To find the area of the triangle $\triangle GHI$ formed by the points $G(4, 6)$, $H(2, -8)$, and $I(-6, 8)$, we can use the shoelace theorem (also called the determinant formula for the area of a polygon).

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

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

Let's plug in the coordinates $G(4, 6)$, $H(2, -8)$, and $I(-6, 8)$:

$\text{Area} = \frac{1}{2} \left| 4(-8 - 8) + 2(8 - 6) + (-6)(6 + 8) \right|$

Now I'll calculate the result.The area of triangle $\triangle GHI$ is $72$ square units.

Would you like further details or have any questions about this calculation?

Here are some related questions for further practice:

1. How do you find the perimeter of this triangle?
2. What is the length of side $GH$ using the distance formula?
3. Can you calculate the slope of line $GI$?
4. How would the area change if one of the points were shifted by a specific amount?
5. What is the equation of the line passing through points $G$ and $H$?

Tip: The shoelace theorem is a quick method to find the area of any polygon if you know the vertices.