Justify the following statement with reasons: The sum of any two sides of a triangle is greater than twice the median drawn to the third side.
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Construction :
Produce AD to E such that AD = DE
and join C and E .
Proof : In ∆ADB and ∆EDC ,
AD = DE [ By construction ]
BD = DC
[ Since, AD is a median of ∆ABC ]
<ADB = <EDC
[ Vertically opposite angles]
Therefore,
∆ADB congruent to ∆EDC
[ SAS Axiom ]
AB = EC [ C.P.C.T ] ----( 1 )
In ∆AEC ,
AC+ EC > AE
[ The sum of any two sides of a
triangle is greater than third side ]
=> AC + AB > AE [ from ( 1 ) ]
=> AB + AC > AE
=> AB + AC > 2AD [ By construction ]
••••
Produce AD to E such that AD = DE
and join C and E .
Proof : In ∆ADB and ∆EDC ,
AD = DE [ By construction ]
BD = DC
[ Since, AD is a median of ∆ABC ]
<ADB = <EDC
[ Vertically opposite angles]
Therefore,
∆ADB congruent to ∆EDC
[ SAS Axiom ]
AB = EC [ C.P.C.T ] ----( 1 )
In ∆AEC ,
AC+ EC > AE
[ The sum of any two sides of a
triangle is greater than third side ]
=> AC + AB > AE [ from ( 1 ) ]
=> AB + AC > AE
=> AB + AC > 2AD [ By construction ]
••••
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