Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angle C corresponds to angle...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC is similar to triangle DEF
  • Angle A corresponds to angle D
  • Angle C corresponds to angle F
  • Angles C and F are right angles
  • \(\mathrm{tan(A)} = \frac{50}{7}\)
  • Need to find: \(\mathrm{tan(E)}\)

2. INFER the angle relationships from similar triangles

• Since the triangles are similar with given correspondences:
  • Angle A = Angle D (corresponding angles in similar triangles are equal)
  • Angle C = Angle F = 90°
  • Angle B = Angle E (the remaining corresponding angles)

• Since \(\mathrm{tan(A)} = \frac{50}{7}\) and angle A = angle D, then \(\mathrm{tan(D)} = \frac{50}{7}\)

3. INFER the complementary angle relationship in right triangle DEF

• In right triangle DEF:
  • F is the right angle (90°)
  • D and E are the two acute angles
  • Therefore: \(\mathrm{D} + \mathrm{E} = 90°\) (complementary angles)

• Key insight: For complementary angles in a right triangle, \(\mathrm{tan(E)} = \frac{1}{\mathrm{tan(D)}}\)

4. SIMPLIFY to find the final answer

• Substitute:

\(\mathrm{tan(E)} = \frac{1}{\mathrm{tan(D)}} = \frac{1}{\frac{50}{7}}\)

• To divide by a fraction, multiply by its reciprocal:

\(\mathrm{tan(E)} = 1 \times \frac{7}{50} = \frac{7}{50}\)

• Convert to decimal if needed:

\(\frac{7}{50} = 0.14\)

Answer: 7/50 or 0.14 or .14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to connect that similar triangles have equal corresponding angles, so they don't establish that \(\mathrm{tan(A)} = \mathrm{tan(D)}\). Instead, they might try to work directly with the given information without recognizing this crucial relationship. This leads to confusion about how to proceed and likely results in guessing.

Second Most Common Error:

Missing conceptual knowledge about complementary angles: Students might correctly establish that \(\mathrm{tan(D)} = \frac{50}{7}\) but fail to recognize that in right triangle DEF, angles D and E are complementary, meaning \(\mathrm{tan(E)} = \frac{1}{\mathrm{tan(D)}}\). Without this insight, they get stuck trying to find \(\mathrm{tan(E)}\) through other means. This may lead them to incorrectly assume \(\mathrm{tan(E)} = \mathrm{tan(D)} = \frac{50}{7}\).

The Bottom Line:

This problem requires connecting two key mathematical relationships: the angle equality property of similar triangles and the reciprocal tangent relationship for complementary angles in right triangles. Students who don't see both connections will struggle to find a clear path to the solution.