To solve this problem, we need to calculate the length of $BD$. Here's the approach:

Problem Breakdown:

1. Given Information:

   • $AK = 6 \, \text{cm}$, $KD = 3 \, \text{cm}$, and $LM = 8 \, \text{cm}$.
   • $AL$ and $LC$ are segments forming a triangle $\triangle ABC$.
   • The question asks for the length of $BD$.

2. Relationships in the Triangle:

   • Since $LM$ is parallel to $AB$, the two triangles $\triangle ALK$ and $\triangle AMC$ are similar by the AA (Angle-Angle) similarity criterion.
   • The ratio of the sides in these similar triangles can be used to calculate missing lengths.

3. Determine Scaling Ratio:

   • $AK : KD = 6 : 3$ (simplifies to $2:1$).
   • Thus, the ratio of the triangles is $2:1$.

4. Use the Scaling Ratio to Find $BD$:

   • The length $AB$ corresponds to $LM + AB$.
   • The total length $BD = AB$, and using the ratio $2:1$, we multiply the scaling factor to extend the calculation.

Calculation:

The total $BD$ length can be found using the similarity properties and corresponding side calculations:

• $LM = 8 \, \text{cm}$
• The ratio ensures $AB = 2 \times LM = 16 \, \text{cm}$.

Answer:

The length of $BD = \mathbf{16 \, \text{cm}}$.

Would you like a more detailed explanation or have other questions? Here are some related explorations:

1. How do we apply the concept of similar triangles in real-world problems?
2. How does the AA similarity criterion work?
3. Can you calculate the height of triangle $\triangle ABC$ using its area?
4. What are the different types of proportionality used in triangles?
5. How do parallel lines impact triangle proportions?

Tip: Always double-check side lengths and proportions to confirm the scaling ratios when working with similar triangles.