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Question
ABCD is quadrilateral.
Is AB + BC + CD + DA > AC + BD?
Solution
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.
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