Let’s solve for the exact values of \( a \) and \( x \) in the given right triangle, using the known properties of a 30°-60°-90° triangle.

### Step 1: Use the Known Ratios for a 30°-60°-90° Triangle
In a 30°-60°-90° triangle, the ratios of the sides are:
- Side opposite the 30° angle = \( \frac{1}{2} \times \) hypotenuse
- Side opposite the 60° angle = \( \frac{\sqrt{3}}{2} \times \) hypotenuse

### Step 2: Solve for the Hypotenuse \( a \)
We are given that the side opposite the 60° angle (which is adjacent to the 30° angle) is \( x = 20 \) cm.

Using the known ratio for the side opposite the 60° angle:
\[
x = \frac{\sqrt{3}}{2} \times a
\]
Substitute \( x = 20 \):
\[
20 = \frac{\sqrt{3}}{2} \times a
\]

Solve for \( a \):
\[
a = \frac{20 \times 2}{\sqrt{3}} = \frac{40}{\sqrt{3}}
\]

### Step 3: Rationalize the Denominator
To express \( a \) in its exact value:
\[
a = \frac{40}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{40\sqrt{3}}{3}
\]
Thus, the exact value of the hypotenuse \( a \) is:
\[
a = \frac{40\sqrt{3}}{3} \text{ cm}
\]

### Step 4: Solve for the Side Opposite the 30° Angle
Now, using the property of a 30°-60°-90° triangle:
\[
\text{Side opposite the 30° angle} = \frac{a}{2} = \frac{40\sqrt{3}}{3} \times \frac{1}{2} = \frac{20\sqrt{3}}{3} \text{ cm}
\]

### Final Exact Values:
- The hypotenuse \( a = \frac{40\sqrt{3}}{3} \) cm.
- The side opposite the 60° angle \( x = 20 \) cm (given).
- The side opposite the 30° angle is \( \frac{20\sqrt{3}}{3} \) cm.

Let me know if you'd like further clarifications!

### Relative Questions:
1. Can you explain how the side ratios in a 30°-60°-90° triangle are derived from the unit circle?
2. How do these side lengths change if the angle measurements were swapped?
3. How can we use these trigonometric relationships in real-world applications?
4. What is the relationship between the hypotenuse and the altitude in a 30°-60°-90° triangle?
5. How would the solution change if we were asked for decimal approximations instead of exact values?

### Tip:
In a 30°-60°-90° triangle, the ratio between the sides remains constant regardless of the triangle's size. This allows you to scale the triangle easily while maintaining the proportions.

Explore how to calculate the exact values of the hypotenuse and opposite sides in a 30°-60°-90° triangle. Step-by-step solution provided with detailed mathematical derivations and real-world application insights. Suitable for Grades 10-12.

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