We are given two similar triangles. The properties of similar polygons tell us that corresponding sides of similar polygons are proportional.

The larger triangle has side lengths of 5 cm, 7 cm, and 9 cm. The smaller triangle has side lengths of 3 cm, $x$ cm (unknown), and 5 cm.

Since the triangles are similar, we can set up a proportion between corresponding sides:

For the side corresponding to $x$: $\frac{x}{7} = \frac{5}{9}$

Now, let's solve for $x$. We will cross-multiply and find the value of $x$.

$x = \frac{7 \times 5}{9}$

I'll calculate that for you now.The value of $x$ is approximately $3.89 \, \text{cm}$.

Do you want more details or have any other questions? Here are five related questions to extend your understanding:

1. How are the side lengths of similar polygons related?
2. Can you calculate the missing side of the larger triangle if only two sides are known?
3. How would you solve for other unknowns in different polygon shapes using proportions?
4. What is the importance of cross-multiplication when solving proportions?
5. How can we verify if two polygons are similar?

Tip: When working with similar figures, always check that the corresponding angles are equal and sides are proportional.