We are given two similar triangles, $\triangle ABC$ and $\triangle PQR$, and are tasked with finding the value of $x$, which represents the length of side $PR$ in $\triangle PQR$.

Step-by-step solution:

Since the triangles are similar, the corresponding sides of the triangles are proportional. This gives us the following relationship:

$\frac{AB}{PQ} = \frac{AC}{PR} = \frac{BC}{QR}$

From the diagram:

• $AB = 8.1 \, \text{cm}$
• $AC = 4.5 \, \text{cm}$
• $BC = \text{(not needed)}$
• $PQ = 9.9 \, \text{cm}$
• $PR = x \, \text{cm}$

Using the ratio of corresponding sides $\frac{AC}{PR} = \frac{AB}{PQ}$, we set up the proportion:

$\frac{4.5}{x} = \frac{8.1}{9.9}$

Now, solve for $x$:

$\frac{4.5}{x} = \frac{8.1}{9.9} \quad \Rightarrow \quad x = \frac{4.5 \times 9.9}{8.1}$

Let me calculate the value of $x$.The value of $x$ is approximately $5.5 \, \text{cm}$.

Would you like more details or have any questions? Here are 5 related questions you might find interesting:

1. How do we identify corresponding sides in similar triangles?
2. Why are the ratios of corresponding sides equal in similar triangles?
3. Can this method be used for non-right triangles as well?
4. What happens if two triangles are not similar?
5. How can we prove that two triangles are similar geometrically?

Tip: Always ensure that the corresponding sides in similar triangles are matched correctly when setting up proportions!