Question

In triangle ABC If AD is a median then show that AB+AC>2AD .

Answer 1

given,
ad is the median of Δabc
to prove,
ab + ac > 2ad
construction,
produce ad to E such that ad = de and join ce
ad = de(Construction)
∠adb = ∠cde(Vertically opposite angles)
bd = dc(ad is the median from A to BC)
∴ Δabd congruent to  Δcde by (SAS rule) 
⇒ ab = ce(cpct) ...(1)
In Δace
ac + ce > AE(Sum of any two sides of a triangle is greater than the third side)
⇒ ac + ab > ad + de [Using (1)]
⇒ ac + AB > ad + ad(Constriction)
⇒ ac + ab > 2ad {proved}

Answer 2

given,

ad is the median of Δabc

to prove,

ab + ac > 2ad

construction,

produce ad to E such that ad = de and join ce

ad = de(Construction)

∠adb = ∠cde(Vertically opposite angles)

bd = dc(ad is the median from A to BC)

∴ Δabd congruent to  Δcde by (SAS rule)  

⇒ ab = ce(cpct) ...(1)

In Δace

ac + ce > AE(Sum of any two sides of a triangle is greater than the third side)

⇒ ac + ab > ad + de [Using (1)]

⇒ ac + AB > ad + ad(Constriction)

⇒ ac + ab > 2ad {proved}