Given a right triangle with one vertex at the origin \((0, 0)\), another vertex on the unit circle, and the third vertex on the x-axis within the unit circle, we can denote the points as follows:
- The vertex at the origin: \(A = (0, 0)\).
- The vertex on the unit circle, which can be represented as \(B = (\cos(\theta), \sin(\theta))\) for some angle \(\theta\).
- The vertex on the x-axis inside the unit circle can be represented as \(C = (x, 0)\), where \(x\) is in the range \([0, 1]\) because it lies within the unit circle.
Since we have a right triangle with a right angle at point \(C\), we can find the lengths of the sides to determine the hypotenuse \(AB\).
The distance \(AB\) (the hypotenuse) can be calculated using the distance formula:
\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \] Substituting \(A\) and \(B\): \[ AB = \sqrt{(\cos(\theta) - 0)^2 + (\sin(\theta) - 0)^2} \] \[ AB = \sqrt{\cos^2(\theta) + \sin^2(\theta)} = \sqrt{1} = 1 \]
Thus, the hypotenuse \(AB\) of the triangle is always equal to \(1\), regardless of the specific angle \(\theta\), because \(B\) lies on the unit circle.
Therefore, the length of the hypotenuse is:
\[ \boxed{1} \]