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Question
In Triangle ABC AD is the median. Prove that AB +AC > 2AD.
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Solution
Given: AD is a median of the 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 [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]
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]
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