What are the coordinates of the centroid of the triangle having vertices as (a, b), (b,b-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 the triangle with vertices at (a, b), (b, b-a), and (c, a-b), we can follow these steps:

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

### Step 2: Use the centroid formula
The formula for the coordinates of the centroid (G) of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) 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: Substitute the coordinates into the formula
Here, we will substitute the coordinates of the vertices into the centroid formula.

For the x-coordinate of the centroid (G_x):
\[ G_x = \frac{a + b + c}{3} \]

For the y-coordinate of the centroid (G_y):
\[ G_y = \frac{b + (b-a) + (a-b)}{3} \]

### Step 4: Simplify the y-coordinate
Now, let's simplify the expression for G_y:
\[ G_y = \frac{b + b - a + a - b}{3} \]
\[ G_y = \frac{b + b - b + a - a}{3} = \frac{b}{3} \]

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

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