To find the coordinates of point $B$, we use the midpoint formula. The formula for the midpoint $M$ of a line segment between points $A(x_1, y_1)$ and $B(x_2, y_2)$ is:

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

Here, you are given:

• The coordinates of point $A(-5, 5)$,
• The midpoint $M(-6, 3)$,
• The coordinates of point $B(x_2, y_2)$, which are unknown.

Now, let's apply the midpoint formula component-wise.

1. For the $x$-coordinate:

$\frac{-5 + x_2}{2} = -6$

Solve for $x_2$:

$-5 + x_2 = -12 \quad \Rightarrow \quad x_2 = -12 + 5 = -7$

2. For the $y$-coordinate:

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

Solve for $y_2$:

$5 + y_2 = 6 \quad \Rightarrow \quad y_2 = 6 - 5 = 1$

Conclusion:

The coordinates of point $B$ are $(-7, 1)$.

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Do you want details on how this formula works, or have any questions?

Here are some related questions:

1. How do you find the distance between points $A$ and $B$?
2. Can you explain how to generalize the midpoint formula for 3D space?
3. If the midpoint and point $B$ were known, how would you find point $A$?
4. How does the midpoint formula relate to vectors?
5. What is the relationship between the midpoint and the centroid of a triangle?

Tip: Always check your solutions by substituting back into the formula to confirm!