In a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the lengths of the sides are in a specific ratio: the side opposite the \(30^\circ\) angle is \(x\), the side opposite the \(60^\circ\) angle is \(x\sqrt{3}\), and the hypotenuse is \(2x\).
If this triangle is drawn inside the unit circle, we know that the radius of the unit circle is \(1\). In this case, we can consider the hypotenuse of the triangle as the radius of the circle because the triangle’s vertices must lie on the circle.
Thus, the hypotenuse of the triangle, which is equal to the radius of the unit circle, is:
\[ \text{Length of the hypotenuse} = 1 \]
Hence, in a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle inscribed in a unit circle, the hypotenuse is of length \(1\).
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