Question

AD is the median of triangle ABC Prove that AB+AC >2AD

Answer 1

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 [construction].

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

3. included angle ADB= included angle CDG.

Therefore, triangles ABD and DGC are congruent. [SAS congruency]

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

In triangle ACG, AC+CG>AG. [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]

Answer 2

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