a line segment has a midpoint of (5,3/2). if one endpoint is (3,2), what is the other end point

To find the other endpoint of a line segment when one endpoint and the midpoint are given, we use the midpoint formula. The midpoint \( M \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

In this case, the midpoint \( M \) is \( (5, \frac{3}{2}) \), and one endpoint \( A \) is \( (3, 2) \). We need to find the other endpoint \( B(x_2, y_2) \).

### Step 1: Set up the equations for the x and y coordinates.
From the midpoint formula, we can write two equations:
\[
\frac{x_1 + x_2}{2} = 5 \quad \text{(for the x-coordinate)}
\]
\[
\frac{y_1 + y_2}{2} = \frac{3}{2} \quad \text{(for the y-coordinate)}
\]

### Step 2: Solve for \( x_2 \) and \( y_2 \).

For the x-coordinate:
\[
\frac{3 + x_2}{2} = 5
\]
Multiply both sides by 2:
\[
3 + x_2 = 10
\]
Solve for \( x_2 \):
\[
x_2 = 7
\]

For the y-coordinate:
\[
\frac{2 + y_2}{2} = \frac{3}{2}
\]
Multiply both sides by 2:
\[
2 + y_2 = 3
\]
Solve for \( y_2 \):
\[
y_2 = 1
\]

### Final Answer:
The other endpoint is \( (7, 1) \).

Would you like more details or have any questions?

Here are some related questions to expand your understanding:
1. How do you find the midpoint if both endpoints are given?
2. What happens if the line segment lies on a diagonal?
3. How does the distance between endpoints relate to the midpoint?
4. Can you find the equation of the line passing through both endpoints?
5. How would this change if one coordinate was negative?

**Tip**: The midpoint formula is essentially the average of the x- and y-coordinates of the endpoints!

A line segment has a midpoint of (5, 3/2). If one endpoint is (3, 2), what is the other endpoint?

Learn how to find the other endpoint of a line segment with one endpoint (3, 2) and midpoint (5, 3/2) using the midpoint formula. This problem covers key concepts in geometry and is suitable for students in Grades 6-8.

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