The coordinates of the vertices of a triangles are (3,1) , (2,3) and (-2,2) . Find the coordinates of the centroid of the triangle ABC :

Options

A. (1,2)
B. (2,3)
C. (4,5)
D. (5,6)

Answer

To find the coordinates of the centroid of triangle ABC with vertices at A(3,1), B(2,3), and C(-2,2), we can follow these steps:

### Step 1: Identify the coordinates of the vertices
The coordinates of the vertices of triangle ABC are:
- A = (3, 1)
- B = (2, 3)
- C = (-2, 2)

### Step 2: Use the formula for the centroid
The formula for the coordinates of the centroid (G) of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) is given by:
\[ G\left(x, y\right) = \left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]

### Step 3: Substitute the coordinates into the formula
Using the coordinates of points A, B, and C:
- \(x1 = 3\), \(y1 = 1\)
- \(x2 = 2\), \(y2 = 3\)
- \(x3 = -2\), \(y3 = 2\)

Now, substitute these values into the formula:
\[ G\left(x, y\right) = \left(\frac{3 + 2 - 2}{3}, \frac{1 + 3 + 2}{3}\right) \]

### Step 4: Calculate the x-coordinate of the centroid
Calculate the x-coordinate:
\[ x = \frac{3 + 2 - 2}{3} = \frac{3}{3} = 1 \]

### Step 5: Calculate the y-coordinate of the centroid
Calculate the y-coordinate:
\[ y = \frac{1 + 3 + 2}{3} = \frac{6}{3} = 2 \]

### Step 6: Write the coordinates of the centroid
Thus, the coordinates of the centroid G are:
\[ G(1, 2) \]

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

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