To find the centroid of the triangle in the image, we use the formula for the centroid of a triangle:

$G(x, y) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$

Where $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are the coordinates of the triangle's vertices.

From the graph:

• Point $A = (6, -3)$
• Point $B = (-3, -3)$
• Point $C = (-6, 6)$

Now, we calculate the centroid:

1. For the x-coordinate: $x_{\text{centroid}} = \frac{6 + (-3) + (-6)}{3} = \frac{-3}{3} = -1$

2. For the y-coordinate: $y_{\text{centroid}} = \frac{-3 + (-3) + 6}{3} = \frac{0}{3} = 0$

Thus, the centroid of the triangle is $(-1, 0)$.

Since none of the options provided match this exactly, there seems to be a mismatch between the available choices and the correct answer $(-1, 0)$.

Let me know if you'd like further clarification or if you have any other questions!

Here are five related questions to explore further:

1. How do you calculate the centroid for different types of triangles?
2. What is the significance of the centroid in geometry?
3. Can the centroid ever lie outside the triangle?
4. How is the centroid formula derived from coordinate geometry principles?
5. How can we apply centroid calculation in real-world scenarios, such as in physics or engineering?

Tip: Always double-check the coordinates of each vertex carefully when calculating geometric properties like the centroid!