In the figure (i) given below, CD || LA and DE || AC. Find the length of CL if BE = 4 cm and EC = 2 cm.

Considering △ABC and △BDE,

∠B = ∠B (Common angle)
∠DEB = ∠ACB (Corresponding angles)

So, by AA rule of similarity △ABC ~ △BDE. Since ratio of corresponding sides is same,

$\Rightarrow \dfrac{BE}{BC} = \dfrac{BD}{BA} \\[1em] \Rightarrow \dfrac{BE}{BE + EC} = \dfrac{BD}{BA} \\[1em] \Rightarrow \dfrac{4}{4 + 2} = \dfrac{BD}{AB}\\[1em] \Rightarrow \dfrac{4}{6} = \dfrac{BD}{AB} \qquad \text{[….Eq 1]} \\[1em]$

Considering △ABL and △BDC,

∠B = ∠B (Common angle)
∠BDC = ∠BAL (Corresponding angles)

So, by AA rule of similarity △ABL ~ △BDC. Since ratio of corresponding sides is same,

$\Rightarrow \dfrac{BD}{BA} = \dfrac{BC}{BL} \\[1em] \Rightarrow \dfrac{BD}{BA} = \dfrac{BE + EC}{BE + EC + CL} \\[1em] \Rightarrow \dfrac{BD}{BA} = \dfrac{4 + 2}{4 + 2 + CL} \\[1em] \Rightarrow \dfrac{BD}{BA} = \dfrac{6}{6 + CL} \qquad \text{[….Eq 2]} \\[1em]$

Solving Eq 1 and Eq 2 we get,

$\Rightarrow \dfrac{6}{6 + CL} = \dfrac{4}{6} \\[1em] \Rightarrow 36 = 4(6 + CL) \\[1em] \Rightarrow 24 + 4CL = 36 \\[1em] \Rightarrow 4CL = 36 - 24 \\[1em] \Rightarrow 4CL = 12 \\[1em] \Rightarrow CL = 3.$

Hence, the length of CL is 3 cm.