https://file.aiquickdraw.com/mathbot/file/1725857933380wxsv0zau.jpg

Given the problem from the image, we need to determine the length of \( y \) using the fact that the triangles are similar. Here's how to approach the solution:

### Step 1: Identify the given information
- The triangles are similar, which implies that the corresponding sides are proportional.
- \( a = 36 \)
- \( b = 16 \)

### Step 2: Set up the proportion
Since the triangles are similar, the ratio of the corresponding sides must be equal. Specifically, if the side lengths \( a \) and \( y \) in one triangle correspond to side lengths \( y \) and \( b \) in the other triangle, then:

\[
\frac{a}{y} = \frac{y}{b}
\]

### Step 3: Substitute the known values
Substituting \( a = 36 \) and \( b = 16 \):

\[
\frac{36}{y} = \frac{y}{16}
\]

### Step 4: Solve for \( y \)
Cross-multiply to get:

\[
36 \times 16 = y^2
\]

\[
y^2 = 576
\]

Taking the square root of both sides:

\[
y = \sqrt{576} = 24
\]

### Final Answer
The length of \( y \) is \( \boxed{24} \).

Would you like further details, or do you have any questions? Here are some related questions you might find helpful:

1. What is the importance of similarity in triangles?
2. How would you solve for \( y \) if different values for \( a \) and \( b \) were provided?
3. How does the Pythagorean theorem relate to similar triangles?
4. What are the properties of similar triangles?
5. Can this method be applied to non-right triangles?

**Tip:** When dealing with similar triangles, always remember that their corresponding sides are proportional, which allows you to set up equations to find missing lengths.

Given that the pair of triangles is similar, find the length y. (Let a = 36 and b = 16.)

Learn how to solve for the length y in a pair of similar triangles. Using the proportion formula a / y = y / b with a = 36 and b = 16, find the unknown side length through cross multiplication and square roots. Suitable for students in Grades 9-10.

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