Consider the triangle ABC, If BD is the angle bisector for angle B, determine the value of x?.

ABC is a traingle and BD is the anglular bisector of \$\angle B\$,
and sides AB = 4, BC = 6, AD = x, and DC = x + 2.

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

\begin{align} \require{cancel} && \frac{AB}{BC} &= \frac{AD}{DC} \\ \end{align}

Substitute the values.

\begin{align} \Rightarrow && \frac{4}{6} &= \frac{x}{x + 2} \\ \end{align}

Multiply.

\begin{align} \Rightarrow && 6x &= 4x + 8 \\ \end{align}

Subtract like terms.

\begin{align} \Rightarrow && 2x &= 8 \\ \end{align}

Cancel out common factor.

\begin{align} \Rightarrow && \cancel{2}^1 x &= \cancel{8}^4 \\ \end{align}

Answer

The value of x = 4.