First, let's analyze triangle ABC. In triangle ABC, we have:
- \( AB = 8 \) (one leg)
- \( BC = 15 \) (the other leg)
- \( AC = 17 \) (the hypotenuse)
The side opposite angle \( C \) is \( AB \), which is 8. The hypotenuse \( AC \) is 17.
Now, we find the ratio of the side opposite \( C \) to the hypotenuse:
\[ \text{Ratio} = \frac{AB}{AC} = \frac{8}{17} \]
Next, we apply this ratio to triangle \( A'B'C' \). The sides of triangle \( A'B'C' \) are given as:
- \( A'B' = 34 \)
- \( B'C' = 30 \)
- \( A'C' = 16 \)
In triangle \( A'B'C' \), we also need to identify which side corresponds to the hypotenuse. It is \( A'C' = 34 \).
To find the length of the side opposite point \( C' \), we can set up a proportion using the ratio determined from triangle \( ABC \):
\[ \text{Side opposite } C' = \text{Ratio} \times \text{Hypotenuse of } A'B'C' \]
Therefore, we have:
\[ \text{Side opposite } C' = \frac{8}{17} \times 34 \]
Calculating this, we get:
\[ \text{Side opposite } C' = \frac{8 \times 34}{17} = \frac{272}{17} = 16 \]
Thus, the length of the side opposite \( C' \) is \( \boxed{16} \) units.