Question

which statement about the two triangles is correct?
(\triangle abcsim\triangle def) because (\frac{6}{4}=\frac{12}{8}=\frac{9}{6})
(\triangle abc) is not similar to (\triangle def), because (12 - 8 = 4), while (9 - 6 = 3).
you cannot tell if the triangles are similar because the angles are not given.
(\triangle abc) is congruent to (\triangle def).

Explanation:

Step1: Check similarity by SSS similarity criterion

For two triangles \(\triangle ABC\) and \(\triangle DEF\), if \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\), then \(\triangle ABC\sim\triangle DEF\).
Here, \(AB = 12\), \(DE=8\), \(BC = 9\), \(EF = 6\), \(AC=6\), \(DF = 4\).
Calculate \(\frac{AB}{DE}=\frac{12}{8}=\frac{3}{2}\), \(\frac{BC}{EF}=\frac{9}{6}=\frac{3}{2}\), \(\frac{AC}{DF}=\frac{6}{4}=\frac{3}{2}\).

Step2: Analyze other options

• Option 2: Subtraction of side - lengths (\(12 - 8\) and \(9 - 6\)) is not a valid criterion for non - similarity. Similarity is based on ratios, not differences.
• Option 3: When the ratios of the corresponding sides of two triangles are equal (as we found \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}=\frac{3}{2}\)), we can tell they are similar without knowing the angles (by SSS similarity criterion).
• Option 4: Congruent triangles require \(AB = DE\), \(BC=EF\), \(AC = DF\) (i.e., ratio \(1\)). Here the ratio is \(\frac{3}{2}\), so they are not congruent.

Answer:

\(\triangle ABC\sim\triangle DEF\) because \(\frac{6}{4}=\frac{12}{8}=\frac{9}{6}\) (first option)