Right triangles ABC and DEF are similar, where vertices A, B, and C correspond to vertices D, E, and F,...

1. TRANSLATE the problem information

• Given information:
  • Right triangles ABC and DEF are similar
  • Vertex correspondence: \(\mathrm{A\leftrightarrow D, B\leftrightarrow E, C\leftrightarrow F}\)
  • Right angles at B and E
  • \(\cos(\mathrm{A}) = \frac{12}{13}\)
• Find: \(\cos(\mathrm{D})\)

2. INFER the key relationship

• Since the triangles are similar with A corresponding to D, angle A equals angle D
• This is the fundamental property of similar triangles: corresponding angles are congruent

3. INFER the trigonometric connection

• Cosine is a function that depends only on the angle measure
• Since \(\angle\mathrm{A} = \angle\mathrm{D}\), we must have \(\cos(\mathrm{A}) = \cos(\mathrm{D})\)
• Therefore: \(\cos(\mathrm{D}) = \frac{12}{13}\)

Answer: C. \(\frac{12}{13}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that similar triangles have equal corresponding angles

Instead of using the correspondence directly, they might try to set up ratios or use the \(\frac{12}{13}\) value to calculate side lengths, thinking they need to 'work' with the given cosine value. This leads to unnecessary calculations and confusion about what the problem is actually asking.

This may lead them to select Choice A (\(\frac{5}{13}\)) if they mistakenly calculate the sine of the complementary angle, or causes them to get stuck and guess.

The Bottom Line:

This problem tests understanding of the most basic property of similar triangles - that corresponding parts are equal. The \(\frac{12}{13}\) cosine value is a red herring that makes students think they need to do trigonometric calculations when the answer follows directly from similarity.