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

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

In this problem:

• Midpoint $M = (4, -4)$
• Point $A = (-3, 1)$
• Point $B = (x_2, y_2)$

We need to find the coordinates of $B(x_2, y_2)$.

Step-by-Step Solution:

1. Calculate the x-coordinate of B:

   $4 = \frac{-3 + x_2}{2}$

   Multiply both sides by 2:

   $8 = -3 + x_2$

   Add 3 to both sides:

   $x_2 = 11$

2. Calculate the y-coordinate of B:

   $-4 = \frac{1 + y_2}{2}$

   Multiply both sides by 2:

   $-8 = 1 + y_2$

   Subtract 1 from both sides:

   $y_2 = -9$

Conclusion:

The coordinates of point $B$ are $(11, -9)$.

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

Here are 5 related questions for further exploration:

1. How do you find the midpoint if both points are given?
2. How can you determine the distance between two points?
3. How do you find the equation of a line given two points?
4. What is the significance of the slope of a line segment?
5. How can the midpoint formula be used in geometry problems?

Tip: Always double-check your calculations by substituting the values back into the formula to verify your results.