Directions - For the special right triangle, find the values of x and y (leave your answer in simplified form = 32553 2​ ​  )x =   y =

To solve for the values of $x$ and $y$ in a special right triangle, we first need to clarify which type of special right triangle is being referred to, as there are two main types: the 45-45-90 triangle and the 30-60-90 triangle.

1. Break Down the Problem

• Let's assume a 45-45-90 triangle where the sides opposite the 45-degree angles are equal, and the side opposite the 90-degree angle is $x\sqrt{2}$.
• For a 30-60-90 triangle, the ratio of the sides is $1:\sqrt{3}:2$.

2. Relevant Concepts

• For a 45-45-90 triangle:

  • If the legs are $a$, then the hypotenuse $h$ is given by: $h = a\sqrt{2}$

• For a 30-60-90 triangle:

  • If the shorter leg (opposite the 30-degree angle) is $a$, the longer leg (opposite the 60-degree angle) is $a\sqrt{3}$, and the hypotenuse is $2a$.

3. Analysis and Detail

• Without additional context or specific values, I'll proceed with the 45-45-90 triangle:

  • Let $x$ be one of the legs of the triangle. Then $y$ (the other leg) will also be $x$.
  • The relation is therefore $y = x$ and the hypotenuse is $x\sqrt{2}$.

• If the problem presents a specific length for one side, use that to solve for $x$ and $y$.

4. Verify and Summarize

• If the triangle in question is a 45-45-90 triangle and no specific lengths are provided, we would confirm that both legs are equal and correspond to $x$ and $y$, hence $x = y$.

Final Answer

If $x$ and $y$ represent the legs of a 45-45-90 triangle, then: $x = y$ If specific lengths were expected, please provide those for exact calculations.