The coordinates of the point equidistant from the points A(0, 0, 0), B(4, 0, 0), C(0, 6, 0) and D(0, 0, 8) is

Options

A. (3, 5, 1)
B. (2, 6, 8)
C. (-2, 3, 4)
D. (2, 3, 4)

Answer

To find the coordinates of the point that is equidistant from the points A(0, 0, 0), B(4, 0, 0), C(0, 6, 0), and D(0, 0, 8), we can follow these steps:

### Step 1: Define the point
Let the coordinates of the point we need to find be \( P(x, y, z) \).

### Step 2: Set up the distance equations
Since point \( P \) is equidistant from points A, B, C, and D, we can set up the following equations based on the distance formula:

1. Distance from \( P \) to \( A \):
\[ PA = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{x^2 + y^2 + z^2} \]

2. Distance from \( P \) to \( B \):
\[ PB = \sqrt{(x - 4)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{(x - 4)^2 + y^2 + z^2} \]

3. Distance from \( P \) to \( C \):
\[ PC = \sqrt{(x - 0)^2 + (y - 6)^2 + (z - 0)^2} = \sqrt{x^2 + (y - 6)^2 + z^2} \]

4. Distance from \( P \) to \( D \):
\[ PD = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - 8)^2} = \sqrt{x^2 + y^2 + (z - 8)^2} \]

### Step 3: Equate distances
Since \( PA = PB \), we can square both sides to eliminate the square root:
\[ x^2 + y^2 + z^2 = (x - 4)^2 + y^2 + z^2 \]
This simplifies to:
\[ x^2 = (x - 4)^2 \]
Expanding the right side:
\[ x^2 = x^2 - 8x + 16 \]
Cancelling \( x^2 \) from both sides gives:
\[ 0 = -8x + 16 \implies 8x = 16 \implies x = 2 \]

### Step 4: Equate \( PA \) and \( PC \)
Now equate \( PA \) and \( PC \):
\[ x^2 + y^2 + z^2 = x^2 + (y - 6)^2 + z^2 \]
This simplifies to:
\[ y^2 = (y - 6)^2 \]
Expanding the right side:
\[ y^2 = y^2 - 12y + 36 \]
Cancelling \( y^2 \) gives:
\[ 0 = -12y + 36 \implies 12y = 36 \implies y = 3 \]

### Step 5: Equate \( PA \) and \( PD \)
Now equate \( PA \) and \( PD \):
\[ x^2 + y^2 + z^2 = x^2 + y^2 + (z - 8)^2 \]
This simplifies to:
\[ z^2 = (z - 8)^2 \]
Expanding the right side:
\[ z^2 = z^2 - 16z + 64 \]
Cancelling \( z^2 \) gives:
\[ 0 = -16z + 64 \implies 16z = 64 \implies z = 4 \]

### Final Coordinates
Thus, the coordinates of the point \( P \) that is equidistant from points A, B, C, and D are:
\[ P(2, 3, 4) \]