Question

1. two triangles have corresponding sides with ratios of \\(\frac{3}{6}\\), \\(\frac{4}{8}\\), and \\(\frac{5}{10}\\). what does this indicate about the two triangles?

a. they are reflections of one another.
b. they cannot be related.
c. they are congruent.
d. they are similar.

Explanation:

Step1: Simplify the ratios

First, simplify each ratio:

• For \(\frac{3}{6}\), simplify to \(\frac{1}{2}\) (divide numerator and denominator by 3).
• For \(\frac{4}{8}\), simplify to \(\frac{1}{2}\) (divide numerator and denominator by 4).
• For \(\frac{5}{10}\), simplify to \(\frac{1}{2}\) (divide numerator and denominator by 5).

Step2: Recall the similarity criterion

The Side - Side - Side (SSS) similarity criterion states that if the corresponding sides of two triangles are in proportion (have the same ratio), then the triangles are similar. Since all three corresponding side ratios simplify to \(\frac{1}{2}\), the sides are proportional.

Step3: Analyze the options

• Option a: Reflection is a transformation, but the ratio of sides doesn't directly imply reflection. Eliminate a.
• Option b: For congruence, the ratio of corresponding sides should be \(1\) (since congruent triangles have equal sides). Here the ratio is \(\frac{1}{2}\), so they are not congruent. Eliminate b.
• Option c: The triangles are related by similarity, so they can be related. Eliminate c.
• Option d: Since the corresponding sides are in proportion (ratio \(\frac{1}{2}\) for all three pairs), by SSS similarity, the triangles are similar.

Answer:

d. They are similar.