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!