The coordinates of the point which is equidistant from the points O(0,0,0) A (a,0,0), B(0,b,0) and C(0,0,c)

Options

A. (a,b,c)
B. `((a)/(2),(b)/(2),(c)/(2))`
C. `((a)/(3),(b)/(3),(c)/(3))`
D. `(-(a)/(2),-(b)/(2),-(c)/(2))`

Answer

To find the coordinates of the point that is equidistant from the points O(0,0,0), A(a,0,0), B(0,b,0), and C(0,0,c), we can follow these steps:

### Step 1: Define the Point
Let the coordinates of the point P be (x, y, z).

### Step 2: Calculate the Distance from Point O
The distance from point O(0,0,0) to point P(x,y,z) is given by the distance formula:
\[ PO = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{x^2 + y^2 + z^2} \]

### Step 3: Calculate the Distance from Point A
The distance from point A(a,0,0) to point P(x,y,z) is:
\[ PA = \sqrt{(x - a)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{(x - a)^2 + y^2 + z^2} \]

### Step 4: Set the Distances Equal
Since point P is equidistant from points O and A, we set the distances equal:
\[ \sqrt{x^2 + y^2 + z^2} = \sqrt{(x - a)^2 + y^2 + z^2} \]

### Step 5: Square Both Sides
Squaring both sides to eliminate the square roots gives:
\[ x^2 + y^2 + z^2 = (x - a)^2 + y^2 + z^2 \]

### Step 6: Simplify the Equation
Expanding the right-hand side:
\[ x^2 + y^2 + z^2 = x^2 - 2ax + a^2 + y^2 + z^2 \]
Now, cancel \(y^2\) and \(z^2\) from both sides:
\[ x^2 = x^2 - 2ax + a^2 \]
This simplifies to:
\[ 0 = -2ax + a^2 \]

### Step 7: Solve for x
Rearranging gives:
\[ 2ax = a^2 \implies x = \frac{a}{2} \]

### Step 8: Repeat for Points B and C
Now, we repeat the process for points B(0,b,0) and C(0,0,c).

1. **For Point B**:
\[ PB = \sqrt{(x - 0)^2 + (y - b)^2 + (z - 0)^2} \]
Setting \(PO = PB\):
\[ \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + (y - b)^2 + z^2} \]
Squaring both sides and simplifying gives:
\[ 0 = -2by + b^2 \implies y = \frac{b}{2} \]

2. **For Point C**:
\[ PC = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - c)^2} \]
Setting \(PO = PC\):
\[ \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + y^2 + (z - c)^2} \]
Squaring both sides and simplifying gives:
\[ 0 = -2cz + c^2 \implies z = \frac{c}{2} \]

### Final Coordinates
Thus, the coordinates of the point P that is equidistant from points O, A, B, and C are:
\[ \left(\frac{a}{2}, \frac{b}{2}, \frac{c}{2}\right) \]