Similarity Theorems: Applying the formula

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 $2$.

To solve the problem, we proceed with the following steps:

• Identify the given side lengths for each triangle.
• Compare the side ratios of each triangle pair to check for similarity.
• Verify the side ratios to affirm the similarity ratio.
• Select the correct multiple-choice answer based on the analysis.

Given side lengths:
Triangle C: $6$, $3$, $3$ (perpendicular and base, as seen in figure).
Triangle B: $4.5$, $3$, $2$ (perpendicular and base, as seen in figure).
Triangle A: $6$, $4$, $3.5$ (perpendicular and base, as seen in figure).

Calculating the ratios:

• For triangles C and B:
  $\frac{6}{4.5} = \frac{3}{2}$ which simplifies to $\frac{3}{2} = 1.5$, indicating that triangles C and B are similar.
• Comparison for other pairs: Triangle A with Triangle B or C reveals no common proportionality.

Therefore, the only pair of similar triangles is C and B with a similarity ratio of $\frac{3}{2}$ or 1.5.

The correct choice is, therefore, C + B are similar with a ratio of 1.5.

Triangle a and triangle b are similar according to the S.S.S (side side side) theorem

And the relationship between the sides is identical:

$\frac{GH}{DE}=\frac{HI}{EF}=\frac{GI}{DF}$

$\frac{9}{6}=\frac{3}{1}=\frac{6}{2}=3$

That is, the ratio between them is 1:3.

To solve this problem, we'll compare the side ratios of the given triangles to determine which pair are similar and find the similarity ratio.

• Step 1: Identify the given triangle side lengths:
  Triangle A: Sides of length $8$, $4$, and $6$.
  Triangle B: Sides of length $10$, $6$, and $8$.
  Triangle C: Sides of length $4$, $2$, and $3$.
• Step 2: Compare the ratios of corresponding sides between pairs of triangles.

Comparing Triangle A and Triangle B:

• Ratio $\frac{8}{10} = 0.8$; $\frac{4}{6} \approx 0.67$; $\frac{6}{8} = 0.75$

Here, the ratios are not equal; hence, triangles A and B are not similar.

Comparing Triangle A and Triangle C:

• Ratio $\frac{8}{4} = 2$; $\frac{4}{2} = 2$; $\frac{6}{3} = 2$

All ratios are equal, so triangles A and C are similar, with a similarity ratio of 2.

Comparing Triangle B and Triangle C:

• Ratio $\frac{10}{4} = 2.5$; $\frac{6}{2} = 3$; $\frac{8}{3} \approx 2.67$

The ratios are not equal, so triangles B and C are not similar.

Therefore, the similar triangles are Triangle A and Triangle C, with a similarity ratio of 2.

The correct answer is A + C are similar with a ratio of 2.