If the coordinates of the mid-points of the sides of a triangle are (1, 1) , (2, -3) and (3, 4). Find the coordinates of the centroid.

Answer

To find the coordinates of the centroid of a triangle when given the midpoints of its sides, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Midpoint Coordinates**:
The midpoints of the sides of the triangle are given as:
- \( M_1(1, 1) \)
- \( M_2(2, -3) \)
- \( M_3(3, 4) \)

2. **Assign Coordinates**:
Let:
- \( M_1 = (x_1, y_1) = (1, 1) \)
- \( M_2 = (x_2, y_2) = (2, -3) \)
- \( M_3 = (x_3, y_3) = (3, 4) \)

3. **Calculate the Centroid Coordinates**:
The formula for the centroid \( G \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

4. **Calculate the x-coordinate of the Centroid**:
\[ x_G = \frac{x_1 + x_2 + x_3}{3} = \frac{1 + 2 + 3}{3} = \frac{6}{3} = 2 \]

5. **Calculate the y-coordinate of the Centroid**:
\[ y_G = \frac{y_1 + y_2 + y_3}{3} = \frac{1 + (-3) + 4}{3} = \frac{1 - 3 + 4}{3} = \frac{2}{3} \]

6. **Combine the Coordinates**:
Thus, the coordinates of the centroid \( G \) are:
\[ G(2, \frac{2}{3}) \]

### Final Answer:
The coordinates of the centroid of the triangle are \( (2, \frac{2}{3}) \).

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