The midpoint of start overline, AB, end overline 
AB
  is M, left parenthesis, minus, 4, comma, 0, right parenthesisM(−4,0). If the coordinates of AA are left parenthesis, minus, 7, comma, 3, right parenthesis(−7,3), what are the coordinates of BB?

To find the coordinates of point \( B \), we use the formula for the midpoint between two points. The midpoint \( M(x_m, y_m) \) of a line segment connecting points \( A(x_a, y_a) \) and \( B(x_b, y_b) \) is given by:

\[
M(x_m, y_m) = \left( \frac{x_a + x_b}{2}, \frac{y_a + y_b}{2} \right)
\]

We know:
- The coordinates of \( M \) are \( (-4, 0) \),
- The coordinates of \( A \) are \( (-7, 3) \).

Let \( B(x_b, y_b) \) represent the unknown coordinates of point \( B \).

Using the midpoint formula:

\[
-4 = \frac{-7 + x_b}{2} \quad \text{and} \quad 0 = \frac{3 + y_b}{2}
\]

Now, solve each equation:

### For the \( x \)-coordinate:
\[
-4 = \frac{-7 + x_b}{2}
\]
Multiply both sides by 2:
\[
-8 = -7 + x_b
\]
Add 7 to both sides:
\[
x_b = -1
\]

### For the \( y \)-coordinate:
\[
0 = \frac{3 + y_b}{2}
\]
Multiply both sides by 2:
\[
0 = 3 + y_b
\]
Subtract 3 from both sides:
\[
y_b = -3
\]

Thus, the coordinates of \( B \) are \( (-1, -3) \).

Would you like more details or have any questions? Here are 5 related questions to deepen your understanding:

1. How do you derive the midpoint formula?
2. What happens if one or both coordinates of \( M \) are negative?
3. Can you apply this method to 3-dimensional points?
4. How can you verify your answer using the distance formula?
5. What if the midpoint is given in terms of variables?

**Tip:** Always check your work by substituting back into the original equations to ensure consistency.

The midpoint of line segment AB is M(−4, 0). If the coordinates of A are (−7, 3), what are the coordinates of B?

Learn how to find the coordinates of point B when given the midpoint M(−4, 0) and point A(−7, 3) using the midpoint formula. This problem explores key geometry concepts and is suitable for students in Grades 8-10.

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