Find the centroid of the triangle with vertices at `(-1,0)` , `(5,-2)` and `(8,2)`

Answer

To find the centroid of the triangle with vertices at \((-1, 0)\), \((5, -2)\), and \((8, 2)\), we can use the formula for the centroid of a triangle given its vertices:

\[ \text{Centroid} (C) = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

### Step-by-Step Solution:

1. **Identify the vertices**:
- Let \( A = (-1, 0) \)
- Let \( B = (5, -2) \)
- Let \( C = (8, 2) \)

2. **Extract the coordinates**:
- From vertex \( A \): \( x_1 = -1 \), \( y_1 = 0 \)
- From vertex \( B \): \( x_2 = 5 \), \( y_2 = -2 \)
- From vertex \( C \): \( x_3 = 8 \), \( y_3 = 2 \)

3. **Calculate the x-coordinate of the centroid**:
\[ x_c = \frac{x_1 + x_2 + x_3}{3} = \frac{-1 + 5 + 8}{3} \]
\[ x_c = \frac{12}{3} = 4 \]

4. **Calculate the y-coordinate of the centroid**:
\[ y_c = \frac{y_1 + y_2 + y_3}{3} = \frac{0 - 2 + 2}{3} \]
\[ y_c = \frac{0}{3} = 0 \]

5. **Combine the coordinates to find the centroid**:
\[ C = (x_c, y_c) = (4, 0) \]

### Final Answer:
The centroid of the triangle is \((4, 0)\).