Question

ABCD is quadrilateral.

Is AB + BC + CD + DA > AC + BD?

Answer 1

In a triangle, the sum of the lengths of either two sides is always greater than the third side.

Considering ΔABC,

AB + BC > CA       ...(i)

In ΔBCD,

BC + CD > DB       ...(ii)

In ΔCDA,

CD + DA > AC       ...(iii)

In ΔDAB,

DA + AB > DB       ...(iv)

Adding equations (i), (ii), (iii) and (iv) we obtain

AB + BC + BC + CD + CD + DA + DA + AB > AC + BD + AC + BD

2AB + 2BC + 2CD + 2DA > 2AC + 2BD

2(AB + BC + CD + DA) > 2(AC + BD)

(AB + BC + CD + DA) > (AC + BD)

Yes, the given expression is true.