Find the centroid of the triangle formed by the points (2,7) (3,-1)(-5,6).

Answer

To find the centroid of the triangle formed by the points (2,7), (3,-1), and (-5,6), we can follow these steps:

### Step 1: Identify the coordinates of the points
Let the points be:
- \( A(2, 7) \) where \( x_1 = 2 \) and \( y_1 = 7 \)
- \( B(3, -1) \) where \( x_2 = 3 \) and \( y_2 = -1 \)
- \( C(-5, 6) \) where \( x_3 = -5 \) and \( y_3 = 6 \)

### Step 2: Use the centroid formula
The formula for the centroid \( G \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

### Step 3: Calculate the x-coordinate of the centroid
Substituting the x-coordinates into the formula:
\[ x_G = \frac{x_1 + x_2 + x_3}{3} = \frac{2 + 3 + (-5)}{3} \]
Calculating the sum:
\[ 2 + 3 - 5 = 0 \]
Now divide by 3:
\[ x_G = \frac{0}{3} = 0 \]

### Step 4: Calculate the y-coordinate of the centroid
Now substituting the y-coordinates into the formula:
\[ y_G = \frac{y_1 + y_2 + y_3}{3} = \frac{7 + (-1) + 6}{3} \]
Calculating the sum:
\[ 7 - 1 + 6 = 12 \]
Now divide by 3:
\[ y_G = \frac{12}{3} = 4 \]

### Step 5: Write the coordinates of the centroid
Thus, the coordinates of the centroid \( G \) are:
\[ G(0, 4) \]

### Final Answer:
The centroid of the triangle formed by the points (2,7), (3,-1), and (-5,6) is \( G(0, 4) \).

---