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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.

Find the area of the triangle formed by points G(4, 6), H(2, -8), and I(-6, 8).

Learn how to calculate the area of a triangle formed by points G(4, 6), H(2, -8), and I(-6, 8) using the Shoelace Theorem. This detailed solution is suitable for high school students (Grades 9-12).

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