Question:

ABCD is quadrilateral. Is AB + BC + CD + DA >AC + BD?
ABCD 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 ΔABCΔABC,
AB+BC>CAAB + BC > CA (i)
In ΔBCDΔBCD,
BC+CD>DBBC + CD > DB (ii)
In ΔCDAΔCDA,
CD+DA>ACCD + DA > AC (iii)
In ΔDABΔDAB,
DA+AB>DBDA + AB > DB (iv)
Adding equations (i), (ii), (iii), and (iv), we obtain
AB+BC+BC+CD+CD+DA+DA+AB>AC+BD+AC+BDAB + BC + BC + CD + CD + DA + DA + AB > AC + BD + AC + BD
2AB+2BC+2CD+2DA>2AC+2BD\Rightarrow 2AB + 2BC + 2CD +2DA > 2AC + 2BD
2(AB+BC+CD+DA)>2(AC+BD)\Rightarrow2(AB + BC + CD + DA) > 2(AC + BD)
(AB+BC+CD+DA)>(AC+BD)\Rightarrow(AB + BC + CD + DA) > (AC + BD) 

Yes, the given expression is true.

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