Question:

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

Updated On: Dec 8, 2023
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Solution and Explanation

In a triangle, the sum of the lengths of either two sides is always greater than the third side.
Considering ΔOABΔOAB,
OA+OB>ABOA + OB > AB (i)
In ΔOBCΔOBC,
OB+OC>BCOB + OC > BC (ii)
In ΔOCDΔOCD,
OC+OD>CDOC + OD > CD (iii)
In ΔODAΔODA,
OD+OA>DAOD + OA > DA (iv)

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

OA+OB+OB+OC+OC+OD+OD+OA>AB+BC+CD+DAOA + OB + OB + OC + OC + OD + OD + OA > AB + BC + CD + DA
\Rightarrow 2  OA+2  OB+2  OC+2  OD>AB+BC+CD+DA2\;OA + 2\;OB + 2\;OC + 2\;OD > AB + BC + CD + DA
\Rightarrow 2  OA+2  OC+2  OB+2  OD>AB+BC+CD+DA2\;OA + 2\;OC + 2\;OB + 2\;OD > AB + BC + CD + DA
\Rightarrow 2(OA+OC)+2(OB+OD)>AB+BC+CD+DA2(OA + OC) + 2(OB + OD) > AB + BC + CD + DA
\Rightarrow 2(AC)+2(BD)>AB+BC+CD+DA2(AC) + 2(BD) > AB + BC + CD + DA
\Rightarrow 2(AC+BD)>AB+BC+CD+DA2(AC + BD) > AB + BC + CD + DA

Yes, the given expression is true.

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