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ABC is a triangle, D is mid-point of BC. Prove that AB+AC=2AD

ABC is a triangle, D is mid-point of BC. Prove that AB+AC=2AD

Grade:12

1 Answers

Arun
25750 Points
6 years ago
i) This can be proved by making a small construction. Extend AD to E such that AD = DE. Join BE and CE. So, AE = 2AD ---------- (1) 

ii) In the quadrilateral, ABEC, the diagonal AE and BC bisect each other at D. [Since by construction AD = DE and as given AD is the median to BC; so D is the midpoint of BC] 
Since diagonals bisect each other, the quadrilateral ABEC is a parallelogram. 
In a parallelogram opposite sides are equal; so BE = AC --------- (2) 

iii) In triangle ABE, AB + BE > AE [Sum of any two sides of a triangle is greater than 
the third side] 

So from (1) & (2) 
AB + AC > 2AD

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