Consider the `Delta ABC` with vertices A( - 2 , 3) , B (3 , 1) and C (1 ,2)
What is the centroid of the `Delta ABC` ?

Options

A. `((2)/(3) , 2)`
B. `((1)/(3) , 2)`
C. `(1 , (2)/(3))`
D. `((1)/(2) , 3)`

Answer

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

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

### Step 2: Calculate the x-coordinates' average
To find the x-coordinate of the centroid, we sum the x-coordinates of the vertices and divide by 3:
\[ x_{centroid} = \frac{x_A + x_B + x_C}{3} = \frac{-2 + 3 + 1}{3} \]
Calculating this gives:
\[ x_{centroid} = \frac{2}{3} \]

### Step 3: Calculate the y-coordinates' average
Next, we calculate the y-coordinate of the centroid by summing the y-coordinates of the vertices and dividing by 3:
\[ y_{centroid} = \frac{y_A + y_B + y_C}{3} = \frac{3 + 1 + 2}{3} \]
Calculating this gives:
\[ y_{centroid} = \frac{6}{3} = 2 \]

### Step 4: Combine the results
Now we can combine the x and y coordinates to find the centroid of triangle ABC:
\[ Centroid = \left( \frac{2}{3}, 2 \right) \]

### Final Answer
The centroid of triangle ABC is \(\left( \frac{2}{3}, 2 \right)\).

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