The midpoint of start overline, AB, end overline 
AB
  is M, left parenthesis, 1, comma, minus, 7, right parenthesisM(1,−7). If the coordinates of AA are left parenthesis, minus, 2, comma, minus, 6, right parenthesis(−2,−6), what are the coordinates of BB?

We are given the midpoint \( M(1, -7) \) of the line segment \( \overline{AB} \), and the coordinates of point \( A(-2, -6) \). We need to find the coordinates of point \( B(x_B, y_B) \).

The formula for the midpoint \( M \) of a line segment between points \( A(x_A, y_A) \) and \( B(x_B, y_B) \) is:

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

Given:
- \( M(1, -7) \)
- \( A(-2, -6) \)

We can set up two equations based on the midpoint formula:

1. For the \( x \)-coordinates:
\[
\frac{x_A + x_B}{2} = 1
\]
Substitute \( x_A = -2 \):
\[
\frac{-2 + x_B}{2} = 1
\]
Multiply both sides by 2:
\[
-2 + x_B = 2
\]
Solve for \( x_B \):
\[
x_B = 4
\]

2. For the \( y \)-coordinates:
\[
\frac{y_A + y_B}{2} = -7
\]
Substitute \( y_A = -6 \):
\[
\frac{-6 + y_B}{2} = -7
\]
Multiply both sides by 2:
\[
-6 + y_B = -14
\]
Solve for \( y_B \):
\[
y_B = -8
\]

Thus, the coordinates of point \( B \) are \( B(4, -8) \).

Would you like further explanation or have any questions?

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Here are 5 related questions to explore:
1. How would the midpoint change if the coordinates of \( A \) were different?
2. How can you calculate the distance between two points \( A \) and \( B \)?
3. What is the formula for finding the slope of a line between two points?
4. How can you determine if a point lies on the line segment between \( A \) and \( B \)?
5. How would you find the equation of the line passing through points \( A \) and \( B \)?

**Tip**: When solving for unknown coordinates, break the problem into separate calculations for the \( x \)- and \( y \)-coordinates.

The midpoint of segment AB is M(1, -7). If the coordinates of A are (-2, -6), what are the coordinates of B?

Learn how to find the coordinates of point B when given the midpoint M(1, -7) and point A(-2, -6) on line segment AB. This detailed solution uses the midpoint formula and is suitable for students in Grades 8-10.

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