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

Solution (By Examveda Team)

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