In the following figure, AD bisects angle BAC. Find the length (in cm) of BD.

Answer: Option A

$$\begin{array}{rl}& AD\text{ is angle bisector of }\angle A\\ & \therefore \frac{AB}{AC}=\frac{BD}{DC}\\ & \Rightarrow \frac{6}{2x-3}=\frac{x-2}{x}\\ & \Rightarrow 6x=2x^2-4x-3x+6\\ & \Rightarrow 2x^2-13x+6=0\\ & \Rightarrow 2x^2-12x-x+6=0\\ & \Rightarrow 2x\left(x-6\right)-1\left(x-6\right)=0\\ & \Rightarrow \left(x-6\right)\left(2x-1\right)=0\\ & x-6=0\\ & x=6\\ & 2x-1=0\\ & x=\frac{1}{2}\left(\text{not valied}\right)\\ & \therefore BD=x-2=6-2=4\end{array}$$