Find the centroid of `Delta`ABC, where A(-4, 6), B(2, -2) and C(2, 5)

Answer

To find the centroid of triangle ABC with vertices A(-4, 6), B(2, -2), and C(2, 5), we can use the formula for the centroid of a triangle. The centroid (G) is given by the coordinates:

\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

### Step-by-step Solution:

1. **Identify the coordinates of the vertices:**
- A(x₁, y₁) = A(-4, 6)
- B(x₂, y₂) = B(2, -2)
- C(x₃, y₃) = C(2, 5)

2. **Substitute the coordinates into the centroid formula:**
- \( x_1 = -4, y_1 = 6 \)
- \( x_2 = 2, y_2 = -2 \)
- \( x_3 = 2, y_3 = 5 \)

The formula for the x-coordinate of the centroid (H) is:
\[ H = \frac{x_1 + x_2 + x_3}{3} = \frac{-4 + 2 + 2}{3} \]

The formula for the y-coordinate of the centroid (K) is:
\[ K = \frac{y_1 + y_2 + y_3}{3} = \frac{6 + (-2) + 5}{3} \]

3. **Calculate the x-coordinate (H):**
- Calculate the sum:
\[ -4 + 2 + 2 = 0 \]
- Divide by 3:
\[ H = \frac{0}{3} = 0 \]

4. **Calculate the y-coordinate (K):**
- Calculate the sum:
\[ 6 + (-2) + 5 = 9 \]
- Divide by 3:
\[ K = \frac{9}{3} = 3 \]

5. **Write the coordinates of the centroid:**
- The centroid G is at the coordinates:
\[ G(0, 3) \]

### Final Answer:
The centroid of triangle ABC is at the point \( G(0, 3) \).