klm are three points on a line such that l lies in between k and m prove that kl+lm=km

To prove that \( KL + LM = KM \) for points \( K \), \( L \), and \( M \) on a line with \( L \) lying between \( K \) and \( M \), we can use the following reasoning:

### Proof:

1. **Position the Points on a Line:**
   - Assume that the points \( K \), \( L \), and \( M \) are on a straight line, and \( L \) lies between \( K \) and \( M \).
   - Let the coordinates of points \( K \), \( L \), and \( M \) be \( x_K \), \( x_L \), and \( x_M \) respectively.

2. **Distance Between Points:**
   - The distance between two points on a line is the absolute difference of their coordinates. 
   - Thus, the distance \( KL \) is given by \( KL = |x_K - x_L| \).
   - The distance \( LM \) is given by \( LM = |x_L - x_M| \).
   - The distance \( KM \) is given by \( KM = |x_K - x_M| \).

3. **Use the Fact that \( L \) Lies Between \( K \) and \( M \):**
   - Since \( L \) lies between \( K \) and \( M \), we have:
     \[
     x_K \leq x_L \leq x_M \quad \text{or} \quad x_M \leq x_L \leq x_K.
     \]

4. **Case 1: \( x_K \leq x_L \leq x_M \):**
   - In this case:
     \[
     KL = x_L - x_K, \quad LM = x_M - x_L, \quad \text{and} \quad KM = x_M - x_K.
     \]
   - Adding \( KL \) and \( LM \) gives:
     \[
     KL + LM = (x_L - x_K) + (x_M - x_L) = x_M - x_K = KM.
     \]
   - Therefore, \( KL + LM = KM \).

5. **Case 2: \( x_M \leq x_L \leq x_K \):**
   - In this case:
     \[
     KL = x_K - x_L, \quad LM = x_L - x_M, \quad \text{and} \quad KM = x_K - x_M.
     \]
   - Adding \( KL \) and \( LM \) gives:
     \[
     KL + LM = (x_K - x_L) + (x_L - x_M) = x_K - x_M = KM.
     \]
   - Therefore, \( KL + LM = KM \).

### Conclusion:
In both cases, we have shown that \( KL + LM = KM \). Therefore, the statement is proven to be true.

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Would you like more details or have any questions? Here are 5 questions to expand your understanding:

1. How would the proof change if the points were not on a straight line?
2. Can you derive the same proof using vector notation?
3. How would you prove this result using the concept of a metric space?
4. What if \( L \) did not lie between \( K \) and \( M \)? How would the equation change?
5. Can this concept be extended to points in higher-dimensional spaces?

**Tip:** When dealing with points on a line, always consider the geometric relationship between them and use absolute values to account for direction.

Learn how to prove that KL + LM = KM for points K, L, and M on a straight line where L lies between K and M. This geometric problem involves using the distance formula and understanding the positioning of points on a line. Suitable for high school students studying geometry.

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