Points O, A, B lie on a line. OA = 12 cm and OB = 9 cm. Find
the distance between the midpoints of segments OA and OB in each case.
Point O lies on segment AB .
Answer:
The distance between the midpoints of segments
OA and OB is
cm

We are given that points \( O \), \( A \), and \( B \) lie on the same line, and the point \( O \) lies on segment \( AB \). The distances between \( O \) and \( A \) is 12 cm, and between \( O \) and \( B \) is 9 cm.

### Step-by-step solution:
1. **Coordinates setup:**
   - Let's place point \( O \) at the origin \( O(0) \).
   - Since \( O \) lies on segment \( AB \), point \( A \) will be at \( A(-12) \), and point \( B \) will be at \( B(9) \).

2. **Midpoints:**
   - The midpoint of \( OA \) is calculated as:
     \[
     M_{OA} = \frac{0 + (-12)}{2} = \frac{-12}{2} = -6
     \]
   - The midpoint of \( OB \) is calculated as:
     \[
     M_{OB} = \frac{0 + 9}{2} = \frac{9}{2} = 4.5
     \]

3. **Distance between the midpoints:**
   The distance between the midpoints of segments \( OA \) and \( OB \) is the absolute difference between \( M_{OA} \) and \( M_{OB} \):
   \[
   d = |M_{OB} - M_{OA}| = |4.5 - (-6)| = |4.5 + 6| = |10.5| = 10.5 \, \text{cm}
   \]

### Final Answer:
The distance between the midpoints of segments \( OA \) and \( OB \) is \( 10.5 \) cm.

Would you like a more detailed explanation, or do you have any further questions?

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Here are 5 related questions to further explore this topic:

1. How would the solution change if point \( O \) were not on segment \( AB \)?
2. Can you find the midpoint of any line segment using the same method?
3. What is the geometric significance of midpoints in coordinate geometry?
4. How would you calculate the distance if the coordinates of \( A \) and \( B \) were given in 2D or 3D space?
5. What if the distance between \( O \) and \( A \) was greater than that between \( O \) and \( B \)?

**Tip:** Midpoints always divide a segment into two equal halves, and the distance formula can be applied between any two points on a number line.

Points O, A, B lie on a line. OA = 12 cm and OB = 9 cm. Find the distance between the midpoints of segments OA and OB in each case. Point O lies on segment AB.

This problem explores how to find the distance between the midpoints of segments OA and OB where OA = 12 cm and OB = 9 cm. Learn the step-by-step solution using midpoint and distance formulas, perfect for students in Grades 7-9.

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