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find the values of a and x in the diagram

To find the values of \(a\) and \(x\) in the given right triangle, we can use trigonometric ratios and the properties of special triangles.

### Step 1: Identify the Special Triangle
This triangle is a 30°-60°-90° triangle, which has known ratio properties:
- The side opposite the 30° angle is half the length of the hypotenuse.
- The side opposite the 60° angle is \(\sqrt{3}\) times the length of the side opposite the 30° angle.

### Step 2: Use the Properties of the 30°-60°-90° Triangle
Given:
- The side adjacent to the 30° angle (which is the side opposite the 60° angle) is 20 cm.

Let’s call:
- \( x \) the side opposite the 60° angle (which is adjacent to the 30° angle).
- \( a \) the hypotenuse.

From the properties of the 30°-60°-90° triangle, we know that:
- \( x = 20 \text{ cm} \) (given).
- The hypotenuse \( a \) is related to the side opposite the 30° angle (half of the hypotenuse).

Thus:
- The side opposite the 30° angle = \( \frac{a}{2} \).
- The side opposite the 60° angle = \( x = 20 \).

### Step 3: Solve for \( a \)
We can use the ratio for the side opposite the 60° angle in terms of the hypotenuse \( a \):
\[
x = \frac{\sqrt{3}}{2} \cdot a
\]
Substitute \( x = 20 \) cm into the equation:
\[
20 = \frac{\sqrt{3}}{2} \cdot a
\]
Solve for \( a \):
\[
a = \frac{20 \times 2}{\sqrt{3}} = \frac{40}{\sqrt{3}}
\]
Rationalizing the denominator:
\[
a = \frac{40 \sqrt{3}}{3} \approx 23.09 \text{ cm}
\]

### Step 4: Verify with 30°-60°-90° Triangle Rules
Now that we have \( a \approx 23.09 \text{ cm} \), we can verify this by checking the other side (which should be half of the hypotenuse):
\[
\text{Side opposite 30°} = \frac{a}{2} = \frac{23.09}{2} \approx 11.55 \text{ cm}
\]

Everything aligns with the properties of the triangle.

### Final Values:
- \( x = 20 \text{ cm} \)
- \( a \approx 23.09 \text{ cm} \)

Would you like more details or have any questions?

### Relative Questions:
1. How would you use trigonometry to verify the values of \( x \) and \( a \)?
2. What are the sine and cosine values for 30° and 60°, and how are they derived from this triangle?
3. How can we generalize the solution for any 30°-60°-90° triangle?
4. What if the length of the side opposite the 60° angle was changed—how would that affect \( a \) and the other sides?
5. How does the Pythagorean Theorem apply to this right triangle?

### Tip:
For 30°-60°-90° triangles, always remember that the ratio of the sides is 1 : \(\sqrt{3}\) : 2, which can simplify calculations greatly.

Learn how to find the values of a and x in a 30°-60°-90° triangle using trigonometric ratios and special triangle properties. Detailed step-by-step solution included.

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