The problem provided is asking for the coordinates of point $B$ given that the midpoint $M$ of segment $\overline{AB}$ is $M(-1, 2)$ and the coordinates of point $A$ are $A(4, 5)$.

To solve this:

We know the formula for the midpoint $M$ of a line segment $\overline{AB}$ is:

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

Where:

• $(x_1, y_1)$ are the coordinates of point $A$,
• $(x_2, y_2)$ are the coordinates of point $B$,
• $M(x_m, y_m)$ is the midpoint.

Given:

• $M = (-1, 2)$,
• $A = (4, 5)$.

Let the coordinates of $B$ be $B(x_2, y_2)$. Using the midpoint formula, we can set up the following equations:

$-1 = \frac{4 + x_2}{2} \quad \text{(for the x-coordinates)}$ $2 = \frac{5 + y_2}{2} \quad \text{(for the y-coordinates)}$

Solving for $x_2$ and $y_2$:

1. For the x-coordinate: $-1 = \frac{4 + x_2}{2}$ Multiply both sides by 2: $-2 = 4 + x_2$ Subtract 4 from both sides: $x_2 = -6$

2. For the y-coordinate: $2 = \frac{5 + y_2}{2}$ Multiply both sides by 2: $4 = 5 + y_2$ Subtract 5 from both sides: $y_2 = -1$

Final Answer:

The coordinates of point $B$ are $(-6, -1)$.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:

1. How do you calculate the distance between two points?
2. What is the formula for finding the slope of a line given two points?
3. How do you determine if two segments are perpendicular or parallel based on their slopes?
4. Can you find the equation of a line given its midpoint and one point on the line?
5. How do you find the area of a triangle given its three vertices?

Tip: When solving midpoint problems, always double-check your algebra by verifying that the computed midpoint matches the given value.