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

Solution

Given: AD is the median of ΔABC.

To prove: AB + AC > 2AD

Construction: Produce AD to E such that AD = DE. Join CE.

Proof: In ΔABD and ΔCDE

AD = DE (Construction)

∠ADB = ∠CDE (Vertically opposite angles)

BD = DC (AD is the median from A to BC)

∴ ΔABD ΔCDE (SAS congruence criterion)

⇒ 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