Question

if abcd is a quad. show that( ab + bc +cd +da ) is greater than (ac+bd)

Answer 1

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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.