The problem asks to find the coordinates of point $B$, given the midpoint $M(0, 1)$ of line segment $AB$ and the coordinates of point $A(-8, 8)$.

We use the midpoint formula: $M(x, y) = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right)$ where $(x_A, y_A)$ are the coordinates of point $A$, and $(x_B, y_B)$ are the coordinates of point $B$.

Given:

• $M(0, 1)$
• $A(-8, 8)$

Using the midpoint formula: $0 = \frac{-8 + x_B}{2} \quad \text{(for the x-coordinate)}$ $1 = \frac{8 + y_B}{2} \quad \text{(for the y-coordinate)}$

Solve for $x_B$:

$0 = \frac{-8 + x_B}{2}$ Multiply both sides by 2: $0 = -8 + x_B$ Add 8 to both sides: $x_B = 8$

Solve for $y_B$:

$1 = \frac{8 + y_B}{2}$ Multiply both sides by 2: $2 = 8 + y_B$ Subtract 8 from both sides: $y_B = -6$

Thus, the coordinates of point $B$ are $(8, -6)$.

Would you like more details or have any questions about this solution?

Here are 5 related questions for further practice:

1. If the midpoint is $M(3, 4)$ and point $A$ is $(1, 2)$, what are the coordinates of point $B$?
2. How do you calculate the distance between two points in a plane?
3. Can you derive the equation of a line given two points?
4. What is the relationship between the slope of a line and its perpendicular bisector?
5. How does the midpoint formula relate to the distance formula?

Tip: The midpoint formula is simply the average of the x-coordinates and y-coordinates of two points.