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,-4,16)`
B. `(1,-2,13)`
C. `(-2,4-2)`
D. `(1,-1,1)`

Answer

To find the coordinates of the fourth vertex of the tetrahedron given the coordinates of three vertices and the centroid, we can follow these steps:

### Step 1: Identify the known coordinates
The coordinates of the three vertices of the tetrahedron are:
- Vertex A: \( A(0, 0, 0) \)
- Vertex B: \( B(6, -5, -1) \)
- Vertex C: \( C(-4, 1, 3) \)

The coordinates of the centroid \( G \) are given as:
- \( G(1, -2, 5) \)

Let the coordinates of the fourth vertex \( D \) be \( D(x_4, y_4, z_4) \).

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

Substituting the known values into the centroid formula, we have:
\[ G_x = \frac{0 + 6 - 4 + x_4}{4} = 1 \]
\[ G_y = \frac{0 - 5 + 1 + y_4}{4} = -2 \]
\[ G_z = \frac{0 - 1 + 3 + z_4}{4} = 5 \]

### Step 3: Solve for \( x_4 \)
From the equation for \( G_x \):
\[ \frac{0 + 6 - 4 + x_4}{4} = 1 \]
\[ \frac{2 + x_4}{4} = 1 \]
Multiplying both sides by 4:
\[ 2 + x_4 = 4 \]
Subtracting 2 from both sides:
\[ x_4 = 2 \]

### Step 4: Solve for \( y_4 \)
From the equation for \( G_y \):
\[ \frac{0 - 5 + 1 + y_4}{4} = -2 \]
\[ \frac{-4 + y_4}{4} = -2 \]
Multiplying both sides by 4:
\[ -4 + y_4 = -8 \]
Adding 4 to both sides:
\[ y_4 = -4 \]

### Step 5: Solve for \( z_4 \)
From the equation for \( G_z \):
\[ \frac{0 - 1 + 3 + z_4}{4} = 5 \]
\[ \frac{2 + z_4}{4} = 5 \]
Multiplying both sides by 4:
\[ 2 + z_4 = 20 \]
Subtracting 2 from both sides:
\[ z_4 = 18 \]

### Step 6: Write the coordinates of the fourth vertex
Thus, the coordinates of the fourth vertex \( D \) are:
\[ D(2, -4, 18) \]

### Final Answer
The coordinates of the fourth vertex of the tetrahedron are \( (2, -4, 18) \).

---