Triangle FGH is similar to triangle JKL, where angle F corresponds to angle J and angles G and K are...

1. TRANSLATE the problem information

• Given information:
  • Triangle FGH ~ Triangle JKL (similar triangles)
  • Angle F corresponds to angle J
  • Angles G and K are right angles
  • \(\sin(\mathrm{F}) = \frac{308}{317}\)
• Find: \(\sin(\mathrm{J})\)

2. INFER the key relationship

• Since the triangles are similar, corresponding angles are congruent
• "Angle F corresponds to angle J" means these angles have the same measure
• Therefore: \(m\angle\mathrm{F} = m\angle\mathrm{J}\)

3. INFER the trigonometric consequence

• If two angles have the same measure, they have identical trigonometric ratios
• Since \(m\angle\mathrm{F} = m\angle\mathrm{J}\), then \(\sin(\mathrm{F}) = \sin(\mathrm{J})\)

4. Apply the given information

• We know \(\sin(\mathrm{F}) = \frac{308}{317}\)
• Since \(\sin(\mathrm{F}) = \sin(\mathrm{J})\), we have \(\sin(\mathrm{J}) = \frac{308}{317}\)

Answer: B. \(\frac{308}{317}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "corresponds to" and think they need to perform calculations using similar triangle ratios or the Pythagorean theorem, rather than recognizing it simply means the angles are equal.

This leads to confusion about what to calculate, causing them to get stuck and guess randomly.

Second Most Common Error:

Missing conceptual knowledge about trigonometric functions: Students confuse sine with cosine or think they need the reciprocal, not recognizing that equal angles simply have equal sine values.

This may lead them to select Choice A (\(\frac{75}{317}\)) if they confuse sine with cosine, or Choice C (\(\frac{317}{308}\)) if they incorrectly take the reciprocal.

The Bottom Line:

This problem tests whether students recognize that similarity creates a direct equality relationship between corresponding angles, making complex calculations unnecessary. The trigonometry is straightforward once you realize the angles are simply equal.