Question

1. determine if the two triangles below are similar. justify your answer.

Explanation:

Identify the side lengths of both triangles

We are given two triangles with three known side lengths each. Let us list the side lengths of each triangle in ascending order to compare corresponding sides.

For the first triangle, the side lengths are:

• Shortest side: \(2\)
• Medium side: \(4\)
• Longest side: \(5\)

For the second triangle, the side lengths are:

• Shortest side: \(1.4\)
• Medium side: \(2.8\)
• Longest side: \(3.5\)

Calculate the ratios of corresponding sides

To determine if the triangles are similar, we check if the ratios of corresponding sides are equal.

Ratio of the shortest sides:

$$ \frac{2}{1.4} = \frac{20}{14} = \frac{10}{7} \approx 1.43 $$

Ratio of the medium sides:

$$ \frac{4}{2.8} = \frac{40}{28} = \frac{10}{7} \approx 1.43 $$

Ratio of the longest sides:

$$ \frac{5}{3.5} = \frac{50}{35} = \frac{10}{7} \approx 1.43 $$

Apply the SSS similarity criterion

Since all three pairs of corresponding sides have the exact same ratio:

$$ \frac{2}{1.4} = \frac{4}{2.8} = \frac{5}{3.5} = \frac{10}{7} $$

the two triangles are similar by the Side-Side-Side (SSS) similarity criterion.

Answer:

Yes, the two triangles are similar.

The ratios of their corresponding sides (ordered from shortest to longest) are equal:

$$ \frac{2}{1.4} = \frac{4}{2.8} = \frac{5}{3.5} = \frac{10}{7} $$

Since the corresponding side lengths are proportional, the triangles are similar by the Side-Side-Side (SSS) similarity criterion.