Obtain the value of BD for the given triangle ABC, if AD is angle bisector for angle \$\angle A\$.

ABC is a traingle and AD is the anglular bisector of \$\angle A\$,
and sides AB = 20, AC = 16, and DC = 4.

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

\begin{align} && \frac{BD}{DC} &= \frac{AB}{AC} \\ \end{align}

Substitute the values.

\begin{align} \Rightarrow && \frac{BD}{4} &= \frac{20}{16} \\ \end{align}

cancel out common factor.

\begin{align} \require{cancel} \Rightarrow && \frac{BD}{4} &= \frac{\cancel{20} ^{5}}{\cancel{16} ^{4}} \\ \end{align}

Multiply.

\begin{align} \Rightarrow && 4BD &= 20 \\ \end{align}

Cancel out common factor.

\begin{align} \require{cancel} \Rightarrow && BD &= \frac{\cancel{20} ^{5}}{\cancel{4} ^1} \\ \\ \end{align}

Answer

The value of BD = 5.