Prove that the point `(x, sqrt(1-x^(2)))` is at a distance of 1 unit from the origin .

Answer

To prove that the point \((x, \sqrt{1 - x^2})\) is at a distance of 1 unit from the origin \((0, 0)\), we will use the distance formula between two points in a Cartesian coordinate system.

### Step-by-Step Solution:

1. **Identify the Points**:
- The point we are considering is \(A(x, \sqrt{1 - x^2})\).
- The origin point is \(O(0, 0)\).

2. **Use the Distance Formula**:
The distance \(d\) between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by the formula:
\[ d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]
In our case, \(A\) is \((x, \sqrt{1 - x^2})\) and \(O\) is \((0, 0)\).

3. **Substitute the Points into the Formula**:
\[ d = \sqrt{(x - 0)^2 + (\sqrt{1 - x^2} - 0)^2} \]
This simplifies to:
\[ d = \sqrt{x^2 + (\sqrt{1 - x^2})^2} \]

4. **Simplify the Expression**:
The term \((\sqrt{1 - x^2})^2\) simplifies to \(1 - x^2\). Therefore, we have:
\[ d = \sqrt{x^2 + (1 - x^2)} \]

5. **Combine Like Terms**:
\[ d = \sqrt{x^2 + 1 - x^2} = \sqrt{1} \]

6. **Final Calculation**:
\[ d = 1 \]

### Conclusion:
We have shown that the distance from the point \((x, \sqrt{1 - x^2})\) to the origin \((0, 0)\) is indeed 1 unit.