The problem involves similar triangles, $\triangle RST \sim \triangle UVW$, and requires finding the length of side $VW$. Since the triangles are similar, the ratios of their corresponding sides are equal.

Given data:

• In $\triangle RST$:
  • $RS = 6 \, \text{cm}$,
  • $ST = 5 \, \text{cm}$,
  • $RT = 6.5 \, \text{cm}$.
• In $\triangle UVW$:
  • $UV = 6.5 \, \text{cm}$,
  • $UW = 9.75 \, \text{cm}$,
  • Find $VW$.

We use the proportion: $\frac{\text{side in } \triangle RST}{\text{corresponding side in } \triangle UVW} = \frac{\text{another side in } \triangle RST}{\text{corresponding side in } \triangle UVW}.$

Steps to solve:

1. Corresponding sides are:

   • $RS \leftrightarrow UV$,
   • $ST \leftrightarrow VW$,
   • $RT \leftrightarrow UW$.

2. Set up the ratio using known corresponding sides: $\frac{RS}{UV} = \frac{ST}{VW}.$

3. Substitute the values: $\frac{6}{6.5} = \frac{5}{VW}.$

4. Solve for $VW$:

   • First, simplify the left side: $\frac{6}{6.5} = 0.923.$
   • Next, use the proportion: $0.923 = \frac{5}{VW}.$
   • Rearrange to find $VW$: $VW = \frac{5}{0.923}.$

5. Calculate $VW$: $VW \approx 5.42 \, \text{cm}.$

Final Answer:

The length of side $VW$ is approximately 5.42 cm.

Would you like to see further details or clarifications?

Here are some related questions you might explore:

1. What other methods can verify similarity of triangles?
2. How can we apply the concept of proportionality to other geometric problems?
3. What are real-life applications of similar triangles?
4. How do you handle problems with unknowns in more than one side?
5. Why are corresponding angles in similar triangles always equal?

Tip: When working with proportions, always ensure you correctly pair corresponding sides to avoid errors.