midpoint of start overline, AB, end overline 
AB
  is M, left parenthesis, minus, 1, comma, 2, right parenthesis M(−1,2). If the coordinates of AA are left parenthesis, 4, comma, 5, right parenthesis(4,5), what are the coordinates of BB?

To find the coordinates of point \( B \), we can use the midpoint formula. The midpoint \( M \) of a line segment \( AB \) is given by:

\[
M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right)
\]

In this case, the midpoint \( M \) is \( (-1, 2) \), and the coordinates of point \( A \) are \( (4, 5) \). Let the coordinates of point \( B \) be \( (x_B, y_B) \).

Using the midpoint formula, we set up the following two equations:

1. \( \frac{4 + x_B}{2} = -1 \)
2. \( \frac{5 + y_B}{2} = 2 \)

### Solving for \( x_B \):
\[
\frac{4 + x_B}{2} = -1
\]
Multiply both sides by 2:
\[
4 + x_B = -2
\]
Subtract 4 from both sides:
\[
x_B = -6
\]

### Solving for \( y_B \):
\[
\frac{5 + y_B}{2} = 2
\]
Multiply both sides by 2:
\[
5 + y_B = 4
\]
Subtract 5 from both sides:
\[
y_B = -1
\]

Thus, the coordinates of point \( B \) are \( (-6, -1) \).

Would you like more details or have any questions?

Here are some related questions:
1. How do you calculate the distance between two points?
2. What is the significance of the midpoint in geometry?
3. How can you generalize the midpoint formula for three dimensions?
4. How would you find the coordinates of a point if the distance and midpoint are given?
5. What are the properties of a line segment bisected by a midpoint?

**Tip:** When dealing with midpoints, always remember that the midpoint coordinates are the averages of the corresponding coordinates of the endpoints.

The midpoint of line segment AB is M(-1, 2). If the coordinates of A are (4, 5), what are the coordinates of B?

Learn how to find the coordinates of point B given the midpoint M(-1, 2) and point A(4, 5) using the midpoint formula. This solution involves fundamental concepts of coordinate geometry and is suitable for students in Grades 6-8.

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