A tetrahedron is three dimensional figure bounded by four non coplanar triangular plane. So a tetrahedron has points A,B,C,D as its vertices, which have coordinates `(x_(1),y_(1),z_(1)) (x_(2), y_(2), z_(2)) , (x _(3), y_(3) , z_(3)) and (x _(4), y _(4), z _(4))` respectively in a rectangular three –dimensional space. Then the coordinates of its centroid are
`((x_(1)+ x_(2) + x _(3) + x_(4))/(4) , (y _(1) + y _(2) + y_(3) + y _(4))/(4), (z_(1) + z_(2) + z_(3)+ z_(4))/(4)).`
The circumcentre of the tetrahedron is the centre of a sphere passing through its vertices. So, the circumcentre is a point equidistant from each of the vertices of tetrahedron.
Let tetrahedron has three of its vertices represented by the points
`(0,0,0) ,(6,-5,-1) and (-4,1,3)` and its centroid lies at the point `(1,-2,5).` Now answer the following questions
The coordinate of the fourth vertex of the tetrahedron is :

Options

A. `(2,-4,18)`
B. `(1,-2,13)`
C. `(-2,4-2)`
D. `(1,-1,1)`

Answer

To find the coordinates of the fourth vertex of the tetrahedron, we will use the formula for the centroid of a tetrahedron and the given coordinates of the other three vertices.

### Step 1: Identify the given vertices and centroid
The vertices of the tetrahedron are:
- A (0, 0, 0)
- B (6, -5, -1)
- C (-4, 1, 3)

The centroid (G) is given as:
- G (1, -2, 5)

### Step 2: Set up the centroid formula
The formula for the centroid of a tetrahedron with vertices A, B, C, and D is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3 + x_4}{4}, \frac{y_1 + y_2 + y_3 + y_4}{4}, \frac{z_1 + z_2 + z_3 + z_4}{4} \right) \]
Where:
- \( (x_1, y_1, z_1) = (0, 0, 0) \)
- \( (x_2, y_2, z_2) = (6, -5, -1) \)
- \( (x_3, y_3, z_3) = (-4, 1, 3) \)
- \( (x_4, y_4, z_4) \) is the unknown vertex D.

### Step 3: Write the equations for the coordinates
Using the centroid formula, we can write three equations based on the x, y, and z coordinates:

1. For the x-coordinate:
\[ \frac{0 + 6 - 4 + x_4}{4} = 1 \]

2. For the y-coordinate:
\[ \frac{0 - 5 + 1 + y_4}{4} = -2 \]

3. For the z-coordinate:
\[ \frac{0 - 1 + 3 + z_4}{4} = 5 \]

### Step 4: Solve for \( x_4 \)
From the first equation:
\[ \frac{2 + x_4}{4} = 1 \]
Multiplying both sides by 4:
\[ 2 + x_4 = 4 \]
Subtracting 2 from both sides:
\[ x_4 = 2 \]

### Step 5: Solve for \( y_4 \)
From the second equation:
\[ \frac{-4 + y_4}{4} = -2 \]
Multiplying both sides by 4:
\[ -4 + y_4 = -8 \]
Adding 4 to both sides:
\[ y_4 = -4 \]

### Step 6: Solve for \( z_4 \)
From the third equation:
\[ \frac{2 + z_4}{4} = 5 \]
Multiplying both sides by 4:
\[ 2 + z_4 = 20 \]
Subtracting 2 from both sides:
\[ z_4 = 18 \]

### Step 7: Write the coordinates of the fourth vertex
Thus, the coordinates of the fourth vertex D are:
\[ D (x_4, y_4, z_4) = (2, -4, 18) \]

### Final Answer:
The coordinate of the fourth vertex of the tetrahedron is \( (2, -4, 18) \).

---