The midpoint of start overline, AB, end overlineAB is M, left parenthesis, minus, 1, comma, 3, right parenthesis*M*(−1,3). If the coordinates of A*A* are left parenthesis, 3, comma, minus, 2, right parenthesis(3,−2), what are the coordinates of B*B*?

To find the coordinates of point \( B \), we'll use the midpoint formula again. The midpoint \( M(x_m, y_m) \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:

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

Given:
- The midpoint \( M(-1, 3) \)
- The coordinates of point \( A(3, -2) \)

We need to find the coordinates of point \( B(x_2, y_2) \).

### Step 1: Set up the midpoint equations

\[
-1 = \frac{3 + x_2}{2}
\]
\[
3 = \frac{-2 + y_2}{2}
\]

### Step 2: Solve for \( x_2 \)

Multiply both sides of the first equation by 2 to eliminate the fraction:

\[
-2 = 3 + x_2
\]

Subtract 3 from both sides:

\[
x_2 = -5
\]

### Step 3: Solve for \( y_2 \)

Multiply both sides of the second equation by 2 to eliminate the fraction:

\[
6 = -2 + y_2
\]

Add 2 to both sides:

\[
y_2 = 8
\]

### Final Answer:
The coordinates of point \( B \) are \( \boxed{(-5, 8)} \).

Do you want details or have any questions?

Here are 5 related questions:
1. How would you find the coordinates of point \( A \) if you only know point \( B \) and the midpoint?
2. What happens to the midpoint if both points are shifted horizontally by the same amount?
3. Can the midpoint formula be used in a coordinate system with different units for the x and y axes?
4. How can the midpoint be useful in finding the center of a rectangle?
5. What does it mean geometrically if the midpoint and one endpoint are the same point?

**Tip:** For any geometry problem, drawing a quick sketch of the points can help you visualize the solution better.

Learn how to find the coordinates of point B using the midpoint formula and the coordinates of point A. This problem covers coordinate geometry concepts and the application of the midpoint formula. Suitable for high school students.

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