Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angles B and E are...

1. TRANSLATE the problem information

• Given information:
  • \(\triangle \mathrm{ABC} \sim \triangle \mathrm{DEF}\) (similar)
  • \(\angle \mathrm{A} \leftrightarrow \angle \mathrm{D}\) (corresponding)
  • Angles B and E are right angles
  • \(\mathrm{Area_{ABC}} = 4 \times \mathrm{Area_{DEF}}\)
  • \(\cos \mathrm{A} = \frac{4}{5}\)
• Find: \(\sin \mathrm{D}\)

2. INFER the key relationship

• Since the triangles are similar and angle A corresponds to angle D, these angles must be equal
• Therefore: \(\angle \mathrm{A} = \angle \mathrm{D}\)
• This means: \(\sin \mathrm{D} = \sin \mathrm{A}\)
• Strategy: Find \(\sin \mathrm{A}\), and that will give us \(\sin \mathrm{D}\)

3. INFER the approach to find sin A

• We know \(\cos \mathrm{A} = \frac{4}{5}\), and we need \(\sin \mathrm{A}\)
• The Pythagorean identity connects these: \(\sin^2 \mathrm{A} + \cos^2 \mathrm{A} = 1\)
• We can solve for \(\sin \mathrm{A}\) using this relationship

4. SIMPLIFY using the Pythagorean identity

• \(\sin^2 \mathrm{A} + \cos^2 \mathrm{A} = 1\)
• \(\sin^2 \mathrm{A} + \left(\frac{4}{5}\right)^2 = 1\)
• \(\sin^2 \mathrm{A} + \frac{16}{25} = 1\)
• \(\sin^2 \mathrm{A} = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}\)
• \(\sin \mathrm{A} = \sqrt{\frac{9}{25}} = \frac{3}{5}\)

5. APPLY the relationship to find the final answer

• Since \(\sin \mathrm{D} = \sin \mathrm{A}\), we have:
• \(\sin \mathrm{D} = \frac{3}{5}\)

Answer: A) 3/5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that corresponding angles in similar triangles are equal

Students might get overwhelmed by the similarity information and the area ratio (4 times), missing the crucial insight that \(\angle \mathrm{A} = \angle \mathrm{D}\). They may try to use the area ratio somehow or get confused about which properties transfer between similar triangles. This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Making calculation errors with the Pythagorean identity

Even after correctly setting up \(\sin^2 \mathrm{A} = 1 - \left(\frac{4}{5}\right)^2\), students might make arithmetic mistakes: calculating \(\left(\frac{4}{5}\right)^2\) incorrectly as \(\frac{8}{25}\) instead of \(\frac{16}{25}\), or incorrectly simplifying \(\sqrt{\frac{9}{25}}\) as \(\frac{9}{5}\) instead of \(\frac{3}{5}\). This may lead them to select Choice C (4/5) or another incorrect option.

The Bottom Line:

This problem tests whether students can identify the essential relationship (corresponding angles are equal) while ignoring the distracting information about areas. The trigonometric calculation is straightforward once the key insight is made.