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To solve for the length of \( KM \) in the given diagram, we'll need to use the segment addition postulate. According to the diagram, \( K \), \( L \), and \( M \) are collinear points with the following segments:

- \( KL = 10 \)
- \( LM = 2x + 19 \)
- \( KM = x + 21 \)

The segment addition postulate states that:
\[
KM = KL + LM
\]
Substituting the given values:
\[
x + 21 = 10 + (2x + 19)
\]
Now, let's solve for \( x \):

1. Simplify the equation:
\[
x + 21 = 10 + 2x + 19
\]

2. Combine like terms on the right side:
\[
x + 21 = 2x + 29
\]

3. Subtract \( x \) from both sides:
\[
21 = x + 29
\]

4. Subtract 29 from both sides to isolate \( x \):
\[
x = 21 - 29
\]
\[
x = -8
\]

Now that we have \( x = -8 \), substitute it back into the expression for \( KM \) to find its length:

\[
KM = x + 21 = -8 + 21 = 13
\]

Thus, the length of \( KM \) is \( \boxed{13} \).

Would you like more details on this solution, or do you have any further questions? Here are some related questions:

1. What happens if you change the values of \( KL \) or \( LM \)?
2. How does segment addition apply to other geometric figures?
3. Can you solve for \( x \) using a different method?
4. How do you verify the solution in this context?
5. What if \( KL \) and \( LM \) were not collinear, how would this affect the solution?

**Tip:** Always double-check your algebraic steps when solving equations to avoid small errors that can lead to incorrect results.

Learn how to solve for the length of KM in a geometry problem using the segment addition postulate. This detailed solution involves applying algebraic equations and segment addition principles to find the length of KM.

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