This is a 45°-45°-90° triangle, which has specific properties. In a 45°-45°-90° triangle, the sides are in the ratio $1:1:\sqrt{2}$, where the hypotenuse is the longest side.

Given:

• The hypotenuse = $\sqrt{2}$

Step 1: Identify the sides $x$ and $y$:

Using the property of the triangle:

• The legs (both $x$ and $y$) are equal.
• To find $x$ or $y$, divide the hypotenuse by $\sqrt{2}$: $x = y = \frac{\text{hypotenuse}}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2}} = 1$

Thus: $x = 1 \quad \text{and} \quad y = 1$

Final Answer:

The correct values for $x$ and $y$ are not included in the given options. If there is a misinterpretation in labeling or formatting in the problem, please clarify for exact confirmation.

Would you like further clarification, details on the process, or exploration of another approach?

Related Questions:

1. How does the $1:1:\sqrt{2}$ ratio relate to other triangle properties?
2. Can this method be extended to other special triangles like 30°-60°-90° triangles?
3. What are practical examples where this concept is used in geometry?
4. How do we rationalize denominators in such cases?
5. How do transformations affect the dimensions of a 45°-45°-90° triangle?

Tip: Always verify that the hypotenuse and leg lengths satisfy the Pythagorean theorem for such triangles.