Question

in triangle, AD is a median. Prove that AB+AC > 2AD.​

Answer 1

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GIVEN- AD is a median of triangle ABC.

REQUIRED TO PROVE- AB+AC>2AD.

CONSTRUCTION- AD is extended to G such that AD=DG. B,G and C,G are joined.

PROOF-

In triangles ABD and DGC,

1. AD=DG [ by construction ].

2. BD=DC [ since AD is a median (given) ]

3. included angle ADB= included angle CDG.

Therefore, triangles ABD and DGC are congruent. [ by S.A.S. congruency. ]

AB=CG, since they are corresponding sides of congruent triangles.

In triangle ACG,

AC+CG>AG. [ since sum of two sides of a triangle is greater than the third side. ]

or, AC+AB>AD+DG. [ since AB=CG ( proved earlier ) ]

or, AB+AC>AD+AD [ since AD=DG ( by construction )]

or, AB+AC> 2AD. [ PROVED ]

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Answer 2

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