What are the coordinates of the centroid of the triangle having vertices as (a,b-c), (b,c-a) and (c,a-b)?

Options

A. (1,1)
B. `((a+b+c)/(3),0)`
C. (0,0)
D. `(0,(b)/(3))`

Answer

To find the coordinates of the centroid of a triangle with vertices at \( (a, b-c) \), \( (b, c-a) \), and \( (c, a-b) \), we will use the formula for the centroid of a triangle given by the coordinates of its vertices.

### Step 1: Identify the vertices
The vertices of the triangle are:
- \( A = (a, b-c) \)
- \( B = (b, c-a) \)
- \( C = (c, a-b) \)

### Step 2: Write down the formula for the centroid
The coordinates of the centroid \( (G_x, G_y) \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} \]

### Step 3: Calculate the x-coordinate of the centroid
Using the x-coordinates of the vertices:
\[ G_x = \frac{a + b + c}{3} \]

### Step 4: Calculate the y-coordinate of the centroid
Using the y-coordinates of the vertices:
\[ G_y = \frac{(b-c) + (c-a) + (a-b)}{3} \]
Now, simplify \( G_y \):
\[ G_y = \frac{b - c + c - a + a - b}{3} \]
Notice that \( b - b + c - c + a - a = 0 \):
\[ G_y = \frac{0}{3} = 0 \]

### Step 5: Combine the results
Thus, the coordinates of the centroid \( G \) are:
\[ G = \left( \frac{a + b + c}{3}, 0 \right) \]

### Final Answer
The coordinates of the centroid of the triangle are:
\[ \left( \frac{a + b + c}{3}, 0 \right) \]

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