the answer is a=40√3/3 and x=40/3




Thank you for pointing that out! Let's correct the solution using the correct approach.

Given:
- The triangle is a 30°-60°-90° triangle.
- The side opposite the 60° angle is 20 cm (this side is labeled as \(x\)).

### Step 1: Relate the sides using trigonometric ratios

For a 30°-60°-90° triangle:
- The side opposite the 30° angle is half the hypotenuse.
- The side opposite the 60° angle (which is \(x\)) is \(\sqrt{3}/2\) times the hypotenuse.

### Step 2: Use the relationship for \(x\)

Since the side opposite the 60° angle is \(x\) and it is given as 20 cm:
\[
x = \frac{20}{\sqrt{3}} = \frac{20 \times \sqrt{3}}{3} \times \frac{1}{\sqrt{3}} = \frac{20 \times 1}{3} \times \sqrt{3} = \frac{20 \times 1}{3} \times \sqrt{3}
\]
Thus:
\[
x = \frac{20}{\sqrt{3}} \approx \frac{20}{1.732} \approx 11.55 \text{ cm}
]

The value of \(a\), the hypotenuse, is:
\[
x = \frac{20}{\sqrt{3}} = 40/\sqrt{3} = 20*sqrt3
]

Thus:

\( x = 20/sqrt{3}\) and \(a\) = 20\)

### Final Answer:
- \(x = 20/sqrt(3)\)
- \(a = 20\) cm

Would you like any additional details, or do you have any questions?

### Related Questions:
1. How do the sides of a 45°-45°-90° triangle relate to each other?
2. What is the general formula for the area of a right triangle?
3. How do you find the hypotenuse in a right triangle given the other two sides?
4. Can you use trigonometry to solve for unknown sides in any right triangle?
5. What is the Pythagorean theorem and how is it applied?

### Tip:
In right triangles, always confirm the side's relationship and property before calculation to ensure a better understanding.

Learn how to solve for side lengths in a 30-60-90 triangle with a detailed step-by-step solution. This problem covers trigonometric concepts and specific triangle properties, including the relationships in a 30-60-90 triangle. Suitable for high school students.

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