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

Answer

To solve the problem step by step, we will use the properties of similar triangles and Thales' theorem.

### Step-by-Step Solution:

1. **Identify the Triangles and Parallel Lines**:
- We have two sets of parallel lines: CD || LA and DE || AC. This indicates that we can use Thales' theorem in both triangles ABL and ABC.

2. **Apply Thales' Theorem in Triangle ABL**:
- According to Thales' theorem, if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
- For triangle ABL, we can write the proportion:
\[ \frac{BD}{DA} = \frac{BC}{CL} \quad \text{(Equation 1)} \]

3. **Apply Thales' Theorem in Triangle ABC**:
- Similarly, for triangle ABC, we can apply Thales' theorem:
\[ \frac{BD}{DA} = \frac{BE}{EC} \quad \text{(Equation 2)} \]

4. **Set the Two Equations Equal**:
- Since both equations equal \(\frac{BD}{DA}\), we can set them equal to each other:
\[ \frac{BC}{CL} = \frac{BE}{EC} \]

5. **Substitute Known Values**:
- We know that \(BE = 4 \, \text{cm}\) and \(EC = 2 \, \text{cm}\). Therefore, we can substitute these values into the equation:
\[ \frac{BC}{CL} = \frac{4}{2} \]
- Simplifying gives:
\[ \frac{BC}{CL} = 2 \]

6. **Express BC in Terms of CL**:
- From the previous step, we can express \(BC\) as:
\[ BC = 2 \cdot CL \]

7. **Substitute BC Back into the Equation**:
- Now we can substitute \(BC\) back into the equation we derived from Thales' theorem:
\[ \frac{BE + EC}{CL} = \frac{4}{2} \]
- Since \(BC = BE + EC\), we have:
\[ \frac{4 + 2}{CL} = 2 \]

8. **Solve for CL**:
- This simplifies to:
\[ \frac{6}{CL} = 2 \]
- Cross-multiplying gives:
\[ 6 = 2 \cdot CL \]
- Dividing both sides by 2 results in:
\[ CL = 3 \, \text{cm} \]

### Final Answer:
The length of CL is **3 cm**.