The coordinates of the centroid G of a triangle with vertices at (3, 7), (5, 5) and ( -3, 2) is

Options

A. (10/3, 14/3)
B. (10/3, 10/3)
C. (5/3 14/3)
D. (11/3, 10/3)

Answer

To find the coordinates of the centroid \( G \) of a triangle with vertices at \( (3, 7) \), \( (5, 5) \), and \( (-3, 2) \), we can use the formula for the centroid of a triangle:

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

where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices of the triangle.

### Step 1: Identify the coordinates of the vertices
The vertices of the triangle are:
- \( (x_1, y_1) = (3, 7) \)
- \( (x_2, y_2) = (5, 5) \)
- \( (x_3, y_3) = (-3, 2) \)

### Step 2: Substitute the coordinates into the centroid formula
Now, we will substitute the coordinates into the centroid formula.

\[ G\left( \frac{3 + 5 + (-3)}{3}, \frac{7 + 5 + 2}{3} \right) \]

### Step 3: Calculate the x-coordinate of the centroid
Now, calculate the x-coordinate:

\[ x = \frac{3 + 5 - 3}{3} = \frac{5}{3} \]

### Step 4: Calculate the y-coordinate of the centroid
Next, calculate the y-coordinate:

\[ y = \frac{7 + 5 + 2}{3} = \frac{14}{3} \]

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

\[ G\left( \frac{5}{3}, \frac{14}{3} \right) \]

### Final Answer
The coordinates of the centroid \( G \) of the triangle are \( \left( \frac{5}{3}, \frac{14}{3} \right) \).

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