The coordinates of the middle points of the sides of a triangle are (4, 2), (3, 3) and (2, 2), then the coordinates of its centroid are

Options

A. `(3,(7)/(3))`
B. (3,3)
C. (4,3)
D. None of these

Answer

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

### Step-by-Step Solution:

1. **Identify the midpoints**: We have the midpoints of the sides of the triangle as follows:
- Midpoint A (between points B and C): (4, 2)
- Midpoint B (between points A and C): (3, 3)
- Midpoint C (between points A and B): (2, 2)

2. **Set up equations for the vertices**:
- Let the vertices of the triangle be \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).
- From the midpoint formula, we can derive the following equations:
- For midpoint A:
\[ \frac{x_2 + x_3}{2} = 4 \quad \text{and} \quad \frac{y_2 + y_3}{2} = 2 \]
This gives us:
\[ x_2 + x_3 = 8 \quad (1) \]
\[ y_2 + y_3 = 4 \quad (2) \]

- For midpoint B:
\[ \frac{x_1 + x_3}{2} = 3 \quad \text{and} \quad \frac{y_1 + y_3}{2} = 3 \]
This gives us:
\[ x_1 + x_3 = 6 \quad (3) \]
\[ y_1 + y_3 = 6 \quad (4) \]

- For midpoint C:
\[ \frac{x_1 + x_2}{2} = 2 \quad \text{and} \quad \frac{y_1 + y_2}{2} = 2 \]
This gives us:
\[ x_1 + x_2 = 4 \quad (5) \]
\[ y_1 + y_2 = 4 \quad (6) \]

3. **Solve the equations for x-coordinates**:
- From equations (1) and (5):
\[ x_2 + x_3 = 8 \quad (1) \]
\[ x_1 + x_2 = 4 \quad (5) \]
- Substituting \( x_2 = 4 - x_1 \) into (1):
\[ (4 - x_1) + x_3 = 8 \implies x_3 = 8 - 4 + x_1 = 4 + x_1 \quad (7) \]
- Now substitute \( x_3 \) from (7) into (3):
\[ x_1 + (4 + x_1) = 6 \implies 2x_1 + 4 = 6 \implies 2x_1 = 2 \implies x_1 = 1 \]
- Substitute \( x_1 = 1 \) back into (5):
\[ 1 + x_2 = 4 \implies x_2 = 3 \]
- Substitute \( x_2 = 3 \) into (1):
\[ 3 + x_3 = 8 \implies x_3 = 5 \]

4. **Solve the equations for y-coordinates**:
- From equations (2) and (6):
\[ y_2 + y_3 = 4 \quad (2) \]
\[ y_1 + y_2 = 4 \quad (6) \]
- Substituting \( y_2 = 4 - y_1 \) into (2):
\[ (4 - y_1) + y_3 = 4 \implies y_3 = y_1 \quad (8) \]
- Now substitute \( y_3 \) from (8) into (4):
\[ y_1 + y_1 = 6 \implies 2y_1 = 6 \implies y_1 = 3 \]
- Substitute \( y_1 = 3 \) back into (6):
\[ 3 + y_2 = 4 \implies y_2 = 1 \]
- Substitute \( y_2 = 1 \) into (2):
\[ 1 + y_3 = 4 \implies y_3 = 3 \]

5. **Coordinates of the vertices**:
- Now we have the coordinates of the vertices:
- \( A(1, 3) \)
- \( B(3, 1) \)
- \( C(5, 3) \)

6. **Calculate the centroid**:
- The formula for the centroid \( G \) of a triangle is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
- Substituting the coordinates:
\[ G\left(\frac{1 + 3 + 5}{3}, \frac{3 + 1 + 3}{3}\right) = G\left(\frac{9}{3}, \frac{7}{3}\right) = G(3, \frac{7}{3}) \]

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