In $\Delta ABC$, $AD$ is the bisector of $\angle A$ and intersects $BC$ at $D$. If $BC=a$, $AC=b$ and $AB=c$, then $BD$ is equal to:

According to the Angle Bisector Theorem, the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.
$$\frac{BD}{DC}=\frac{AB}{AC}=\frac{c}{b}$$
We are given that $BC=a$, and $D$ lies on $BC$, so $BD+DC=a$.Using the ratio:
$$BD=\frac{c}{b+c}\times BC=\frac{ac}{b+c}$$