To find the coordinates of point $B$, given that $M(0, 2)$ is the midpoint of $\overline{AB}$ and $A(5, -1)$ is one of the endpoints, we use the midpoint formula.

The midpoint formula states that for two points $A(x_1, y_1)$ and $B(x_2, y_2)$, the midpoint $M(x_m, y_m)$ is given by:

$x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}$

Given:

• Midpoint $M(0, 2)$
• Point $A(5, -1)$

Let's denote the coordinates of point $B$ as $(x_2, y_2)$.

Step 1: Set up the equations

Using the midpoint formula, we have: $0 = \frac{5 + x_2}{2} \quad \text{(for the x-coordinate)}$ $2 = \frac{-1 + y_2}{2} \quad \text{(for the y-coordinate)}$

Step 2: Solve for $x_2$ and $y_2$

1. Solving for $x_2$: $0 = \frac{5 + x_2}{2}$ Multiply both sides by 2: $0 = 5 + x_2$ Subtract 5 from both sides: $x_2 = -5$

2. Solving for $y_2$: $2 = \frac{-1 + y_2}{2}$ Multiply both sides by 2: $4 = -1 + y_2$ Add 1 to both sides: $y_2 = 5$

Final Answer

The coordinates of point $B$ are $(-5, 5)$.

Would you like more details, or do you have any questions?

Here are 5 related questions to explore:

1. How would you find the distance between points $A$ and $B$?
2. What is the slope of the line segment $\overline{AB}$?
3. How would you find the equation of the line that passes through points $A$ and $B$?
4. If $B$ was at $(-5, 5)$, what would be the length of the line segment $\overline{AB}$?
5. How would the midpoint formula change in three dimensions?

Tip: The midpoint formula can also be extended to higher dimensions, such as 3D, where you average the coordinates of all dimensions.