Question

△abc∼△def
which statement is a proportionality statement for the ratio of the sides of the triangles?
○ $\frac{bc}{ef} = \frac{ac}{df} = \frac{ab}{de}$
○ $\frac{ab}{ef} = \frac{ac}{df} = \frac{bc}{de}$
○ $\frac{ac}{df} = \frac{ab}{ef} = \frac{bc}{de}$

Explanation:

Step1: Recall Similar Triangles Property

When two triangles \( \triangle ABC \sim \triangle DEF \), the corresponding sides are proportional. So, \( \frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF} \). We can rearrange these ratios to check the options.

Step2: Analyze Each Option

• Option 1: \( \frac{BC}{EF}=\frac{AC}{DF}=\frac{AB}{DE} \). From the similarity, \( \frac{BC}{EF}=\frac{AC}{DF}=\frac{AB}{DE} \) (since \( \frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF} \) can be reordered as \( \frac{BC}{EF}=\frac{AC}{DF}=\frac{AB}{DE} \)).
• Option 2: \( \frac{AB}{EF}=\frac{AC}{DF}=\frac{BC}{DE} \). Here, \( AB \) corresponds to \( DE \), not \( EF \), so this is incorrect.
• Option 3: \( \frac{AC}{DF}=\frac{AB}{EF}=\frac{BC}{DE} \). \( AB \) should correspond to \( DE \), \( BC \) to \( EF \), so this is incorrect.

Answer:

The first option ( \( \boldsymbol{\frac{BC}{EF}=\frac{AC}{DF}=\frac{AB}{DE}} \) )