To solve this problem, let's analyze the given information step-by-step:

1. We have a right triangle.
2. One of the acute angles is $60^\circ$.
3. The length of the opposite side to the $60^\circ$ angle is 8 cm.
4. We need to find the length of the hypotenuse.

In a right triangle, we can use trigonometric ratios. Specifically, the sine function of an angle in a right triangle is defined as:

$\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$

Step 1: Set up the equation

Here, $\theta = 60^\circ$, and the opposite side is 8 cm. Let $h$ be the length of the hypotenuse. Then:

$\sin(60^\circ) = \frac{8}{h}$

Step 2: Substitute $\sin(60^\circ)$

The value of $\sin(60^\circ)$ is $\frac{\sqrt{3}}{2}$. So we have:

$\frac{\sqrt{3}}{2} = \frac{8}{h}$

Step 3: Solve for $h$

Now, cross-multiply to find $h$:

$h = \frac{8 \times 2}{\sqrt{3}} = \frac{16}{\sqrt{3}}$

Step 4: Rationalize the denominator (optional)

To express this without a radical in the denominator, multiply both the numerator and the denominator by $\sqrt{3}$:

$h = \frac{16 \sqrt{3}}{3} \approx 9.24 \, \text{cm}$

Final Answer:

The length of the hypotenuse is approximately 9.24 cm.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. How would you calculate the length of the other leg in this triangle?
2. What if the given angle was $30^\circ$? How would that change the solution?
3. How can we use the cosine function to find the hypotenuse in a similar problem?
4. Why is rationalizing the denominator useful in trigonometric calculations?
5. How would the problem change if the known side was the hypotenuse instead of the opposite side?

Tip: Remember that in a 30-60-90 triangle, the ratio of the sides is always $1 : \sqrt{3} : 2$, which can simplify calculations in problems like this.