Question

In a triangle ABC ab=ac and trange ADC = 90.prove that AB +AC =2AD

Answer 1

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]