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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?

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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.

En un triángulo rectángulo, si uno de sus ángulos agudos mide 60°, ¿cuánto mide la hipotenusa si el cateto opuesto mide 8 cm?

Learn how to find the hypotenuse of a right triangle where one acute angle is 60° and the opposite side is 8 cm. This problem utilizes trigonometric ratios, specifically the sine function, to solve for the hypotenuse. Suitable for students in Grades 9-10.

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