Let's analyze the given diagram and the information provided:

• $KP$, $LQ$, $MR$, $NS$, and $BC$ are all parallel to each other.
• The segments $KN = LB = 3KL$ and $M$ is the midpoint of $LN$.
• The length $PQ = x$.

We are asked to find $PR$, $PS$, $QS$, and $RC$, and the answers provided are $2x$, $3x$, $2x$, and $2x$, respectively.

Steps to solve the problem:

1. Similar Triangles: Since all the segments are parallel, the triangles formed between the different lines (e.g., $\triangle AKP$, $\triangle ALQ$, $\triangle BMC$, etc.) are similar by the Basic Proportionality Theorem (Thales' Theorem).

2. Applying the Midpoint Theorem:

   • Given $M$ is the midpoint of $LN$, and since $MR \parallel BC$, the ratio of $PR$ to $PQ$ will be 2:1.
   • This implies $PR = 2x$.

3. Finding $PS$:

   • Similarly, $S$ is on the next parallel line, so the proportionality between $PS$ and $PQ$ gives $PS = 3x$.

4. Finding $QS$:

   • Since $QS \parallel BC$ and lies on the next segment after $PQ$, it will also have a proportionality of $2x$, like $PR$.

5. Finding $RC$:

   • $RC$, being a continuation in the same proportionality, will again be $2x$, as it corresponds to the segment under the parallel lines.

Thus, the values of the segments are confirmed as:

• $PR = 2x$
• $PS = 3x$
• $QS = 2x$
• $RC = 2x$

Would you like a detailed explanation of any of the steps, or do you have any further questions?

Additional Related Questions:

1. What is the Basic Proportionality Theorem, and how does it apply in this case?
2. How can we verify that triangles formed by parallel lines are similar?
3. If the value of $PQ$ were given, how would you calculate the other segment lengths numerically?
4. Why does the midpoint condition $M$ affect the ratios between the lines?
5. Can we find the total length of $BC$ if $PQ = x$?

Tip:

When dealing with similar triangles, always check for parallel lines and midpoint conditions to apply the Basic Proportionality Theorem for dividing the sides proportionally.