Similar Triangles: Find the Similarity Ratio Among Three Labeled Triangles I, II, and III

We will analyze the given triangles to establish which ones are similar:

• Triangle I: Sides are $8$, $6$, and $4$.
• Triangle II: Sides are $4$, $3$, and $2$.
• Triangle III: Sides are $6$, $4$, and $2$.

To check for similarity using the Side-Side-Side (SSS) criterion, we compare the ratios of the corresponding sides of each triangle:

• For Triangle I and II:
  $\frac{8}{4} = 2$, $\frac{6}{3} = 2$, $\frac{4}{2} = 2$
  All sides are in the ratio $2:1$.
• For Triangle I and III:
  The ratios of sides will be:
  $\frac{8}{6} \neq \frac{6}{4} \neq \frac{4}{2}$
  These do not confirm similarity as the ratios differ.
• For Triangle II and III:
  $\frac{6}{4} = 1.5$, $\frac{4}{2} = 2$, which are not equal in proportions resulting in no similarity.

The only pair of triangles meeting the similarity condition based on the SSS criterion is Triangle II and Triangle III, with a similarity ratio of $2:1$.

Therefore, Triangles II and III are similar with a similarity ratio of 2.

This matches with the correct given answer, choice 4: $II, III, 2$.