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.