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 $(-7, -6)$.

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.