Angle Bisector Theorem

In $DABC$, $AD$ is the bisector of $\angle A$, also $AB=8 \mathrm{cm}, BD=5 \mathrm{cm}, DC=4 \mathrm{cm}$, find the length of $AC$.

Note: This question given in the book seems to have errors and the modified question should be as below.

In a $∆ABC$, if point $D$ is on the side $BC$ such that $\frac{AB}{AC}=\frac{BD}{DC}$ and $\angle B=70°, \angle C=50°$. Find $\angle BAD$.

In figure, if $AB=3.4 \mathrm{cm}, BD=4 \mathrm{cm}, BC=10 \mathrm{cm}$, then the measure of $AC$ is:

(Note: $\mathrm{AD}$ is the angle bisector of $\mathrm{A}$).

In the figure, $AD$ is the bisector of $\angle A$, if $AB=6 \mathrm{cm}, BD=8 \mathrm{cm}, DC=6 \mathrm{cm}$, then the length of $AC$ will be:

In triangle $XYZ, YM$ is the angle bisector of $\angle Y$ and $\angle Y$ is $100°$
Find $\angle XYM$  and $\angle ZYM$.

In $△\mathrm{ABC}, \mathrm{AD}$ is the bisector of $\angle \mathrm{A}$ meeting side $\mathrm{BC}$ at $\mathrm{D},$ if $\mathrm{AB}=10 \mathrm{cm}$, $\mathrm{AC}=14 \mathrm{cm}$ and $\mathrm{BC}=6 \mathrm{cm},$ find $\mathrm{BD}$ and $\mathrm{DC}$.

The three angle bisectors of a triangle are concurrent. Their point of concurrence is called the