The coordinates of the mid-points of the sides of a triangle are (4, 2), (3, 3) and (2, 2). What will be the coordinates of the centroid of the triangle?

Options

A. `(3,(7)/(3))`
B. `(-3,(-7)/(3))`
C. `(3,(-7)/(3))`
D. `(3,(7)/(3))`

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:
- Midpoint of AB: (4, 2)
- Midpoint of BC: (3, 3)
- Midpoint of AC: (2, 2)

### Step 2: Set up equations for the vertices
Let the vertices of the triangle be A(x1, y1), B(x2, y2), and C(x3, y3). The coordinates of the midpoints can be expressed in terms of the vertices:
1. For midpoint AB:
\[ \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right) = (4, 2) \]
This gives us two equations:
\[ x1 + x2 = 8 \quad (1) \]
\[ y1 + y2 = 4 \quad (2) \]

2. For midpoint BC:
\[ \left(\frac{x2 + x3}{2}, \frac{y2 + y3}{2}\right) = (3, 3) \]
This gives us:
\[ x2 + x3 = 6 \quad (3) \]
\[ y2 + y3 = 6 \quad (4) \]

3. For midpoint AC:
\[ \left(\frac{x3 + x1}{2}, \frac{y3 + y1}{2}\right) = (2, 2) \]
This gives us:
\[ x3 + x1 = 4 \quad (5) \]
\[ y3 + y1 = 4 \quad (6) \]

### Step 3: Solve the equations for x-coordinates
Now, we have three equations for the x-coordinates:
1. \(x1 + x2 = 8\) (1)
2. \(x2 + x3 = 6\) (3)
3. \(x3 + x1 = 4\) (5)

We can add all three equations:
\[ (x1 + x2) + (x2 + x3) + (x3 + x1) = 8 + 6 + 4 \]
This simplifies to:
\[ 2x1 + 2x2 + 2x3 = 18 \]
Dividing by 2:
\[ x1 + x2 + x3 = 9 \quad (7) \]

### Step 4: Solve the equations for y-coordinates
Now, we have three equations for the y-coordinates:
1. \(y1 + y2 = 4\) (2)
2. \(y2 + y3 = 6\) (4)
3. \(y3 + y1 = 4\) (6)

Adding these equations:
\[ (y1 + y2) + (y2 + y3) + (y3 + y1) = 4 + 6 + 4 \]
This simplifies to:
\[ 2y1 + 2y2 + 2y3 = 14 \]
Dividing by 2:
\[ y1 + y2 + y3 = 7 \quad (8) \]

### Step 5: Calculate the centroid
The coordinates of the centroid (G) of the triangle can be found using the formula:
\[ G\left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]
Substituting the values from equations (7) and (8):
\[ G\left(\frac{9}{3}, \frac{7}{3}\right) = (3, \frac{7}{3}) \]

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