5. In a quadrilateral OPQR, show that :
(i) sum of its sides > sum of its diagonals, and
(ii) sum of its sides < 2 (sum of its diagonals)
Answers
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Answer:
2(AB+BC+CD+AD)>2(AC+BD)
AB+BC+CD+DA>AC+BD
Step-by-step explanation:
Consider a quadrilateral ABCD
Consider ΔACD
By triangle inequality
AC<AD+CD⋯(1)
Consider ΔACB
By triangle inequality
AC<AB+CB⋯(2)
Consider ΔBCD
By triangle inequality
BD<BC+CD⋯(3)
Consider ΔADB
By triangle inequality
BD<AD+AB⋯(4)
Adding all the equations
2(AB+BC+CD+AD)>2(AC+BD)
AB+BC+CD+DA>AC+BD
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