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