Similar Triangles Analysis: Determining Similarity Ratio Among Three Given Triangles

To determine which triangles are similar, we will compare the side ratios of each pair of triangles systematically.

First, let us consider Triangle A-B-C and Triangle D-E-F:

• Triangle A-B-C: The side lengths are $8$, $4$, and $6$.
• Triangle D-E-F: The side lengths are $4$, $2$, and $3.5$.
• Ratio comparison: $\frac{8}{4} = 2$, $\frac{4}{2} = 2$, and $\frac{6}{3.5} \approx 1.71$.
• Since the last comparison does not yield equal ratios, Triangles A-B-C and D-E-F are not similar.

Next, consider Triangle A-B-C and Triangle G-H-I:

• Triangle A-B-C: Side lengths are already known.
• Triangle G-H-I: The side lengths are $3$, $2$, and $4$.
• Ratio comparison: $\frac{8}{4} = 2$, $\frac{4}{2} = 2$, and $\frac{6}{3} = 2$.
• All corresponding side ratios are equal at $2$, making these triangles similar by SSS criterion.

To confirm, check Triangle B-C and Triangle G-H-I for the same ratio conformity.

Therefore, the similar triangles are Triangle G-H-I and Triangle A-B-C, with a similarity ratio of $2$.