The midpoint of start overline, AB, end overline
AB
is M, left parenthesis, 0, comma, minus, 1, right parenthesisM(0,−1). If the coordinates of AA are left parenthesis, 8, comma, minus, 3, right parenthesis(8,−3), what are the coordinates of BB?

To find the coordinates of point \( B \), we can use the midpoint formula. 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:

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

Given:
- \( M(0, -1) \)
- \( A(8, -3) \)

Let the coordinates of \( B \) be \( B(x_2, y_2) \).

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

\[
\frac{8 + x_2}{2} = 0
\]
\[
\frac{-3 + y_2}{2} = -1
\]

Now, solve these equations:

### For the \( x \)-coordinate:
\[
\frac{8 + x_2}{2} = 0
\]
Multiply both sides by 2:
\[
8 + x_2 = 0
\]
Solve for \( x_2 \):
\[
x_2 = -8
\]

### For the \( y \)-coordinate:
\[
\frac{-3 + y_2}{2} = -1
\]
Multiply both sides by 2:
\[
-3 + y_2 = -2
\]
Solve for \( y_2 \):
\[
y_2 = 1
\]

So, the coordinates of point \( B \) are \( (-8, 1) \).

Would you like further details or have any questions?

Here are five related questions for further practice:
1. What would the midpoint be if the coordinates of \( B \) were changed to \( (-10, 5) \)?
2. If the midpoint of another segment is \( (2, -3) \) and one endpoint is \( (6, -7) \), what is the other endpoint?
3. How does the midpoint formula relate to finding the average of two numbers?
4. What is the distance between points \( A(8, -3) \) and \( B(-8, 1) \)?
5. If the midpoint of a segment is on the x-axis, what can be said about the y-coordinates of the endpoints?

**Tip:** When solving midpoint problems, always double-check by plugging the found coordinates back into the midpoint formula to ensure accuracy.

The midpoint of line segment AB is M(0,−1). If the coordinates of A are (8,−3), what are the coordinates of B?

Discover how to find the coordinates of point B given the midpoint M(0,−1) and point A(8,−3). This problem demonstrates the application of the midpoint formula, suitable for students in Grades 8-10.

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