The centroid of a `Delta ABC` with vertices A(-4, 6), B(2,-2) and C(2, 5) is:

Options

A. (3, 0)
B. `((8)/(3), 3)`
C. `(3, (8)/(3))`
D. (0, 3)

Answer

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

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

### Step 1: Identify the coordinates of the vertices
- A = (-4, 6)
- B = (2, -2)
- C = (2, 5)

### Step 2: Substitute the coordinates into the centroid formula
Using the formula for the x-coordinate of the centroid:

\[ x_g = \frac{x_1 + x_2 + x_3}{3} = \frac{-4 + 2 + 2}{3} \]

### Step 3: Calculate the x-coordinate
\[ x_g = \frac{-4 + 2 + 2}{3} = \frac{0}{3} = 0 \]

### Step 4: Substitute the coordinates into the centroid formula for y-coordinate
Using the formula for the y-coordinate of the centroid:

\[ y_g = \frac{y_1 + y_2 + y_3}{3} = \frac{6 + (-2) + 5}{3} \]

### Step 5: Calculate the y-coordinate
\[ y_g = \frac{6 - 2 + 5}{3} = \frac{9}{3} = 3 \]

### Step 6: Combine the results
The coordinates of the centroid G are:

\[ G(0, 3) \]

Thus, the centroid of triangle ABC is at the point (0, 3).

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

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