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To solve for \( y \) in the given right triangle:

### Step 1: Identify the triangle type
This is a right triangle with one of the angles being \( 45^\circ \). Therefore, it is a 45-45-90 triangle, where the two legs are equal in length.

### Step 2: Use the properties of a 45-45-90 triangle
In a 45-45-90 triangle, the ratio of the legs to the hypotenuse is:
\[
\text{Leg} : \text{Leg} : \text{Hypotenuse} = 1 : 1 : \sqrt{2}
\]
Let \( y \) be the length of each leg and \( x \) be the hypotenuse.

### Step 3: Use the given hypotenuse value
Here, the hypotenuse \( x = \frac{7\sqrt{2}}{2} \).

Using the ratio \( \text{Leg} : \text{Hypotenuse} = 1 : \sqrt{2} \), we can express \( y \) as:
\[
y = \frac{x}{\sqrt{2}}
\]

### Step 4: Substitute the value of \( x \)
\[
y = \frac{\frac{7\sqrt{2}}{2}}{\sqrt{2}} = \frac{7\sqrt{2}}{2\sqrt{2}}
\]

### Step 5: Simplify the expression
\[
y = \frac{7\sqrt{2}}{2\sqrt{2}} = \frac{7}{2}
\]

Thus, \( y = \frac{7}{2} \) or \( y = 3.5 \).

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Would you like more details on any of these steps, or do you have any questions? 

Here are some related questions:

1. How do you find the length of the hypotenuse in a 45-45-90 triangle if only the leg is known?
2. What is the significance of the \( \sqrt{2} \) ratio in a 45-45-90 triangle?
3. How would you verify the solution for \( y \) using the Pythagorean Theorem?
4. How would you solve for \( y \) if the hypotenuse were given in a different form?
5. What are the steps to rationalize a denominator, as seen in simplifying \( y \)?
6. Can you generalize the relationship between the sides of any 45-45-90 triangle?
7. How would the approach differ if the triangle was not a 45-45-90 triangle?
8. How can trigonometric identities be used to find the sides of a right triangle?

**Tip:** Remember, in a 45-45-90 triangle, the legs are always equal, and the hypotenuse is \( \sqrt{2} \) times the length of a leg.

Learn how to find the length of y in a 45-45-90 triangle using the Pythagorean Theorem and properties of 45-45-90 triangles. Step-by-step solution provided with clear explanations and calculations. Suitable for high school students.

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