To solve for the coordinates of point $B$, we can use the midpoint formula. The midpoint $M$ of a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by:

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

Here, we know:

• The midpoint $M(-2, -5)$
• The coordinates of $A(-3, -2)$
• We need to find the coordinates of $B(x_2, y_2)$

Using the midpoint formula, we can set up the following equations:

$\frac{-3 + x_2}{2} = -2$ $\frac{-2 + y_2}{2} = -5$

Solving for $x_2$:

$\frac{-3 + x_2}{2} = -2$ Multiply both sides by 2: $-3 + x_2 = -4$ Add 3 to both sides: $x_2 = -1$

Solving for $y_2$:

$\frac{-2 + y_2}{2} = -5$ Multiply both sides by 2: $-2 + y_2 = -10$ Add 2 to both sides: $y_2 = -8$

So, the coordinates of $B$ are $\mathbf{(-1, -8)}$.

Would you like any further explanation or have any other questions?

Related Questions:

1. How is the midpoint formula derived from basic coordinate geometry?
2. What are some applications of the midpoint formula in real life?
3. How do you find the distance between two points if you know their coordinates?
4. What happens to the midpoint if one point is moved further away from the other?
5. Can you use the midpoint formula in three-dimensional space? How?

Tip:

Always double-check your calculations, especially when solving for multiple variables, to ensure the solution is consistent with the given information.