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find teh act values of a and x in this diagram

This image shows a right-angled triangle with angles of 30° and 60°, and sides labeled \(a\) and \(x\). The base is given as 20 cm. Let's find the actual values of \(a\) and \(x\).

### Step 1: Understand the triangle
This is a special right triangle, known as a 30°-60°-90° triangle. The properties of such triangles are:
- The side opposite the 30° angle is half the hypotenuse.
- The side opposite the 60° angle is \(\sqrt{3}/2\) times the hypotenuse.

Given:
- The angle opposite side \(x\) is 60°.
- The angle opposite side \(a\) is 30°.
- The base of 20 cm is the side opposite the 60° angle.

### Step 2: Use the properties of the triangle
1. **Side \(x\)**:
   \[
   x = \frac{20}{\sqrt{3}} \approx \frac{20}{1.732} \approx 11.55 \text{ cm}
   \]

2. **Side \(a\)**:
   - Since \(a\) is opposite the 30° angle, and using the property that the side opposite the 30° angle is half of the hypotenuse (which is \(20\) cm):
   \[
   a = 2 \times 10 = 20 \text{ cm}
   \]

### Final Answer:
- \(x \approx 11.55\) cm
- \(a = 20\) 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:
Remember that in a 30°-60°-90° triangle, the hypotenuse is always twice the length of the shorter leg.

Learn how to find the values of sides a and x in a 30°-60°-90° triangle with a detailed step-by-step solution. This problem involves understanding the properties of special right triangles and applying trigonometric ratios. Suitable for students in Grades 10-12.

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