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To find the coordinates of point \( B \), we can use the midpoint formula. Given that the midpoint \( M \) of \( \overline{AB} \) is at \( M(-3, 1) \) and the coordinates of \( A \) are \( (1, 8) \), we need to determine the coordinates of \( B(x, y) \).

The midpoint formula for a line segment \( AB \) with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Since \( M = (-3, 1) \) and \( A = (1, 8) \), we set up equations for \( x \) and \( y \) as follows:

1. For the \( x \)-coordinate:
   \[
   \frac{1 + x}{2} = -3
   \]
   Solving for \( x \):
   \[
   1 + x = -6
   \]
   \[
   x = -7
   \]

2. For the \( y \)-coordinate:
   \[
   \frac{8 + y}{2} = 1
   \]
   Solving for \( y \):
   \[
   8 + y = 2
   \]
   \[
   y = -6
   \]

Thus, the coordinates of point \( B \) are \( (-7, -6) \).

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Would you like further details on this solution or have any questions?

Here are 5 related questions you may find interesting:

1. How would you calculate the midpoint if both points \( A \) and \( B \) are known?
2. What would the coordinates of \( B \) be if the midpoint \( M \) were at \( (0, 0) \) instead?
3. Can you find the length of the line segment \( AB \) using the coordinates of \( A \) and \( B \)?
4. How does the midpoint formula apply in three-dimensional space?
5. What if the midpoint and \( B \) were known; how would you find the coordinates of \( A \)?

**Tip:** Always double-check each step when solving for coordinates, as simple sign errors can lead to incorrect answers.

The midpoint of \( \overline{AB} \) is \( M(-3, 1) \). If the coordinates of \( A \) are \( (1, 8) \), what are the coordinates of \( B \)?

Learn how to find the coordinates of point B when given the midpoint M(-3, 1) and coordinates of point A (1, 8). This problem uses the midpoint formula and covers key concepts in coordinate geometry, suitable for students in Grades 7-9.

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