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

Given:

- Midpoint \( M(1, 3) \)
- Coordinates of point \( A(8, 1) \)
- Coordinates of point \( B(x, y) \) (unknown)

The midpoint formula for a segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:

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

Substituting the known values:

\[
\left(\frac{8 + x}{2}, \frac{1 + y}{2}\right) = (1, 3)
\]

We can equate the coordinates:

1. For the x-coordinate:

   \[
   \frac{8 + x}{2} = 1
   \]

   Solving for \( x \):

   \[
   8 + x = 2 \cdot 1
   \]

   \[
   8 + x = 2
   \]

   \[
   x = 2 - 8
   \]

   \[
   x = -6
   \]

2. For the y-coordinate:

   \[
   \frac{1 + y}{2} = 3
   \]

   Solving for \( y \):

   \[
   1 + y = 2 \cdot 3
   \]

   \[
   1 + y = 6
   \]

   \[
   y = 6 - 1
   \]

   \[
   y = 5
   \]

So, the coordinates of point \( B \) are:

\[
\boxed{(-6, 5)}
\]

Would you like more details or have any questions? Here are some related questions:

1. How is the midpoint formula derived?
2. Can we find the distance between points A and B using their coordinates?
3. What if the midpoint were given differently? How would that affect point B?
4. How would the coordinates change if we rotated the segment AB around the midpoint?
5. What are some real-world applications of finding midpoints?

**Tip:** Always double-check your calculations, especially when dealing with negative numbers to avoid errors!

Learn how to find the coordinates of point B using the midpoint formula with a detailed step-by-step solution. This problem involves coordinate geometry and the application of the midpoint formula. Suitable for students in Grades 9-12.

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