The problem asks for the coordinates of vertex D of rectangle ABCD, given the coordinates of vertices A, B, and C. To find vertex D, we can use the property that the diagonals of a rectangle are equal in length and bisect each other.

Step-by-step process:

1. The coordinates of A, B, and C are given as:

   • A(-4, 7)
   • B(5, 7)
   • C(5, -8)

2. To find the coordinates of D, use the midpoint formula for diagonals. The midpoint of diagonal AC should be the same as the midpoint of diagonal BD. This is because diagonals of a rectangle bisect each other.

The midpoint formula is: $\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

3. First, calculate the midpoint of diagonal AC: $\text{Midpoint of AC} = \left(\frac{-4 + 5}{2}, \frac{7 + (-8)}{2}\right) = \left(\frac{1}{2}, \frac{-1}{2}\right)$

4. The midpoint of diagonal BD must be the same, so use the midpoint formula for B and D: $\text{Midpoint of BD} = \left(\frac{5 + x_D}{2}, \frac{7 + y_D}{2}\right)$ Equating the two midpoints: $\left(\frac{1}{2}, \frac{-1}{2}\right) = \left(\frac{5 + x_D}{2}, \frac{7 + y_D}{2}\right)$

This gives us the system of equations: $\frac{5 + x_D}{2} = \frac{1}{2} \quad \text{and} \quad \frac{7 + y_D}{2} = \frac{-1}{2}$

5. Solve for $x_D$ and $y_D$: From the first equation: $5 + x_D = 1 \quad \Rightarrow \quad x_D = 1 - 5 = -4$

From the second equation: $7 + y_D = -1 \quad \Rightarrow \quad y_D = -1 - 7 = -8$

Thus, the coordinates of vertex D are $(-4, -8)$.

Would you like further clarification or additional problems? Here are some related questions:

1. How do you calculate the distance between two points?
2. What are the properties of a rectangle in coordinate geometry?
3. How can you find the area of a rectangle using its vertices?
4. How do you use the distance formula to find the length of the diagonals of a rectangle?
5. What is the relationship between the slopes of opposite sides of a rectangle?

Tip: Always verify your result by checking if the diagonals bisect each other and have the same length!