`A (1, 4), B (3, 2) and C (7,5)`  are vertices of a triangle ABC. Find :
  the co-ordinates of the centroid of triangle ABC.

Answer

To find the coordinates of the centroid of triangle ABC with vertices A(1, 4), B(3, 2), and C(7, 5), we will follow these steps:

### Step 1: Identify the coordinates of the vertices
The coordinates of the vertices are:
- A (x₁, y₁) = (1, 4)
- B (x₂, y₂) = (3, 2)
- C (x₃, y₃) = (7, 5)

### Step 2: Use the formula for the centroid
The formula for the centroid (G) of a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by:
\[ G\left(x, y\right) = \left(\frac{x₁ + x₂ + x₃}{3}, \frac{y₁ + y₂ + y₃}{3}\right) \]

### Step 3: Calculate the x-coordinate of the centroid
Substituting the x-coordinates of the vertices into the formula:
\[ x = \frac{x₁ + x₂ + x₃}{3} = \frac{1 + 3 + 7}{3} = \frac{11}{3} \]

### Step 4: Calculate the y-coordinate of the centroid
Now substituting the y-coordinates of the vertices into the formula:
\[ y = \frac{y₁ + y₂ + y₃}{3} = \frac{4 + 2 + 5}{3} = \frac{11}{3} \]

### Step 5: Write the coordinates of the centroid
Thus, the coordinates of the centroid G are:
\[ G\left(\frac{11}{3}, \frac{11}{3}\right) \]

### Final Answer:
The coordinates of the centroid of triangle ABC are \(\left(\frac{11}{3}, \frac{11}{3}\right)\).

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