Points O, A, B lie on the same line. OA = 12 cm, OB = 9 cm. Find

the distance between the midpoints of segments OA and OB if:

Point O lies on the segment AB

Let's solve the problem step by step.

### Step 1: Understand the problem setup
Points \( O \), \( A \), and \( B \) lie on the same line, with \( O \) lying between \( A \) and \( B \). We are given:
- \( OA = 12 \) cm
- \( OB = 9 \) cm

We need to find the distance between the midpoints of segments \( OA \) and \( OB \).

### Step 2: Assign positions on a coordinate line
Assume that point \( O \) is at the origin of the coordinate system, i.e., \( O = 0 \) on the number line.

- Since \( OA = 12 \) cm, the coordinates of \( A \) will be \( A = 12 \) cm.
- Since \( OB = 9 \) cm, and \( O \) lies between \( A \) and \( B \), the coordinates of \( B \) will be \( B = -9 \) cm.

### Step 3: Find the midpoints of segments \( OA \) and \( OB \)
The midpoint \( M_1 \) of segment \( OA \) is given by:
\[
M_1 = \frac{O + A}{2} = \frac{0 + 12}{2} = 6 \text{ cm}
\]

The midpoint \( M_2 \) of segment \( OB \) is given by:
\[
M_2 = \frac{O + B}{2} = \frac{0 + (-9)}{2} = -4.5 \text{ cm}
\]

### Step 4: Calculate the distance between \( M_1 \) and \( M_2 \)
The distance between \( M_1 \) and \( M_2 \) is:
\[
\text{Distance} = |M_1 - M_2| = |6 - (-4.5)| = |6 + 4.5| = 10.5 \text{ cm}
\]

So, the distance between the midpoints of segments \( OA \) and \( OB \) is \( 10.5 \) cm.

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How would the solution change if point \( O \) were outside the segment \( AB \)?
2. What is the significance of the midpoint in geometry, and how is it used in other problems?
3. Can you find the distance between the midpoints of two arbitrary segments if the points are not collinear?
4. If \( OB = 15 \) cm and \( O \) still lies on segment \( AB \), how would the distance between midpoints change?
5. What are the properties of the midpoint that make it useful in geometric constructions?

**Tip:** When solving problems involving midpoints, always consider using a coordinate system to simplify calculations.

Learn how to find the distance between the midpoints of segments OA and OB when points O, A, and B lie on the same line. This problem involves using the midpoint formula and the distance formula in geometry. Suitable for students in Grades 7-9.

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