(iv)

Adding eq. (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 > 2(AC + DB)

⇒⇒ 2(AB + BC + AD + DC) > 2(AC + DB)

⇒⇒ AB + BC + AD + DC > AC + DB

⇒⇒ AB + BC + CD + DA > AC + DB