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Question

In quadrilateral ABCD, prove that
(i) ¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯BC+¯¯¯¯¯¯¯¯¯AD+¯¯¯¯¯¯¯¯¯DC=2¯¯¯¯¯¯¯¯AC
(ii)¯¯¯¯¯¯¯¯¯AD+¯¯¯¯¯¯¯¯BC=2¯¯¯¯¯¯¯¯¯¯¯MN, where M and N are the mid points of side AB and CD respectively.

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Solution

(i) AB+BC=AC(1)
AD+DC=AC(2)
Adding (1) and (2) ,
AB+BC+AD+DC=2AC
(ii) MN+NC=MC,MB+BC=MC
MN+NC=MB+BC(1)
MA+AD=MD,MN+ND=MD
MA+AD=MN+ND(2)
From (1) , MNND=MB+BC[NC=ND]
MNMB=ND+BC
MN+MA=ND+BC[MA=MB]
MN+MN+NDAD=ND+BC [From (2), MA=MN+ND ]
2MN+ND=ND+AD+BC
2MN=AD+BC

968047_1000965_ans_9db1764ece6341d2babddee91c4a8c33.png

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