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We are asked to find the value of \(x\) and the length of \( \overline{KM} \) in the given diagram. The line segment is divided into two parts:

- \( KL = 12 \)
- \( LM = x - 5 \)
- The total length of \( \overline{KM} \) is given as \( 2x - 5 \)

Since the total length of \( KM \) is equal to the sum of \( KL \) and \( LM \), we can set up the equation:

\[
KM = KL + LM
\]

Substitute the given values:

\[
2x - 5 = 12 + (x - 5)
\]

Now, solve for \(x\).

### Step 1: Simplify the equation
\[
2x - 5 = 12 + x - 5
\]
\[
2x - 5 = x + 7
\]

### Step 2: Subtract \(x\) from both sides
\[
2x - x = 7 + 5
\]
\[
x = 12
\]

### Step 3: Find the length of \( KM \)
Now that we have \(x = 12\), substitute it into the expression for \( KM \):

\[
KM = 2x - 5 = 2(12) - 5 = 24 - 5 = 19
\]

### Final Answer:
- \(x = 12\)
- Length of \( \overline{KM} = 19 \)

Would you like further details or have any questions?

Here are some related questions to consider:
1. What if the total length \(KM\) was given in a different form?
2. How would the problem change if \(LM\) had a different expression?
3. How do you apply the segment addition postulate in different scenarios?
4. What happens if we rearrange points \(K\), \(L\), and \(M\) on the line?
5. How would you handle this if it was a three-dimensional geometry problem?

**Tip:** When solving geometric problems, it's important to clearly identify the known values and the relationships between segments to set up the correct equations.

Find the value of x and the length of KM in the given line segment with KL = 12, LM = x - 5, and KM = 2x - 5.

Learn how to find the value of x and the length of line segment KM using the Segment Addition Postulate. The given problem includes KL = 12, LM = x - 5, and KM = 2x - 5. Suitable for students in Grades 8-10.

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