In the given figure, AB = AC, then :

1. BD = BC

2. BD = CD

3. BD < CD

4. BD > CD

In △ ABC,

⇒ AB = AC (Given)

⇒ ∠C = ∠B (Angles opposite to equal sides are equal)

In △ ABD,

⇒ AD = AC - CD

⇒ AD < AB

⇒ ∠ABD < ∠ADB (If two sides of a triangle are unequal, the greater side has the greater angle opposite to it.)

In △ BDC,

⇒ ∠DBC = ∠B - ∠ABD = ∠C - ∠ABD

∴ ∠DBC < ∠C

⇒ CD < BD

⇒ BD > CD.

Hence, Option 4 is the correct option.