We are given the coordinates of points $D(-2, 8)$, $E(2, 0)$, and $F(-2, -4)$, and we need to find the area of triangle $\Delta DEF$.

The formula for the area of a triangle given its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is:

$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Substituting the coordinates $D(-2, 8)$, $E(2, 0)$, and $F(-2, -4)$ into this formula, we get:

$\text{Area} = \frac{1}{2} \left| (-2)(0 - (-4)) + 2((-4) - 8) + (-2)(8 - 0) \right|$

Let me calculate that for you.The area of triangle $\Delta DEF$ is $24$ square units.

Do you want further details or have any questions?

Here are some related questions for practice:

1. What is the perimeter of triangle $\Delta DEF$?
2. Can you find the centroid (center of mass) of triangle $\Delta DEF$?
3. How would the area change if point $D$ was shifted to $(-3, 8)$?
4. What is the slope of line $DE$ in $\Delta DEF$?
5. How can you use the distance formula to calculate the lengths of the sides of $\Delta DEF$?

Tip: When working with triangles on a coordinate plane, the formula for the area becomes much simpler when you know the vertices!