https://file.aiquickdraw.com/mathbot/file/1728731242614i7y0u8us.jpg

The diagram shows two triangles with some sides labeled, and we are tasked with finding the unknown lengths \( x \) and \( y \).

### Step-by-step Approach:
- **Similar Triangles**: The two triangles share the same angles, meaning they are similar triangles. The corresponding sides of similar triangles are proportional.

#### Triangles:
1. **Smaller Triangle** (with \( x \), 3 mm, and 2 mm):
   - Side lengths: \( x \), 3 mm, 2 mm.

2. **Larger Triangle** (with \( y \), 9 mm, and 5 mm):
   - Side lengths: \( y \), 9 mm, 5 mm.

### Set up Proportions:
Since the triangles are similar, the ratio of corresponding sides must be equal:

\[
\frac{x}{y} = \frac{3}{9} = \frac{1}{3}
\]

Also:

\[
\frac{2}{5} = \frac{x}{y}
\]

Now, solving for \( x \) and \( y \):

### Solve for \( y \):
Using the proportion of the corresponding sides:

\[
\frac{2}{5} = \frac{x}{y}
\]

Multiply both sides by \( y \):

\[
x = \frac{2}{5} y
\]

Substitute this expression into the earlier proportion \( \frac{x}{y} = \frac{1}{3} \):

\[
\frac{2}{5} = \frac{1}{3}
\]

At first glance we can

Work out the unknown lengths, x and y, in the shape below. Give each answer as an integer or as a fraction in its simplest form.

Learn how to find unknown lengths x and y in a geometric figure using the properties of similar triangles. This problem covers key concepts like proportional reasoning and the Similar Triangles Theorem, suitable for students in Grades 8-10.

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