The mid-points if the sides of a triangle are `(3,2,(3)/(2)),(1,(3)/(2),3)` and `(2,(5)/(2),(5)/(2))`. Find the coordinates of its centroid.

Answer

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

### Step 1: Identify the Midpoints
The midpoints of the sides of the triangle are given as:
- \( P(3, 2, \frac{3}{2}) \)
- \( Q(1, \frac{3}{2}, 3) \)
- \( R(2, \frac{5}{2}, \frac{5}{2}) \)

### Step 2: Use the Centroid Formula
The formula for the centroid \( (G) \) of a triangle with vertices \( (x_1, y_1, z_1) \), \( (x_2, y_2, z_2) \), and \( (x_3, y_3, z_3) \) is given by:

\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]

### Step 3: Substitute the Coordinates
Substituting the coordinates of the midpoints into the centroid formula:

1. **For x-coordinate:**
\[ x_G = \frac{3 + 1 + 2}{3} \]

2. **For y-coordinate:**
\[ y_G = \frac{2 + \frac{3}{2} + \frac{5}{2}}{3} \]

3. **For z-coordinate:**
\[ z_G = \frac{\frac{3}{2} + 3 + \frac{5}{2}}{3} \]

### Step 4: Calculate Each Coordinate

1. **Calculating x-coordinate:**
\[ x_G = \frac{3 + 1 + 2}{3} = \frac{6}{3} = 2 \]

2. **Calculating y-coordinate:**
\[ y_G = \frac{2 + \frac{3}{2} + \frac{5}{2}}{3} = \frac{2 + 4}{3} = \frac{6}{3} = 2 \]

3. **Calculating z-coordinate:**
\[ z_G = \frac{\frac{3}{2} + 3 + \frac{5}{2}}{3} = \frac{\frac{3}{2} + \frac{6}{2} + \frac{5}{2}}{3} = \frac{\frac{14}{2}}{3} = \frac{7}{3} \]

### Step 5: Combine the Results
Thus, the coordinates of the centroid \( G \) are:
\[ G = \left( 2, 2, \frac{7}{3} \right) \]

### Final Answer
The coordinates of the centroid of the triangle are \( (2, 2, \frac{7}{3}) \).

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