The image shows a triangle labeled $\triangle ABC$ with vertices at $A(2, 8)$, $B(5, 5)$, and $C(2, 2)$, and the task asks to find the area of the triangle.

To find the area of a triangle given the coordinates of its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, we use the following formula:

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

Let's calculate this now. The coordinates of the vertices are:

• $A(2, 8)$
• $B(5, 5)$
• $C(2, 2)$

Substituting these values into the formula:The area of the triangle $\triangle ABC$ with vertices at $A(2, 8)$, $B(5, 5)$, and $C(2, 2)$ is $9$ square units.

Would you like more details or have any further questions?

Here are some related questions to expand your understanding:

1. How can we verify the area of a triangle using the base and height method?
2. What is the formula for calculating the perimeter of a triangle?
3. How do you find the centroid of a triangle given its vertices?
4. Can you use this method for any type of triangle?
5. How do you calculate the area of a triangle on a non-Cartesian plane?

Tip: For any triangle in a coordinate plane, always double-check vertex coordinates when using area formulas to avoid sign errors.