Find the co- ordinates of the centroid of the tetrahedron whose vertices are (0,0,0) ,(a,0,0),(0,b,0) and (0,0,c)

Answer

To find the coordinates of the centroid of the tetrahedron with vertices at (0,0,0), (a,0,0), (0,b,0), and (0,0,c), we can follow these steps:

### Step 1: Identify the vertices of the tetrahedron
The vertices of the tetrahedron are given as:
- A = (0, 0, 0)
- B = (a, 0, 0)
- C = (0, b, 0)
- D = (0, 0, c)

### Step 2: Use the formula for the centroid of a tetrahedron
The centroid (G) of a tetrahedron with vertices \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), \( C(x_3, y_3, z_3) \), and \( D(x_4, y_4, z_4) \) is given by the formula:
\[ G = \left( \frac{x_1 + x_2 + x_3 + x_4}{4}, \frac{y_1 + y_2 + y_3 + y_4}{4}, \frac{z_1 + z_2 + z_3 + z_4}{4} \right) \]

### Step 3: Substitute the coordinates into the formula
Substituting the coordinates of the vertices into the formula:
- For the x-coordinate:
\[ x_G = \frac{0 + a + 0 + 0}{4} = \frac{a}{4} \]

- For the y-coordinate:
\[ y_G = \frac{0 + 0 + b + 0}{4} = \frac{b}{4} \]

- For the z-coordinate:
\[ z_G = \frac{0 + 0 + 0 + c}{4} = \frac{c}{4} \]

### Step 4: Write the coordinates of the centroid
Thus, the coordinates of the centroid G are:
\[ G = \left( \frac{a}{4}, \frac{b}{4}, \frac{c}{4} \right) \]

### Final Answer
The coordinates of the centroid of the tetrahedron are \( \left( \frac{a}{4}, \frac{b}{4}, \frac{c}{4} \right) \).

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