Question

In $\Delta ABC,$ if AD is a median, which of the following relations holds good?

• A. AB+AC 2AD
• B. AB+AC < 2AD
• C. AB+AC=2AD
• D. AB+AC>2AD

Answer 1

The correct option is D

AB+AC>2AD

Given - AD is a median of triangle ABC.
Let us extend AD to G such that AD=DG. B,G and C,G are joined.
Now:
In $\Delta$ ABD and $\Delta$ CGD:
1. AD = DG [by construction].
2. $\angle$ ADB = $\angle$ CDG (vertically opposite angles)
3. BD = DC (since AD is a median (given)
Therefore, $\Delta$ ABD $\cong \Delta$ DGC (by S.A.S. congruency)

AB = CG [since they are corresponding sides of congruent triangles.]
Now In triangle ACG:
AC + CG > AG [since, sum of two sides of a triangle is greater than the third side.]
$\Rightarrow$ AC + AB > AD + DG [since, AB=CG ( proved earlier )]
$\Rightarrow$ AB + AC > AD + AD [since, AD=DG (by construction)]
$\Rightarrow$ AB + AC > 2AD.