What are the co-ordinates of the centroid of a triangle whose vertices are A (2, 5) , B(-4, 0) and C (5, 4) ?

Options

A. (-1, 3)
B. (1,3)
C. (1,-3)
D. (-1, -3)

Answer

To find the coordinates of the centroid of a triangle with vertices A(2, 5), B(-4, 0), and C(5, 4), we can use the formula for the centroid (G) of a triangle, which is given by:

\[ G\left(x, y\right) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices of the triangle.

### Step 1: Identify the coordinates of the vertices
- A(2, 5) → \(x_1 = 2\), \(y_1 = 5\)
- B(-4, 0) → \(x_2 = -4\), \(y_2 = 0\)
- C(5, 4) → \(x_3 = 5\), \(y_3 = 4\)

### Step 2: Calculate the x-coordinate of the centroid
Using the formula for the x-coordinate of the centroid:

\[ x = \frac{x_1 + x_2 + x_3}{3} = \frac{2 + (-4) + 5}{3} \]

Calculating the numerator:

\[ 2 - 4 + 5 = 3 \]

Now divide by 3:

\[ x = \frac{3}{3} = 1 \]

### Step 3: Calculate the y-coordinate of the centroid
Using the formula for the y-coordinate of the centroid:

\[ y = \frac{y_1 + y_2 + y_3}{3} = \frac{5 + 0 + 4}{3} \]

Calculating the numerator:

\[ 5 + 0 + 4 = 9 \]

Now divide by 3:

\[ y = \frac{9}{3} = 3 \]

### Step 4: Combine the x and y coordinates to find the centroid
Thus, the coordinates of the centroid G are:

\[ G(1, 3) \]

### Final Answer
The coordinates of the centroid of the triangle are \( (1, 3) \).

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