Given that \( S'T' \) is the image of \( ST \) after a dilation with a scale factor of \( \frac{3}{2} \), we can use the relationship between the original length and the dilated length.
The formula for the length of the dilated segment is:
\[ \text{Length of \( S'T' \)} = \text{Length of \( ST \)} \times \text{Scale Factor} \]
Let \( L \) be the length of \( ST \). From the problem, we know that the length of \( S'T' \) is \( 10 \) units, and the scale factor is \( \frac{3}{2} \). Thus, we have:
\[ 10 = L \times \frac{3}{2} \]
To find \( L \), we can rearrange the equation:
\[ L = \frac{10}{\frac{3}{2}} = 10 \times \frac{2}{3} = \frac{20}{3} \text{ units} \]
Now, if we want to summarize the final answer, \( S'T' \) is \( 10 \) units long, as stated in the original problem.
In conclusion, the length of \( S'T' \) is \( 10 \) units.