Similar Triangles: Applying the formula

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: $II, III, 2$.

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.

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.

Triangle ABC is isosceles, therefore angle B is equal to angle C. We can calculate them since the sum of the angles of a triangle is 180:

$180-50=130$

$130:2=65$

As the triangles are similar, DE is parallel to BC

Angles B and D are corresponding and, therefore, are equal.

B=D=65