The centroid of the triangle whose vertices are (3, 10), (7, 7), (-2, 1) is

Options

A. (8/3, 6)
B. (6, 8/3)
C. (- 4, - 7/3)
D. None of these

Answer

To find the centroid of the triangle with vertices at (3, 10), (7, 7), and (-2, 1), we can use the formula for the centroid of a triangle. The centroid (G) can be calculated using the following formulas:

\[ G_x = \frac{x_1 + x_2 + x_3}{3} \]
\[ G_y = \frac{y_1 + y_2 + y_3}{3} \]

Where:
- \( (x_1, y_1) = (3, 10) \)
- \( (x_2, y_2) = (7, 7) \)
- \( (x_3, y_3) = (-2, 1) \)

### Step 1: Calculate the x-coordinate of the centroid
Substituting the x-coordinates into the formula:

\[ G_x = \frac{3 + 7 - 2}{3} \]

Calculating the numerator:

\[ 3 + 7 - 2 = 8 \]

Now, divide by 3:

\[ G_x = \frac{8}{3} \]

### Step 2: Calculate the y-coordinate of the centroid
Now, substituting the y-coordinates into the formula:

\[ G_y = \frac{10 + 7 + 1}{3} \]

Calculating the numerator:

\[ 10 + 7 + 1 = 18 \]

Now, divide by 3:

\[ G_y = \frac{18}{3} = 6 \]

### Final Result
Thus, the coordinates of the centroid of the triangle are:

\[ \left( \frac{8}{3}, 6 \right) \]

### Conclusion
The centroid of the triangle with vertices (3, 10), (7, 7), and (-2, 1) is \(\left( \frac{8}{3}, 6 \right)\).

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