Find AD and DC.

ABC is a traingle and BD is the anglular bisector of \$\angle B\$

From the given \$\triangle ABC\$, according to the angle bisector theorem, we have

\$\frac{AD}{DC} = \frac{AB}{BC} \$

Substitute the values.

\begin{align} \Rightarrow && \frac{x}{10 - x} &= \frac{9}{6} \\ \end{align}

Cancel out common factor.

\begin{align} \require{cancel} \Rightarrow && \frac{x}{10 - x} &= \frac{\cancel{9} ^3 }{\cancel{6} ^ 2} \\ \end{align}

Multiply.

\begin{align} \Rightarrow && 2x &= 30 - 3x \\ \end{align}

Add the like terms.

\begin{align} \Rightarrow && 5x &= 30 \\ \end{align}

Cancel out common factor.

\begin{align} \require{cancel} \Rightarrow && \cancel{5}^1 x &= \cancel{30}^6 \\ \end{align}

The value of AD.

\$x = 6\$.

The value of DC.

\$DC = 10 - x\$

Substitute 6 for x.

\$DC =10 - 6\$

Subtract.

\$DC = 4\$

Answer

\$AD = 6\$ and \$ DC = 4\$.