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

Options

A. a. `(2, 2/3)`
B. b. `(7/2 , 5)`
C. c. `(2sqrt3 , 5)`
D. d. none of these

Answer

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

### Step 1: Understand the Midpoint Formula
The coordinates of the midpoints of the sides of a triangle are given as (1, 1), (2, -3), and (3, 4). Let's denote these midpoints as:
- M1 = (1, 1)
- M2 = (2, -3)
- M3 = (3, 4)

### Step 2: Set Up the Equations
The midpoints of the sides of a triangle can be expressed in terms of the vertices of the triangle. If we denote the vertices of the triangle as A(x1, y1), B(x2, y2), and C(x3, y3), the midpoints can be expressed as:
- M1 = ((x1 + x2)/2, (y1 + y2)/2)
- M2 = ((x2 + x3)/2, (y2 + y3)/2)
- M3 = ((x1 + x3)/2, (y1 + y3)/2)

From this, we can derive the following equations based on the midpoints provided:

1. From M1:
\[ \frac{x1 + x2}{2} = 1 \quad \text{and} \quad \frac{y1 + y2}{2} = 1 \]
This simplifies to:
\[ x1 + x2 = 2 \quad \text{(Equation 1)} \]
\[ y1 + y2 = 2 \quad \text{(Equation 2)} \]

2. From M2:
\[ \frac{x2 + x3}{2} = 2 \quad \text{and} \quad \frac{y2 + y3}{2} = -3 \]
This simplifies to:
\[ x2 + x3 = 4 \quad \text{(Equation 3)} \]
\[ y2 + y3 = -6 \quad \text{(Equation 4)} \]

3. From M3:
\[ \frac{x1 + x3}{2} = 3 \quad \text{and} \quad \frac{y1 + y3}{2} = 4 \]
This simplifies to:
\[ x1 + x3 = 6 \quad \text{(Equation 5)} \]
\[ y1 + y3 = 8 \quad \text{(Equation 6)} \]

### Step 3: Solve the System of Equations
Now, we have a system of equations to solve for \(x1\), \(x2\), and \(x3\) as well as \(y1\), \(y2\), and \(y3\).

**For x-coordinates:**
Adding Equation 1, Equation 3, and Equation 5:
\[ (x1 + x2) + (x2 + x3) + (x1 + x3) = 2 + 4 + 6 \]
This simplifies to:
\[ 2x1 + 2x2 + 2x3 = 12 \quad \Rightarrow \quad x1 + x2 + x3 = 6 \quad \text{(Equation 7)} \]

**For y-coordinates:**
Adding Equation 2, Equation 4, and Equation 6:
\[ (y1 + y2) + (y2 + y3) + (y1 + y3) = 2 - 6 + 8 \]
This simplifies to:
\[ 2y1 + 2y2 + 2y3 = 4 \quad \Rightarrow \quad y1 + y2 + y3 = 2 \quad \text{(Equation 8)} \]

### Step 4: Calculate the Centroid
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:
\[ G\left(\frac{6}{3}, \frac{2}{3}\right) = G(2, \frac{2}{3}) \]

### Final Answer
The centroid of the triangle is \( G(2, \frac{2}{3}) \).