https://file.aiquickdraw.com/mathbot/file/1725911560845p4lit1zo.jpg

This is a right triangle problem where you need to determine the value of \( y \).

From the image, we can see that the diagram is composed of two right triangles:

1. **Larger right triangle** with legs \( y \) and \( 4 \) (since the total horizontal leg is \( y + 2 \)), and hypotenuse \( 5 \).
2. **Smaller right triangle** with legs \( 2 \) and \( 2 \), and we can solve for the hypotenuse using the Pythagorean theorem.

### Step 1: Solve for the hypotenuse of the smaller triangle
The Pythagorean theorem states that in a right triangle:
\[
a^2 + b^2 = c^2
\]
For the smaller triangle:
\[
2^2 + 2^2 = c^2 \implies 4 + 4 = c^2 \implies c = \sqrt{8} = 2\sqrt{2}
\]
However, this value is not directly needed for the problem.

### Step 2: Use the Pythagorean theorem for the larger triangle
Now we focus on the larger triangle, which has:
- One leg as \( y \)
- The other leg as \( 4 \)
- The hypotenuse as \( 5 \)

We apply the Pythagorean theorem again:
\[
y^2 + 4^2 = 5^2
\]
\[
y^2 + 16 = 25
\]
\[
y^2 = 9 \implies y = 3
\]

### Conclusion:
The value of \( y \) is \( 3 \).

Do you want further clarification on any step, or have any additional questions?

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Here are 5 related questions you might consider:
1. How does the Pythagorean theorem apply to non-right triangles?
2. How can you find missing angles in a right triangle using trigonometry?
3. What are the applications of similar triangles in geometry?
4. How can you calculate the area of a triangle if you know all its sides?
5. How do you use trigonometric ratios (sine, cosine, tangent) in right triangles?

**Tip:** For any right triangle, the Pythagorean theorem is a fundamental tool to relate the sides when two of them are known.

Determine the value of y in a right triangle with sides 2 and 2 in a smaller triangle, and y and 4 in the larger triangle, where the hypotenuse of the larger triangle is 5.

This problem involves determining the value of y in a right triangle using the Pythagorean theorem. The larger triangle has a hypotenuse of 5 and legs y and 4, while the smaller triangle has legs of 2 and 2. The problem is suitable for students in Grades 8-10.

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