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The problem asks to find the coordinates of point \( B \) given that the midpoint of segment \( AB \) is \( (-3, -1) \) and point \( A \) is \( (6, 5) \).

To solve this, we use the formula for the midpoint of a line segment:

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

Where:
- \( M \) is the midpoint
- \( (x_1, y_1) \) are the coordinates of point \( A \)
- \( (x_2, y_2) \) are the coordinates of point \( B \)

We know the midpoint \( M = (-3, -1) \) and \( A = (6, 5) \). Let's substitute these values into the midpoint formula:

\[
(-3, -1) = \left( \frac{6 + x_2}{2}, \frac{5 + y_2}{2} \right)
\]

Now, solve for \( x_2 \) and \( y_2 \).

1. For the \( x \)-coordinate:
\[
\frac{6 + x_2}{2} = -3
\]
Multiply both sides by 2:
\[
6 + x_2 = -6
\]
Solve for \( x_2 \):
\[
x_2 = -6 - 6 = -12
\]

2. For the \( y \)-coordinate:
\[
\frac{5 + y_2}{2} = -1
\]
Multiply both sides by 2:
\[
5 + y_2 = -2
\]
Solve for \( y_2 \):
\[
y_2 = -2 - 5 = -7
\]

Thus, the coordinates of point \( B \) are \( (-12, -7) \).

Let me know if you need any details or have further questions.

Here are 5 related questions to expand on this topic:
1. How do you derive the midpoint formula for a line segment?
2. What would happen if point \( A \) was at the origin, \( (0, 0) \)?
3. How would you calculate the distance between points \( A \) and \( B \)?
4. Can the midpoint formula be extended to three-dimensional space? If so, how?
5. What is the significance of the midpoint in geometry?

**Tip:** When solving problems involving midpoints or centroids, always double-check your algebraic steps, especially when working with fractions or equations.

The midpoint of AB is at (-3, -1). If A = (6, 5), find B.

Solve the problem of finding the coordinates of point B, given the midpoint at (-3, -1) and point A at (6, 5), using the midpoint formula. This geometry problem is suitable for students in grades 8-10.

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