https://file.aiquickdraw.com/mathbot/file/1725938169441aamepdt5.jpg

The image contains two right triangles, and we need to solve for the unknown side lengths \( x \) and \( y \) for both triangles. 

### First Triangle:
This is a **45°-45°-90° triangle**, also known as an isosceles right triangle.

- In a 45°-45°-90° triangle, the legs (which are the two shorter sides) are equal, and the hypotenuse is related to the leg by a factor of \( \sqrt{2} \).
- Given that the hypotenuse is 1, we can use the ratio \( 1 : 1 : \sqrt{2} \), meaning the legs are \( \frac{1}{\sqrt{2}} \).
  
To simplify:
\[
x = y = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\]

Thus, for the first triangle:
- \( x = \frac{\sqrt{2}}{2} \)
- \( y = \frac{\sqrt{2}}{2} \)

### Second Triangle:
This is a **30°-60°-90° triangle**.

- In a 30°-60°-90° triangle, the ratio of the sides is \( 1 : \sqrt{3} : 2 \), where:
  - The shortest side (opposite the 30° angle) is given as 1.
  - The hypotenuse is twice the shortest side.
  - The side opposite the 60° angle is \( \sqrt{3} \) times the shortest side.
  
Using the ratio:
- The hypotenuse \( y = 2 \times 1 = 2 \)
- The side opposite the 60° angle \( x = \sqrt{3} \times 1 = \sqrt{3} \)

Thus, for the second triangle:
- \( x = \sqrt{3} \)
- \( y = 2 \)

### Final Answers:
- First Triangle: \( x = \frac{\sqrt{2}}{2}, y = \frac{\sqrt{2}}{2} \)
- Second Triangle: \( x = \sqrt{3}, y = 2 \)

Would you like further clarification or any additional details?

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Here are five related questions for practice:
1. What are the side lengths of a 45°-45°-90° triangle if the hypotenuse is \( \sqrt{2} \)?
2. In a 30°-60°-90° triangle, if the side opposite 30° is \( \frac{1}{2} \), what are the lengths of the other sides?
3. What is the area of the 45°-45°-90° triangle from the first problem?
4. How do you derive the side ratios for special right triangles?
5. How would the side lengths change if the hypotenuse of the second triangle was 4?

**Tip:** Always remember that 45°-45°-90° and 30°-60°-90° triangles have fixed side ratios that can help you find any unknown side when given one.

Enter the side lengths for each triangle shown below. Write your answers as SIMPLIFIED integers, radicals, or fractions, NOT decimal approximations. It is not necessary to rationalize your answers.

Learn how to find the side lengths of special right triangles, specifically 45°-45°-90° and 30°-60°-90° triangles. This problem involves using specific side ratios and the Pythagorean Theorem. Ideal for students in Grades 8-10.

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