If the middle points of the sides of a triangle are `(1,1),(2,-3)` and `(3,2)` then the centroid of the triangle is (i) `(-2,0)` (ii) `(0,2)` (iii) `(3,2)` (iv) `(2,0)`

Options

A. `(-2,0)`
B. `(0,2)`
C. `(3,2)`
D. `(2,0)`

Answer

To find the centroid of the 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:
- \( A(1, 1) \)
- \( B(2, -3) \)
- \( C(3, 2) \)

### Step 2: Use the centroid formula
The formula for the centroid \( G(x, y) \) of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by:
\[ G(x, y) = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

### Step 3: Substitute the coordinates into the formula
Here, we will substitute the coordinates of the midpoints into the centroid formula:
- \( x_1 = 1, y_1 = 1 \)
- \( x_2 = 2, y_2 = -3 \)
- \( x_3 = 3, y_3 = 2 \)

Now, we calculate:
\[ G(x, y) = \left( \frac{1 + 2 + 3}{3}, \frac{1 + (-3) + 2}{3} \right) \]

### Step 4: Calculate the x-coordinate of the centroid
Calculating the x-coordinate:
\[ G_x = \frac{1 + 2 + 3}{3} = \frac{6}{3} = 2 \]

### Step 5: Calculate the y-coordinate of the centroid
Calculating the y-coordinate:
\[ G_y = \frac{1 - 3 + 2}{3} = \frac{0}{3} = 0 \]

### Step 6: Write the final coordinates of the centroid
Thus, the coordinates of the centroid \( G \) are:
\[ G(2, 0) \]

### Step 7: Identify the correct option
From the options given:
(i) \((-2, 0)\)
(ii) \((0, 2)\)
(iii) \((3, 2)\)
(iv) \((2, 0)\)

The correct answer is option (iv) \((2, 0)\).

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