The centroid of a triangle, whose vertices are (2, 1), (5, 2) and (3, 4) is

Options

A. `((8)/(3),(7)/(3))`
B. `((10)/(3),(7)/(3))`
C. `(-(10)/(3),(7)/(3))`
D. `((10)/(3),-(7)/(3))`

Answer

To find the centroid of a triangle given its vertices, we can use the formula for the centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):

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

Given the vertices of the triangle:
- \( (x_1, y_1) = (2, 1) \)
- \( (x_2, y_2) = (5, 2) \)
- \( (x_3, y_3) = (3, 4) \)

### Step 1: Calculate the x-coordinate of the centroid
Using the formula for the x-coordinate:

\[ x_G = \frac{x_1 + x_2 + x_3}{3} = \frac{2 + 5 + 3}{3} \]

Calculating the sum:

\[ x_G = \frac{10}{3} \]

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

\[ y_G = \frac{y_1 + y_2 + y_3}{3} = \frac{1 + 2 + 4}{3} \]

Calculating the sum:

\[ y_G = \frac{7}{3} \]

### Step 3: Combine the x and y coordinates to find the centroid
Now, we can express the centroid \( G \):

\[ G = \left( \frac{10}{3}, \frac{7}{3} \right) \]

Thus, the centroid of the triangle with vertices \( (2, 1) \), \( (5, 2) \), and \( (3, 4) \) is:

\[ \boxed{\left( \frac{10}{3}, \frac{7}{3} \right)} \]