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To solve for \( x \) and \( y \) in this triangle problem, we can use the properties of a 30-60-90 triangle.

### Properties of a 30-60-90 Triangle
In a 30-60-90 triangle:
- The side opposite the 30° angle (shorter leg) is half the hypotenuse.
- The side opposite the 60° angle (longer leg) is the shorter leg multiplied by \( \sqrt{3} \).

### Given Information
The hypotenuse of the triangle is 3, and we need to find \( x \) (the side opposite the 30° angle) and \( y \) (the side opposite the 60° angle).

1. **Finding \( x \):**
   - Since \( x \) is opposite the 30° angle, it is half of the hypotenuse.
   - \( x = \frac{3}{2} = 1.5 \).

2. **Finding \( y \):**
   - \( y \) is opposite the 60° angle, so it is \( x \) multiplied by \( \sqrt{3} \).
   - \( y = 1.5 \times \sqrt{3} \).

### Solution
Using the properties above, we find:
- \( x = 1.5 \)
- \( y = 1.5 \sqrt{3} \)

### Answer
The correct answer is:
**Option A**: \( x = 1.5 \), \( y = 1.5 \sqrt{3} \).

Would you like more details on this solution, or do you have any further questions?

1. How would the solution change if the hypotenuse were different?
2. Can you explain why the properties of a 30-60-90 triangle work this way?
3. How does this approach generalize to other special triangles, like 45-45-90?
4. What trigonometric ratios can we use instead of these properties?
5. What if we didn’t know the hypotenuse but knew \( x \) and wanted to find \( y \)?

**Tip:** Memorize the side ratios of special triangles like 30-60-90 and 45-45-90; they’re frequently used in geometry.

Find x and y in a 30-60-90 triangle with a hypotenuse of 3, where x is opposite the 30° angle and y is opposite the 60° angle.

Learn how to find the values of x and y in a 30-60-90 triangle with a hypotenuse of 3. This problem uses the properties of special right triangles, specifically the 30-60-90 triangle theorem. Suitable for students in Grades 8-10.

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