Find the co-ordinates of a centroid of a triangle whose vertices are (3, -5) , (-7 , 4) and (10, -2) .

Answer

To find the coordinates of the centroid of a triangle with vertices at (3, -5), (-7, 4), and (10, -2), we can use the formula for the centroid of a triangle. The centroid (G) is given by the coordinates:

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

Where:
- \( (x_1, y_1) \) = (3, -5)
- \( (x_2, y_2) \) = (-7, 4)
- \( (x_3, y_3) \) = (10, -2)

### Step 1: Identify the coordinates of the vertices
We have the vertices:
- \( (x_1, y_1) = (3, -5) \)
- \( (x_2, y_2) = (-7, 4) \)
- \( (x_3, y_3) = (10, -2) \)

### Step 2: Calculate the x-coordinate of the centroid
Using the formula for the x-coordinate of the centroid:

\[ x_G = \frac{x_1 + x_2 + x_3}{3} = \frac{3 + (-7) + 10}{3} \]

Calculating the sum:

\[ 3 - 7 + 10 = 6 \]

Now, divide by 3:

\[ x_G = \frac{6}{3} = 2 \]

### Step 3: Calculate the y-coordinate of the centroid
Using the formula for the y-coordinate of the centroid:

\[ y_G = \frac{y_1 + y_2 + y_3}{3} = \frac{-5 + 4 + (-2)}{3} \]

Calculating the sum:

\[ -5 + 4 - 2 = -3 \]

Now, divide by 3:

\[ y_G = \frac{-3}{3} = -1 \]

### Step 4: Combine the coordinates to get the centroid
Thus, the coordinates of the centroid G are:

\[ G(2, -1) \]

### Final Answer
The coordinates of the centroid of the triangle are \( (2, -1) \).

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