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

Options

A. (3, 7/3)
B. (3, 3)
C. (4, 3)
D. (3, 4)

Answer

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

### Step 1: Set Up the Midpoint Equations
Given the midpoints of the sides of the triangle:
- Midpoint of AB: \( D(4, 2) \)
- Midpoint of BC: \( E(3, 3) \)
- Midpoint of AC: \( F(2, 2) \)

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 set up the following equations:
1. For midpoint \( D \):
\[ \frac{x_1 + x_2}{2} = 4 \quad \text{and} \quad \frac{y_1 + y_2}{2} = 2 \]
This gives us:
\[ x_1 + x_2 = 8 \quad \text{(Equation 1)} \]
\[ y_1 + y_2 = 4 \quad \text{(Equation 2)} \]

2. For midpoint \( E \):
\[ \frac{x_2 + x_3}{2} = 3 \quad \text{and} \quad \frac{y_2 + y_3}{2} = 3 \]
This gives us:
\[ x_2 + x_3 = 6 \quad \text{(Equation 3)} \]
\[ y_2 + y_3 = 6 \quad \text{(Equation 4)} \]

3. For midpoint \( F \):
\[ \frac{x_1 + x_3}{2} = 2 \quad \text{and} \quad \frac{y_1 + y_3}{2} = 2 \]
This gives us:
\[ x_1 + x_3 = 4 \quad \text{(Equation 5)} \]
\[ y_1 + y_3 = 4 \quad \text{(Equation 6)} \]

### Step 2: Solve the System of Equations
Now, we have a system of equations to solve for \( x_1, x_2, x_3, y_1, y_2, y_3 \).

**From Equations 1, 3, and 5:**
1. From Equation 1: \( x_1 + x_2 = 8 \)
2. From Equation 3: \( x_2 + x_3 = 6 \)
3. From Equation 5: \( x_1 + x_3 = 4 \)

We can solve these equations step by step:
- From Equation 1, express \( x_2 \):
\[ x_2 = 8 - x_1 \quad \text{(Substituting into Equation 3)} \]
\[ (8 - x_1) + x_3 = 6 \implies x_3 = 6 - 8 + x_1 = x_1 - 2 \quad \text{(Substituting into Equation 5)} \]
\[ x_1 + (x_1 - 2) = 4 \implies 2x_1 - 2 = 4 \implies 2x_1 = 6 \implies x_1 = 3 \]
Now substituting \( x_1 = 3 \) back:
\[ x_2 = 8 - 3 = 5 \]
\[ x_3 = 3 - 2 = 1 \]

**Now for the y-coordinates using Equations 2, 4, and 6:**
1. From Equation 2: \( y_1 + y_2 = 4 \)
2. From Equation 4: \( y_2 + y_3 = 6 \)
3. From Equation 6: \( y_1 + y_3 = 4 \)

Following the same steps:
- From Equation 2, express \( y_2 \):
\[ y_2 = 4 - y_1 \quad \text{(Substituting into Equation 4)} \]
\[ (4 - y_1) + y_3 = 6 \implies y_3 = 6 - 4 + y_1 = y_1 + 2 \quad \text{(Substituting into Equation 6)} \]
\[ y_1 + (y_1 + 2) = 4 \implies 2y_1 + 2 = 4 \implies 2y_1 = 2 \implies y_1 = 1 \]
Now substituting \( y_1 = 1 \) back:
\[ y_2 = 4 - 1 = 3 \]
\[ y_3 = 1 + 2 = 3 \]

### Step 3: Calculate the Centroid
The coordinates of the vertices are:
- \( A(3, 1) \)
- \( B(5, 3) \)
- \( C(1, 3) \)

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 values:
\[ G\left(\frac{3 + 5 + 1}{3}, \frac{1 + 3 + 3}{3}\right) = G\left(\frac{9}{3}, \frac{7}{3}\right) = G(3, \frac{7}{3}) \]

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

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